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BEGIN:VEVENT
SUMMARY:Michael Roop
DTSTART:20200427T120000Z
DTEND:20200427T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/1
DESCRIPTION:Title: Sho
ck waves in Euler flows of gases\nby Michael Roop as part of Geometry
of differential equations seminar\n\n\nAbstract\nNon-stationary one-dimens
ional Euler flows of gases are studied. The system of differential equatio
ns describing such flows can be represented by means of 2-forms on zero-je
t space and we get some exact solutions by means of such a representation.
Solutions obtained are multivalued and we provide a method of finding cau
stics\, as well as wave front displacement. The method can be applied to a
ny model of thermodynamic state as well as to any thermodynamic process. W
e illustrate the method on adiabatic ideal gas flows.\n
LOCATION:https://researchseminars.org/talk/GDEq/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20200504T120000Z
DTEND:20200504T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/2
DESCRIPTION:Title: On
structure of linear differential operators of the first order\nby Vale
ntin Lychagin as part of Geometry of differential equations seminar\n\n\nA
bstract\nWe'll discuss the equivalence problem (local as well as global) f
or linear differential operators of the first order\, acting in vector bun
dles.\n\nThe slides will be in English and if preferred by anyone in the a
udience the talk itself can be switched from Russian to English.\n
LOCATION:https://researchseminars.org/talk/GDEq/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valery Yumaguzhin
DTSTART:20200511T120000Z
DTEND:20200511T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/3
DESCRIPTION:Title: Inv
ariants of forth order linear differential operators\nby Valery Yumagu
zhin as part of Geometry of differential equations seminar\n\n\nAbstract\n
The report is devoted to linear scalar differential operators of the fourt
h order on 2-dimensional manifolds. The field of rational differential inv
ariants of such operators will be described and their application to the e
quivalence problem with respect to the group of diffeomorphisms of the man
ifold will be shown.\n\nAlthough the talk will be in Russian\, the slides
will be in English and the discussion will be in both languages.\n
LOCATION:https://researchseminars.org/talk/GDEq/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Samokhin
DTSTART:20200518T120000Z
DTEND:20200518T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/4
DESCRIPTION:Title: Usi
ng the KdV conserved quantities in problems of splitting of initial data a
nd reflection / refraction of solitons in varying dissipation and/or dis
persion media\nby Alexey Samokhin as part of Geometry of differential
equations seminar\n\n\nAbstract\nAn arbitrary compact-support initial datu
m for the Korteweg-de Vries equation asymptotically splits into solitons a
nd a radiation tail\, moving in opposite direction. We give a simple metho
d to predict the number and amplitudes of resulting solitons and some inte
gral characteristics of the tail using only conservation laws.\n\nA simila
r technique allows to predict details of the behavior of a soliton which\
, while moving in non-dissipative and dispersion-constant medium encounter
s a finite-width barrier with varying dissipation and/or dispersion\; be
yond the layer dispersion is constant (but not necessarily of the same val
ue) and dissipation is null. The process is described with a special typ
e generalized KdV-Burgers equation $u_t=(u^2+f(x)u_{xx})_x$.\n\nThe transm
itted wave either retains the form of a soliton (though of different param
eters) or scatters a into a number of them. And a reflection wave may be n
egligible or absent. This models a situation similar to a light passing fr
om a humid air to a dry one through the vapor saturation/condensation area
. Some rough estimations for a prediction of an output are given using the
relative decay of the KdV conserved quantities\; in particular a formula
for a number of solitons in the transmitted signal is given.\n
LOCATION:https://researchseminars.org/talk/GDEq/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Tychkov
DTSTART:20200525T120000Z
DTEND:20200525T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/5
DESCRIPTION:Title: Con
tinuum mechanics of media with inner structures\nby Sergey Tychkov as
part of Geometry of differential equations seminar\n\n\nAbstract\nWe propo
se a geometrical approach to the mechanics of continuous media equipped wi
th inner structures and give the basic equations of their motion: the mass
conservation law\, the Navier-Stokes equation and the energy conservation
law.\n\nThis is a joint work with Anna Duyunova and Valentin Lychagin.\n
LOCATION:https://researchseminars.org/talk/GDEq/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleks Kleyn
DTSTART:20200601T120000Z
DTEND:20200601T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/6
DESCRIPTION:Title: Sys
tem of differential equations over quaternion algebra\nby Aleks Kleyn
as part of Geometry of differential equations seminar\n\n\nAbstract\nThe t
alk is based on the file\nhttps://gdeq.org/files/Aleks_Kleyn-2020.06.01.En
glish.pdf (Russian transl.: https://gdeq.org/files/Aleks_Kleyn-2020.06.01.
Russian.pdf)\n\nIn order to study homogeneous system of linear differentia
l equations\, I considered vector space over division D-algebra and the th
eory of eigenvalues in non commutative division D-algebra. I started from
section 1 dedicated to product of matrices. Since product in algebra is no
n-commutative\, I considered two forms of product of matrices and two form
s of eigenvalues (section 4). In sections 5\, 6\, 7\, I considered solving
of homogeneous system of differential equations. In the section 8\, I con
sidered the system of differential equations which has infinitely many fun
damental solutions. Following sections are dedicated to analysis of soluti
ons of system of differential equations. In particular\, if a system of di
fferential equations has infinitely many fundamental solutions\, then each
solution is envelope of a family of solutions of considered system of dif
ferential equations.\n
LOCATION:https://researchseminars.org/talk/GDEq/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irina Bobrova (HSE\, Moscow)
DTSTART:20200608T120000Z
DTEND:20200608T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/7
DESCRIPTION:Title: On
the second Painlevé equation and its higher analogues\nby Irina Bobro
va (HSE\, Moscow) as part of Geometry of differential equations seminar\n\
n\nAbstract\nSix Painlevé equations were obtained by Paul Painlevé and h
is school during the classification of ODE's of the form $w'' = P (z\, w\,
w')$\, where the function $P (z\, w\, w')$ is a polynomial in $w$ and $w'
$ and is an analytic function of $z$. These equations are widely used in p
hysics and have beautiful mathematical structures. My talk is devoted to t
he second Painlevé equation.\n\nWe will discuss the integrability of this
equation and introduce its Hamiltonian representation in terms of the Kaz
uo Okamoto variables. On the other hand\, the PII equation is integrable i
n the sense of the Lax pair and the isomonodromic representation\, that I
will present.\n\nThe Bäcklund transformation and the affine Weyl group ar
e another interesting question. Using these symmetries\, we are able to co
nstruct various rational solutions for the integer parameter PII equation.
\n\nThe second Painlevé equation has one more important representation in
terms of $\\sigma$-coordinates which are $log$-symplectic.\n\nThere are h
igher analogues of the PII equation\, which we will obtain by self-similar
reduction of the modified Korteveg-de Vries hierarchy.\n
LOCATION:https://researchseminars.org/talk/GDEq/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hovhannes Khudaverdian
DTSTART:20200615T120000Z
DTEND:20200615T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/8
DESCRIPTION:Title: Non
-linear homomorphisms and thick morphisms\nby Hovhannes Khudaverdian a
s part of Geometry of differential equations seminar\n\n\nAbstract\nIn 201
4\, Voronov introduced the notion of thick morphisms of (super)manifolds a
s a tool for constructing $L_{\\infty}$-morphisms of homotopy Poisson alge
bras. Thick morphisms generalise ordinary smooth maps\, but are not maps t
hemselves. Nevertheless\, they induce pull-backs on $C^{\\infty}$ function
s. These pull-backs are in general non-linear maps between the algebras o
f functions which are so-called "non-linear homomorphisms". By definition\
, this means that their differentials are algebra homomorphisms in the usu
al sense. The following conjecture was formulated: an arbitrary non-linear
homomorphism of algebras of smooth functions is generated by some thick m
orphism. We prove here this conjecture in the class of formal functionals.
In this way\, we extend the well-known result for smooth maps of manifold
s and algebra homomorphisms of $C^{\\infty}$ functions and\, more generall
y\, provide an analog of classical "functional-algebraic duality" in the n
on-linear setting.\n
LOCATION:https://researchseminars.org/talk/GDEq/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Morozov
DTSTART:20200622T120000Z
DTEND:20200622T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/9
DESCRIPTION:Title: Lax
representations via extensions and deformations of Lie symmetry algebras<
/a>\nby Oleg Morozov as part of Geometry of differential equations seminar
\n\n\nAbstract\nThe challenging problem in the theory of integrable partia
l differential equations is to find conditions that are formulated in inhe
rent terms of a PDE under study and ensure existence of a Lax representati
on. The talk will present the technique for constructing Lax representatio
ns via extensions of the contact symmetry algebras of PDEs. Also I will s
how examples that use deformations of infinite-dimensional Lie algebras fo
r searching new integrable PDEs.\n
LOCATION:https://researchseminars.org/talk/GDEq/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20200629T120000Z
DTEND:20200629T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/10
DESCRIPTION:Title: Ex
tendable symplectic structures and the inverse problem of the calculus of
variations for systems of equations written in an extended Kovalevskaya fo
rm\nby Konstantin Druzhkov as part of Geometry of differential equatio
ns seminar\n\n\nAbstract\nThe talk is devoted to extendable symplectic str
uctures for systems of equations written in an extended Kovalevskaya form.
\n\nIt is shown\, that each extension of a symplectic structure to jets is
related to an extension of a special form.\n\nComplete description of all
extendable symplectic structures is obtained. Relation of this result wit
h the inverse problem of the calculus of variations is discussed.\n\nIt is
shown\, that each variational formulation for a system of evolution equat
ions is related to a two-sided invertible variational operator of a specia
l form.\n
LOCATION:https://researchseminars.org/talk/GDEq/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev (Lebedev Physical Institute\, Institute for Theore
tical and Mathematical Physics of Moscow State University)
DTSTART:20200706T120000Z
DTEND:20200706T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/11
DESCRIPTION:Title: Pr
esymplectic structures and intrinsic Lagrangians\nby Maxim Grigoriev (
Lebedev Physical Institute\, Institute for Theoretical and Mathematical Ph
ysics of Moscow State University) as part of Geometry of differential equa
tions seminar\n\n\nAbstract\nIt is well-known that a Lagrangian induces a
compatible presymplectic form on the equation manifold (stationary surface
\, understood as a submanifold of the respective jet-space). Given an equa
tion manifold and a compatible presymplectic form therein\, we define the
first-order Lagrangian system which is formulated in terms of the intrinsi
c geometry of the equation manifold. It has a structure of a presymplectic
AKSZ sigma model for which the equation manifold\, equipped with the pres
ymplectic form and the horizontal differential\, serves as the target spac
e. For a wide class of systems (but not all) we show that if the presymple
ctic structure originates from a given Lagrangian\, the proposed first-ord
er Lagrangian is equivalent to the initial one and hence the Lagrangian pe
r se can be entirely encoded in terms of the intrinsic geometry of its sta
tionary surface. If the compatible presymplectic structure is generic\, th
e proposed Lagrangian is only a partial one in the sense that its stationa
ry surface contains the initial equation manifold but does not necessarily
coincide with it. I also plan to briefly discuss extension of this constr
uction to gauge PDEs (gauge theories in BV framework).\n
LOCATION:https://researchseminars.org/talk/GDEq/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20200713T120000Z
DTEND:20200713T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/12
DESCRIPTION:Title: Po
lynomial Poisson algebras associated with elliptic curves. Part 1\nby
Vladimir Rubtsov (Université d'Angers) as part of Geometry of differentia
l equations seminar\n\n\nAbstract\nI shall give an introduction in a study
of Poisson algebras which are quasi classical limit of Sklyanin-Odesskii-
Feigin elliptic algebras. I will restrict my description to the algebras w
ith a "small" number of generators (n = 3\,4\,5).\n\nThe results are (almo
st) not new. The talk is based on my old papers with A. Odesskii\, G. Orte
nzi and S. Tagne Pelap.\n
LOCATION:https://researchseminars.org/talk/GDEq/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20200720T120000Z
DTEND:20200720T140000Z
DTSTAMP:20260422T225928Z
UID:GDEq/13
DESCRIPTION:Title: Po
lynomial Poisson algebras associated with elliptic curves. Part 2\nby
Vladimir Rubtsov (Université d'Angers) as part of Geometry of differentia
l equations seminar\n\n\nAbstract\nI shall give an introduction in a study
of Poisson algebras which are quasi classical limit of Sklyanin-Odesskii-
Feigin elliptic algebras. I will restrict my description to the algebras w
ith a "small" number of generators (n = 3\,4\,5).\n\nThe results are (almo
st) not new. The talk is based on my old papers with A. Odesskii\, G. Orte
nzi and S. Tagne Pelap.\n
LOCATION:https://researchseminars.org/talk/GDEq/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Krasil'shchik (Independent University of Moscow)
DTSTART:20200930T162000Z
DTEND:20200930T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/14
DESCRIPTION:Title: No
nlocal conservation laws of PDEs possessing differential coverings\nby
Joseph Krasil'shchik (Independent University of Moscow) as part of Geomet
ry of differential equations seminar\n\nLecture held in room 303 of the In
dependent University of Moscow.\n\nAbstract\nIn his 1892 paper "Sulla tra
sformazione di Bäcklund per le superfici pseudosferiche" (Rend. Mat. Acc.
Lincei\, s. 5\, v. 1 (1892) 2\, pp. 3-12\; Opere\, vol. 5\, pp. 163-173)
Luigi Bianchi noticed\, among other things\, that quite simple transformat
ions of the formulas that describe the Bäcklund transformation of the sin
e-Gordon equation lead to what is called a nonlocal conservation law in mo
dern language. Using the techniques of differential coverings [I.S. Krasil
'shchik\, A.M. Vinogradov\, Nonlocal trends in the geometry of differentia
l equations: symmetries\, conservation laws\, and Bäcklund transformation
s\, Acta Appl. Math. 15 (1989) 161-209]\, we show that this observation is
of a quite general nature. We describe the procedures to construct such c
onservation laws and present a number of illustrative examples.\n
LOCATION:https://researchseminars.org/talk/GDEq/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Markus Dafinger (University of Jena\, Germany)
DTSTART:20201021T162000Z
DTEND:20201021T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/15
DESCRIPTION:Title: A
converse to Noether's theorem\nby Markus Dafinger (University of Jena\
, Germany) as part of Geometry of differential equations seminar\n\n\nAbst
ract\nThe classical Noether's theorem states that symmetries of a variatio
nal functional lead to conservation laws of the corresponding Euler-Lagran
ge equation. It is a well-known statement to physicists with many applicat
ions. In the talk we investigate a reverse statement\, namely that a diffe
rential equation which satisfies sufficiently many symmetries and correspo
nding conservation laws leads to a variational functional whose Euler-Lagr
ange equation is the given differential equation. The aim of the talk is t
o provide some background of the so-called inverse problem of the calculus
of variations and then to discuss some new results\, for example\, how to
prove the reverse statement.\n
LOCATION:https://researchseminars.org/talk/GDEq/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Sheftel
DTSTART:20201104T162000Z
DTEND:20201104T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/17
DESCRIPTION:Title: No
nlocal symmetry of CMA generates ASD Ricci-flat metric with no Killing vec
tors\nby Mikhail Sheftel as part of Geometry of differential equations
seminar\n\n\nAbstract\nThe complex Monge-Ampère equation (CMA) in a two-
component form is treated as a bi-Hamiltonian system. I present explicitly
the first nonlocal symmetry flow in each of the two hierarchies of this s
ystem. An invariant solution of CMA with respect to these nonlocal symmetr
ies is constructed which\, being a noninvariant solution in the usual sens
e\, does not undergo symmetry reduction in the number of independent varia
bles. I also construct the corresponding 4-dimensional anti-self-dual (ASD
) Ricci-flat metric with either Euclidean or neutral signature. It admits
no Killing vectors which is one of characteristic features of the famous g
ravitational instanton K3.\n
LOCATION:https://researchseminars.org/talk/GDEq/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierandrea Vergallo (University of Salento)
DTSTART:20201111T162000Z
DTEND:20201111T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/18
DESCRIPTION:Title: Hy
drodynamic-type systems and homogeneous Hamiltonian operators: a necessary
condition of compatibility\nby Pierandrea Vergallo (University of Sal
ento) as part of Geometry of differential equations seminar\n\n\nAbstract\
nUsing the theory of coverings\, it is presented a necessary condition to
write a hydrodynamic-type system in Hamiltonian formulation. Explicit cond
itions for first\, second and third order homogeneous Hamiltonian operator
s are shown. In particular\, an alternative proof of Tsarev's theorem abou
t compatibility conditions for first order operators is obtained by using
this method.\n\nThen\, analogous conditions are presented for non local h
omogeneous Hamiltonian operators of first and third order.\n\nFinally\, it
is discussed the projective invariance for second and third order operato
rs.\n\nThe talk is based on a joint work with Raffaele Vitolo.\n
LOCATION:https://researchseminars.org/talk/GDEq/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrey Losev
DTSTART:20201118T162000Z
DTEND:20201118T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/19
DESCRIPTION:Title: Ta
u theory\, d=10 N=1 SUSY and BV\nby Andrey Losev as part of Geometry o
f differential equations seminar\n\n\nAbstract\nPlease\, see https://gdeq.
org/Losev for the abstract.\n
LOCATION:https://researchseminars.org/talk/GDEq/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20201125T162000Z
DTEND:20201125T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/20
DESCRIPTION:Title: Di
fferential equations\, their symmetries\, invariants and quotients\nb
y Valentin Lychagin as part of Geometry of differential equations seminar\
n\n\nAbstract\nWe'll discuss quotients of PDEs by their symmetry algebras
and show their applications for integrations.\n
LOCATION:https://researchseminars.org/talk/GDEq/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20201202T162000Z
DTEND:20201202T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/21
DESCRIPTION:Title: Re
al Monge-Ampère operators and (almost) complex structures\nby Vladimi
r Rubtsov (Université d'Angers) as part of Geometry of differential equat
ions seminar\n\n\nAbstract\nWe observe some interesting geometric structur
es which are naturally linked with the geometric approach to Monge-Ampère
operators developed by Lychagin in late 70th.\n\nAmong them are: (almost
) complex\, (almost) product\, generalized complex\, hyperkahler\, hypersy
mplectic and many other geometric structures.\n\nI hope (if I have time) t
o show few interesting examples of its applications.\n
LOCATION:https://researchseminars.org/talk/GDEq/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Mikhailov (University of Leeds)
DTSTART:20201209T162000Z
DTEND:20201209T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/22
DESCRIPTION:Title: Qu
antisation ideals of nonabelian integrable systems\nby Alexander Mikha
ilov (University of Leeds) as part of Geometry of differential equations s
eminar\n\n\nAbstract\nIn my talk I'll discuss a new approach to the proble
m of quantisation of dynamical systems\, introduce the concept of quantisa
tion ideals and show meaningful examples. Traditional quantisation theorie
s start with classical Hamiltonian systems with dynamical variables taking
values in commutative algebras and then study their non-commutative defor
mations\, such that the commutators of observables tend to the correspondi
ng Poisson brackets as the (Planck) constant of deformation goes to zero.
I am proposing to depart from systems defined on a free associative algebr
a. In this approach the quantisation problem is reduced to a description o
f two-sided ideals which define the commutation relations (the quantisatio
n ideals) in the quotient algebras and which are invariant with respect to
the dynamics of the system. Surprisingly this idea works rather efficient
ly and in a number of cases I have been able to quantise the system\, i.e.
to find consistent commutation relations for the system. To illustrate t
his approach I'll consider the quantisation problem for the non-abelian Bo
goyavlensky N-chains and other examples\, including quantisation of nonabe
lian integrable ODEs on free associative algebras.\n\nThe talk is based on
: AVM\, Quantisation ideals of nonabelian integrable systems\, arXiv prepr
int arXiv:2009.01838\, 2020
(Published in Russ. Math. Surv. v.75:5\, pp 199-200\, 2020).\n
LOCATION:https://researchseminars.org/talk/GDEq/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Khavkine (Czech Academy of Sciences)
DTSTART:20201216T162000Z
DTEND:20201216T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/23
DESCRIPTION:Title: Ki
lling compatibility complex on Kerr spacetime\nby Igor Khavkine (Czech
Academy of Sciences) as part of Geometry of differential equations semina
r\n\n\nAbstract\nThe Killing operator $K_{ab}[v] = \\nabla_a v_b + \\nabla
_b v_a$ on a Lorentzian spacetime $(M\,g)$ plays an important role in Gene
ral Relativity (GR): it generates infinitesimal gauge symmetries of the th
eory. Gauge symmetry invariants play the role of physical observables. In
PDE language\, this translates to the following: the components of a comp
atibility operator for $K_{ab}$ generate all local observables for lineari
zed GR on the background $(M\,g)$. In arXiv:1910.08756 we have explicitly constructed such a compatib
ility operator (indeed\, a full compatibility complex) on the astrophysica
lly interesting Kerr spacetime of a rotating black hole. I will motivate a
nd explain our approach and describe the complexity of the construction.\n
LOCATION:https://researchseminars.org/talk/GDEq/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Kruglikov (UiT the Arctic University of Norway)
DTSTART:20201223T162000Z
DTEND:20201223T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/24
DESCRIPTION:Title: Di
spersionless integrable hierarchies and GL(2) geometry\nby Boris Krugl
ikov (UiT the Arctic University of Norway) as part of Geometry of differen
tial equations seminar\n\n\nAbstract\n(joint work with Evgeny Ferapontov)\
n\nParaconformal or GL(2) geometry on an n-dimensional manifold M is defin
ed by a field of rational normal curves of degree n - 1 in the projectiviz
ed cotangent bundle $\\mathbb{P}T^*M$. In dimension n=3 this is nothing bu
t a Lorentzian metric. GL(2) geometry is known to arise on solution spaces
of ODEs with vanishing Wünschmann invariants.\n\nWe show that GL(2) stru
ctures also arise on solutions of dispersionless integrable hierarchies of
PDEs such as the dispersionless Kadomtsev-Petviashvili (dKP) hierarchy. I
n fact\, they coincide with the characteristic variety (principal symbol)
of the hierarchy. GL(2) structures arising in this way possess the propert
y of involutivity. For n=3 this gives the Einstein-Weyl geometry.\n\nThus
we are dealing with a natural generalization of the Einstein-Weyl geometry
. Our main result states that involutive GL(2) structures are governed by
a dispersionless integrable system whose general local solution depends on
2n - 4 arbitrary functions of 3 variables. This establishes integrability
of the system of Wünschmann conditions.\n
LOCATION:https://researchseminars.org/talk/GDEq/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Chetverikov
DTSTART:20210203T162000Z
DTEND:20210203T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/25
DESCRIPTION:Title: Co
verings and multivector pseudosymmetries of differential equations\nby
Vladimir Chetverikov as part of Geometry of differential equations semina
r\n\n\nAbstract\nFinite-dimensional coverings from systems of differential
equations are investigated. This problem is of interest in view of its re
lationship with the computation of differential substitution\, nonlocal sy
mmetries\, recursion operators\, and Backlund transformations. We show tha
t the distribution specified by the fibers of a covering is determined by
an integrable pseudosymmetry of the system. Conversely\, every integrable
pseudosymmetry of a system defines a covering from this system. The vertic
al component of the pseudosymmetry is a matrix analog of the evolution dif
ferentiation. The corresponding generating matrix satisfies a matrix analo
g of the linearization of the equation. We consider also the exterior prod
uct of vector fields defining a pseudosymmetry. The definition of pseudosy
mmetry is rewritten in the language of the Schouten bracket of multivector
fields and total derivatives with respect to the independent variables of
the system. A method for constructing coverings is given and demonstrated
by the examples of the Laplace equation and the Kapitsa pendulum system.\
n
LOCATION:https://researchseminars.org/talk/GDEq/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Samokhin
DTSTART:20210210T162000Z
DTEND:20210210T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/26
DESCRIPTION:Title: On
monotonic pattern in periodic boundary solutions of cylindrical and spher
ical Kortweg-de Vries-Burgers equations\nby Alexey Samokhin as part of
Geometry of differential equations seminar\n\n\nAbstract\nWe studied\, fo
r the Kortweg-de Vries Burgers equations on cylindrical and spherical wave
s\, the development of a regular profile starting from an equilibrium unde
r a periodic perturbation at the boundary.\n\nThe regular profile at the v
icinity of perturbation looks like a periodical chain of shock fronts with
decreasing amplitudes. Further on\, shock fronts become decaying smooth q
uasi periodic oscillations. After the oscillations cease\, the wave develo
ps as a monotonic convex wave\, terminated by a head shock of a constant h
eight and equal velocity. This velocity depends on integral characteristic
s of a boundary condition and on spatial dimensions.\n\nThe explicit asymp
totic formulas for the monotonic part\, the head shock and a median of the
oscillating part are found.\n
LOCATION:https://researchseminars.org/talk/GDEq/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petr Pushkar
DTSTART:20210217T162000Z
DTEND:20210217T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/27
DESCRIPTION:Title: Mo
rse theory\, Bruhat cells and Unitriangular geometry\nby Petr Pushkar
as part of Geometry of differential equations seminar\n\n\nAbstract\nStron
g Morse function is a Morse function with pairwise different critical valu
es. For such a function we construct a collection of numbers\, which is a
(smooth) topological invariant of the strong Morse function.\n\nAlgebraica
lly our construction is a close relative of the construction of Bruhat cel
ls and belongs to Unitriangular geometry. We will present a generalization
of determinant of any linear map between finite dimensional vector spaces
.\n\nTalk based on a joint work with Misha Temkin.\n
LOCATION:https://researchseminars.org/talk/GDEq/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Sokolov (Landau Institute for Theoretical Physics\, Chern
ogolovka\, Russia)
DTSTART:20210224T162000Z
DTEND:20210224T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/28
DESCRIPTION:Title: No
n-Abelian generalizations of integrable PDEs and ODEs\nby Vladimir Sok
olov (Landau Institute for Theoretical Physics\, Chernogolovka\, Russia) a
s part of Geometry of differential equations seminar\n\n\nAbstract\nA gene
ral procedure for nonabelinization of given integrable polynomial differen
tial equation is described. We consider NLS type equations as an example.
We also find nonabelinizations of the Euler top and of the Painleve-2 equa
tion.\n\nAlthough the talk will be in Russian\, the slides will be in Engl
ish and the discussion will be in both languages.\n
LOCATION:https://researchseminars.org/talk/GDEq/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20210303T162000Z
DTEND:20210303T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/29
DESCRIPTION:Title: Re
al Monge-Ampère operators and (almost) complex structures. Part 2\nby
Vladimir Rubtsov (Université d'Angers) as part of Geometry of differenti
al equations seminar\n\n\nAbstract\nWe observe some interesting geometric
structures which are naturally linked with the geometric approach to Monge
-Ampère operators developed by Lychagin in late 70th. I shall concentrate
my attention on the Hitchin generalized complex structure\, hyper-Kahler/
symplectic and hope to show few interesting examples of its relations with
the Monge-Ampère operators and applications.\n
LOCATION:https://researchseminars.org/talk/GDEq/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Pavlov
DTSTART:20210310T162000Z
DTEND:20210310T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/30
DESCRIPTION:Title: Ne
w variational principles for one-dimensional gas dynamics and for Egorov h
ydrodynamic type systems\nby Maxim Pavlov as part of Geometry of diffe
rential equations seminar\n\n\nAbstract\nThe Statement. If some Egorov hyd
rodynamic type system has one local Hamiltonian structure of Dubrovin-Novi
kov type\, then such a system possesses infinitely many: local Hamiltonian
structures of all odd orders\, and infinitely many local Lagrangian repre
sentations.\n
LOCATION:https://researchseminars.org/talk/GDEq/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladislav Zhvick
DTSTART:20210317T162000Z
DTEND:20210317T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/31
DESCRIPTION:Title: No
nlocal conservation law in a submerged jet\nby Vladislav Zhvick as par
t of Geometry of differential equations seminar\n\nLecture held in room 30
3 of the Independent University of Moscow.\n\nAbstract\nLandau was the fir
st to obtain the exact solution of Navier-Stokes equations for an axisymme
tric submerged jet generated by a point momentum source. The Landau jet is
the main term of a coordinate expansion of the flow far field in the case
when the flow is generated by a finite size source (for example\, a tube
with flow). The next term of the expansion was calculated by Rumer. This t
erm has an indefinite coefficient. To determine this coefficient we need a
conservation law connecting the jet far field with the source. Well-known
conservation laws of mass\, momentum\, and angular momentum fail to calcu
late the coefficient. In my talk\, I will solve this problem for low visco
sity. In this case\, the flow satisfies the boundary layer equations that
possess a nonlocal conservation law closing the problem. The problem for a
n arbitrary viscosity remains open.\n
LOCATION:https://researchseminars.org/talk/GDEq/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anatolij Prykarpatski
DTSTART:20210324T162000Z
DTEND:20210324T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/32
DESCRIPTION:Title: On
integrability of some Riemann type hydrodynamical systems and Dubrovin in
tegrability classification of perturbed Korteweg-de Vries type equations\nby Anatolij Prykarpatski as part of Geometry of differential equations
seminar\n\n\nAbstract\nIn our report we will stop on two closely related
to each other integrability theory aspects. The first one concerns the obt
ained integrability results\, based on the gradient-holonomic integrabilit
y scheme\, devised and applied by me jointly with Maxim Pavlov and collabo
rators to a virtually new important Riemann type hierarchy $D_{t}^{N-1}u=z
_{x}^{s}$\, $D_{t}z=0$\, where $s$\, \;$N\\in N$ are arbitrary natural
numbers\, and proposed in our work (M. Pavlov\, A. Prykarpatsky\, at al.\
, arXiv:1108.0878) as a nont
rivial generalization of the infinite hierarchy of the Riemann type flows\
, suggested before by M. Pavlov and D. Holm in the form of dynamical syste
ms $D_{t}^{N}u=0$\, defined on a $2\\pi$-periodic functional manifold $M^{
N}\\subset C^{\\infty}( R/2\\pi Z\; R^{N})$\, the vector $(u\,D_{t}u\,D_{t
}²u\,...\,D_{t}^{N-1}u\,z)^{\\intercal}\\in M^{N}$\, the differentiations
$D_{x}:=\\partial/\\partial x$\, $D_{t}:=\\partial/\\partial t+u\\partial
/\\partial x$ satisfy as above the Lie-algebraic commutator relationship $
[D_{x}\,D_{t}]=u_{x}D_{x}$ and t\\in R is an evolution parameter. The seco
nd aspect of our report concerns the integrability results obtained by B.
Dubrovin jointly with Y. Zhang and collaborators\, devoted to classificati
on of a special perturbation of the Korteweg-de Vries equation in the form
$u_{t}=uu_{x}+\\epsilon^2[f_{31}(u)u_{xxx}+f_{32}(u)u_{xx}u_{x}+f_{33}(u)
u_{x}^3]$\, where $f_{jk}(u)\,~j=3\,~k=1\,~3$\, are some smooth functions
and \\epsiln\\in R is a real parameter. We will deal with classification s
cheme of evolution equations of a special type suspicious on being integra
ble which was devised some years ago by untimely passed away Prof. Boris D
ubrovin (19 March 2019) and developed with his collaborators\, mainly with
Youjin Zhang. We have reanalyzed in detail their interesting results on i
ntegrability classification of a suitably perturbed KdV type equation with
in our gradient-holonomic integrability scheme\, devised many years ago an
d developed by me jointly with Maxim Pavlov and collaborators\, and found
out that the Dubrovin's scheme has missed at least a one very interesting
integrable equation\, whose natural reduction became similar to the well-k
nown Krichever-Novikov equation\, yet different from it. As a consequence
of the analysis\, we presented one can firmly claim that the Dubrovin-Zhan
g integrability criterion inherits some important part of the mentioned ab
ove gradient-holonomic integrability scheme properties\, coinciding with t
he statement about the necessary existence of suitably ordered reduction e
xpansions with coefficients to be strongly homogeneous differential polyno
mials.\n\nJoint with Alex A. Balinsky\, Radoslaw Kycia and Yarema A. Pryka
rpatsky.\n\nAlthough the talk will be in Russian\, the slides will be in E
nglish and the discussion will be in both languages.\n
LOCATION:https://researchseminars.org/talk/GDEq/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hovhannes Khudaverdian
DTSTART:20210331T162000Z
DTEND:20210331T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/33
DESCRIPTION:Title: Od
d symplectic geometry in the BV-formalism\nby Hovhannes Khudaverdian a
s part of Geometry of differential equations seminar\n\nLecture held in ro
om 304 of the Independent University of Moscow.\n\nAbstract\nOdd symplecti
c geometry was considered by physicists as an exotic counterpart of even s
ymplectic geometry. Batalin and Vilkovisky changed this\npoint of view by
the seminal work considering the quantisation of general theory in Lagrang
ian framework\, where they considered odd symplectic superspace of fields
and antifields. [In the case of Lie group of symmetries BV receipt is redu
ced to the standard Faddeev-Popov method.]\n\nThe main ingredient of the t
heory\, the exponent of the master action\, is defined by the function $f$
such that $\\Delta f=0$\, where $\\Delta$ is second order differential op
erator of the second order: $\\Delta=\\frac{\\partial^2}{\\partial x^i \\p
artial\\theta_i}$\, ($x^i\,\\theta_j$ are the Darboux coordinates of an od
d symplectic superspace.) This operator has no analogy in the standard sym
plectic geometry.\n\nI consider in this talk the main properties of the BV
-formalism geometry.\n\nThe $\\Delta$-operator is defined in geometrical c
lear way\, and this operator depends on the volume form.\n\nIt is suggeste
d the canonical operator $\\Delta$ on half-densities. This operator is the
proper framework for BV geometry. We also study the groupoid property of
BV master-equation\; this leads us to the notion of BV groupoid. We also d
iscuss some constructions of invariants for odd symplectic structure.\n
LOCATION:https://researchseminars.org/talk/GDEq/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20210407T162000Z
DTEND:20210407T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/34
DESCRIPTION:Title: On
dynamics of molecular media and generalization of Navier-Stokes equations
\nby Valentin Lychagin as part of Geometry of differential equations s
eminar\n\nLecture held in room 304 of the Independent University of Moscow
.\n\nAbstract\nThis talk is a prolongation of my previous talk that was de
voted to continuum mechanics of media possessing inner structure.\n\nHere
we'll consider molecular media\, its geometry and thermodynamics.\n\nThe m
ain goal of this talk is to present in the explicit form necessary geometr
ical structures and to give the explicit form of the Navier-Stokes equatio
ns.\n
LOCATION:https://researchseminars.org/talk/GDEq/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Morozov
DTSTART:20210421T162000Z
DTEND:20210421T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/35
DESCRIPTION:Title: La
x representations via twisted extensions of infinite-dimensional Lie algeb
ras: some new results\nby Oleg Morozov as part of Geometry of differen
tial equations seminar\n\nLecture held in room 304 of the Independent Univ
ersity of Moscow.\n\nAbstract\nI will discuss the technique for constructi
ng integrable differential equations via twisted extensions of infinite-di
mensional Lie algebras. Examples will include a 3D generalization of the H
unter-Saxton equation with the special value of the parameter and the "deg
enerate heavenly equation".\n
LOCATION:https://researchseminars.org/talk/GDEq/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Taras Skrypnyk
DTSTART:20210428T162000Z
DTEND:20210428T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/36
DESCRIPTION:Title: As
ymmetric variable separation for the Clebsch model\nby Taras Skrypnyk
as part of Geometry of differential equations seminar\n\n\nAbstract\nIn th
e present talk we present our result on separation of variables (SoV) for
the Clebsch model.\n\nIn particular\, we report on the development of two
methods in the variable separation theory:\n\n - the method of the
differential separability conditions\;
\n - the method of the vecto
r fields $Z$.
\n
\nUsing these two methods we construct an asymmet
ric variable separation for the Clebsch model. Our SoV is unusual: it is c
haracterized by two different curves of separation. We explicitly construc
t coordinates and momenta of separation\, the reconstruction formulae and
the Abel-type quadratures for the Clebsch system. The solution of the non-
standard Abel-Jacobi inversion problem is briefly discussed.\n
LOCATION:https://researchseminars.org/talk/GDEq/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuri Sachkov
DTSTART:20210414T162000Z
DTEND:20210414T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/37
DESCRIPTION:Title: Su
b-Riemannian geometry on the group of motions of the plane\nby Yuri Sa
chkov as part of Geometry of differential equations seminar\n\n\nAbstract\
nWe will discuss the unique\, up to local isometries\, contact sub-Riemann
ian struc\nture on the group SE(2) of proper motions of the plane (aka gro
up of rototransla\ntions).\nThe following questions will be addressed:\n\n \; geodesics\,\n &nb
sp\; their local and global optimality\,\n \; cut time\, cu
t locus\, and spheres\,\n \; infinite geodesics\,\n <
li> \; bicycle transform and relation of geodesics with Euler elastica
e\,\n \; group of isometries and homogeneous geodesics\,\n \; applications to imaging and robotics.\n\nJoint
work with Andrei Ardentov.\n
LOCATION:https://researchseminars.org/talk/GDEq/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugene Ferapontov (Loughborough University)
DTSTART:20210505T162000Z
DTEND:20210505T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/38
DESCRIPTION:Title: Se
cond-order PDEs in 3D with Einstein-Weyl conformal structure\nby Eugen
e Ferapontov (Loughborough University) as part of Geometry of differential
equations seminar\n\n\nAbstract\nI will discuss a general class of second
-order PDEs in 3D whose characteristic conformal structure satisfies the E
instein-Weyl conditions on every solution.\n\nThis property is known to be
equivalent to the existence of a dispersionless Lax pair\, as well as to
other equivalent definitions of dispersionless integrability.\n\nI will de
monstrate that (a) the Einstein-Weyl conditions can be viewed as an effici
ent contact-invariant test of dispersionless integrability\, (b) show some
partial classification results\, and (c) formulate a rigidity conjecture
according to which any second-order PDE with Einstein-Weyl conformal struc
ture can be reduced to a dispersionless Hirota form via a suitable contact
transformation.\n\nBased on joint work with S. Berjawi\, B. Kruglikov\, V
. Novikov.\n
LOCATION:https://researchseminars.org/talk/GDEq/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anton Zabrodin
DTSTART:20210519T162000Z
DTEND:20210519T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/39
DESCRIPTION:Title: Ka
domtsev-Petviashvili hierarchies of types B and C\nby Anton Zabrodin a
s part of Geometry of differential equations seminar\n\nLecture held in ro
om 303 or 304 of the Independent University of Moscow.\n\nAbstract\nThis i
s a short review of the Kadomtsev-Petviashvili hierarchies of types B and
C. The main objects are the $L$-operator\, the wave operator\, the auxilia
ry linear problems for the wave function\, the bilinear identity for the w
ave function and the tau-function. All of them are discussed in the paper.
The connections with the usual Kadomtsev-Petviashvili hierarchy (of the t
ype A) are clarified. Examples of soliton solutions and the dispersionless
limit of the hierarchies are also considered.\n
LOCATION:https://researchseminars.org/talk/GDEq/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20210512T162000Z
DTEND:20210512T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/40
DESCRIPTION:Title: Op
erations on universal enveloping algebra and the "argument shift" method\nby Georgy Sharygin as part of Geometry of differential equations semin
ar\n\nLecture held in room 303 or 304 of the Independent University of Mos
cow.\n\nAbstract\nIf a vector field X is given on a Poisson manifold M suc
h that the square of the Lie derivative in the X direction "kills" the Poi
sson bivector\, then there is a well-known simple method of "shifting the
argument" (along X) to construct a commutative subalgebra (with respect to
the Poisson bracket) inside the algebra of functions on M. In a particula
r case\, this method can be applied to the Poisson-Lie bracket on the symm
etric algebra of an arbitrary Lie algebra and gives (according to a well-k
nown result\, the proven Mishchenko-Fomenko conjecture) maximal commutativ
e subalgebras in the symmetric algebra. However\, the lifting of these alg
ebras to commutative subalgebras in the universal enveloping algebra\, alt
hough possible\, is based on very nontrivial results from the theory of in
finite-dimensional Lie algebras. In my talk\, I will describe partial resu
lts that allow one to construct on the universal enveloping algebra of the
algebra $gl_{n}$ the operators of "quasidifferentiation" and with thei
r help\, in some cases\, construct a commutative subalgebra in $Ugl_{n}$
. I will also describe how\, in the general case\, this question is red
uced to the combinatorial question of commuting a certain set of operators
in tensor powers $\\mathbb {R} ^{n}$. The story is based on collaborat
ions with Dmitry Gurevich\, Pavel Saponov and Ikeda Yasushi.\n
LOCATION:https://researchseminars.org/talk/GDEq/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20210922T162000Z
DTEND:20210922T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/41
DESCRIPTION:Title: On
metric invariants of spherical harmonics\nby Valentin Lychagin as par
t of Geometry of differential equations seminar\n\nLecture held in room 30
3 of the Independent University of Moscow.\n\nAbstract\nWe'll discuss the
algebraic and differential SO(3)-invariants of spherical harmonics and giv
e a description of fields of rational algebraic and rational differential
invariants together with their application to the description of regular S
O(3)-orbits of spherical harmonics.\n
LOCATION:https://researchseminars.org/talk/GDEq/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20211006T162000Z
DTEND:20211006T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/43
DESCRIPTION:Title: WD
VV equations and invariant bi-Hamiltonian formalism\nby Raffaele Vitol
o (Università del Salento) as part of Geometry of differential equations
seminar\n\n\nAbstract\nThe WDVV equations are central in Topological Field
Theory and Integrable Systems. We prove that in low dimensions the WDVV e
quations are bi-Hamiltonian. The invariance of the bi-Hamiltonian formalis
m is proved for N = 3. More examples in higher dimensions show that the re
sult might hold in general. The invariance group of the bi-Hamiltonian pai
rs is the group of projective reciprocal transformations. The significance
of projective invariance of WDVV equations is discussed in detail. Comput
er algebra programs that were used for calculations throughout the paper a
re provided at https://github.com/Jakub-Vasicek/WDVV-computations/.\n\nBas
ed on a joint work with Jakub Vašíček.\n
LOCATION:https://researchseminars.org/talk/GDEq/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Khavkine (Czech Academy of Sciences)
DTSTART:20211020T162000Z
DTEND:20211020T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/44
DESCRIPTION:Title: Tr
iangular decoupling of systems of differential equations\, with applicatio
n to separation of variables on Schwarzschild spacetime\nby Igor Khavk
ine (Czech Academy of Sciences) as part of Geometry of differential equati
ons seminar\n\n\nAbstract\nCertain tensor wave equations admit a complete
separation of variables on the Schwarzschild spacetime (asymptotically fla
t\, static\, spherically symmetric black hole in 4d)\, resulting in compli
cated systems of radial mode ODEs. Almost none of the important questions
about these radial mode equations can be answered in their original form.
I will discuss a drastic simplification of these ODE systems to sparse upp
er triangular form\, which uncovers their general properties. Essential to
this simplification are geometric properties of the original tensor wave
equations\, ideas from homological algebra and from the theory of ODEs wit
h rational coefficients. Based on https://arxiv.org/abs/1711.00585 \, http
s://arxiv.org/abs/1801.09800 \, https://arxiv.org/abs/2004.09651\n
LOCATION:https://researchseminars.org/talk/GDEq/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Morozov
DTSTART:20211027T162000Z
DTEND:20211027T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/45
DESCRIPTION:Title: In
tegrable PDEs and extensions of Lie-Rinehart algebras\nby Oleg Morozov
as part of Geometry of differential equations seminar\n\nLecture held in
room 303 of the Independent University of Moscow.\n\nAbstract\nI will disc
uss extensions of Lie-Rinehart algebras and their application to the probl
em of recognizing whether a given PDE admits a Lax representation.\n
LOCATION:https://researchseminars.org/talk/GDEq/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugene Ferapontov (Loughborough University)
DTSTART:20211201T162000Z
DTEND:20211201T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/46
DESCRIPTION:Title: Se
cond order integrable Lagrangians and WDVV equations\nby Eugene Ferapo
ntov (Loughborough University) as part of Geometry of differential equatio
ns seminar\n\n\nAbstract\nI will discuss integrability of 2D and 3D Euler-
Lagrange equations for second-order Lagrangians. A link to WDVV equations
will be established. Based on joint work with Maxim Pavlov and Lingling Xu
e.\n
LOCATION:https://researchseminars.org/talk/GDEq/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Kotov
DTSTART:20211013T162000Z
DTEND:20211013T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/48
DESCRIPTION:Title: Ri
emannian Cartan-Lie algebroids and groupoids and curved Yang-Mills-Higgs m
odels\nby Alexei Kotov as part of Geometry of differential equations s
eminar\n\n\nAbstract\nIn this talk the generalization of the Yang-Mills-Hi
ggs model will be presented\, based upon the notion of Cartan structures a
nd compatible metrics on Lie algebroids and groupoids.\n
LOCATION:https://researchseminars.org/talk/GDEq/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Doubrov
DTSTART:20211124T162000Z
DTEND:20211124T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/49
DESCRIPTION:Title: On
Cartan's C-class differential equations\nby Boris Doubrov as part of
Geometry of differential equations seminar\n\nLecture held in room 303 of
the Independent University of Moscow.\n\nAbstract\nWe consider a very spec
ial class of differential equations\, which is characterized by the condit
ion that all its local differential invariants (under the action of a suit
able Lie pseudogroup) become first integrals when restricted to the equati
on manifold. Such differential equations were introduced in a short note o
f Elie Cartan (Les espaces généralisés et l'intégration de certaines c
lasses d'équations différentielles\, C.R.\, 1938\, V.206\, N.23\, 1689-1
693)\, who characterized them in two simplest cases: scalar 2nd order ODEs
viewed under the pseudogroup of point transformations and scalar 3rd orde
r ODEs under the group of contact transformations. We show how these resul
ts generalize to any systems of ODEs and\, more generally\, differential e
quations of finite type. The same question for arbitrary systems of PDEs s
till remains open.\n
LOCATION:https://researchseminars.org/talk/GDEq/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eivind Schneider
DTSTART:20211110T162000Z
DTEND:20211110T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/50
DESCRIPTION:Title: Di
fferential invariants of Kundt spacetimes\nby Eivind Schneider as part
of Geometry of differential equations seminar\n\n\nAbstract\nWe compute g
enerators for the algebra of rational scalar differential invariants of ge
neral and degenerate Kundt spacetimes. Special attention is given to dimen
sions 3 and 4 since in those dimensions the degenerate Kundt metrics are k
nown to be exactly the Lorentzian metrics that can not be distinguished by
polynomial curvature invariants constructed from the Riemann tensor and i
ts covariant derivatives.\n\nThe talk is based on joint work with Boris Kr
uglikov.\n
LOCATION:https://researchseminars.org/talk/GDEq/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Duyunova and Sergey Tychkov
DTSTART:20211103T162000Z
DTEND:20211103T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/51
DESCRIPTION:Title: Th
e Euler system on a space curve\nby Anna Duyunova and Sergey Tychkov a
s part of Geometry of differential equations seminar\n\n\nAbstract\nWe con
sider flows of an inviscid medium on a space curve in a constant gravitati
onal field (the Euler system). We discuss symmetries and differential inva
riants of the Euler system\, and give their classification based on symmet
ries group of the system. Using differential invariants for this system\,
we obtain its quotient. The solutions of the quotient equation that are co
nstant along characteristic vector field provide some solutions of the Eul
er system.\n\nJoint work with Valentin Lychagin.\n
LOCATION:https://researchseminars.org/talk/GDEq/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Igonin
DTSTART:20211229T162000Z
DTEND:20211229T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/52
DESCRIPTION:Title: Al
gebra and geometry of Lax representations and Bäcklund transformations fo
r (1+1)-dimensional partial differential and differential-difference equat
ions\nby Sergei Igonin as part of Geometry of differential equations s
eminar\n\n\nAbstract\nSee IgoninSeminar20211229abstract.pdf\n
LOCATION:https://researchseminars.org/talk/GDEq/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hynek Baran
DTSTART:20211208T162000Z
DTEND:20211208T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/53
DESCRIPTION:Title: Je
ts\, a computer algebra on diffieties\nby Hynek Baran as part of Geome
try of differential equations seminar\n\n\nAbstract\nJets is a set of Mapl
e procedures to facilitate solution of differential equations in total der
ivatives on diffieties. Otherwise said\, Jets is a tool to compute symmetr
ies\, conservation laws\, zero-curvature representations\, recursion opera
tors\, any many other invariants of systems of partial differential equati
ons.\n
LOCATION:https://researchseminars.org/talk/GDEq/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev
DTSTART:20211117T162000Z
DTEND:20211117T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/54
DESCRIPTION:Title: Pr
esymplectic gauge PDEs and Batalin-Vilkovisky formalism\nby Maxim Grig
oriev as part of Geometry of differential equations seminar\n\nLecture hel
d in room 303 of the Independent University of Moscow.\n\nAbstract\nGauge
PDE is a geometrical object underlying what physicists call a local gauge
field theory defined at the level of equations of motion (i.e. without sp
ecifying Lagranian) in terms of BV-BRST formalism. Although gauge PDE can
be defined as a PDE equipped with extra structures\, the generalization is
not entirely straightforward as\, for instance\, two gauge PDEs can be eq
uivalent even if the underlying PDEs are not. As far as Lagrangian gauge s
ystems are concerned the powerful framework is provided by the BV formalis
m on jet-bundles. However\, just like in the case of usual PDEs it is diff
icult to encode the BV extension of the Lagrangian in terms of the intrins
ic geometry of the equation manifold while working on jet-bundles is often
very restrictive\, especially in analyzing boundary behaviour\, e.g.\, in
the context of AdS/CFT correspondence. We show that BV Lagrangian (or its
weaker analogs) can be encoded in the compatible graded presymplectic str
ucture on the gauge PDE. In the case of genuine Lagrangian systems this pr
esymplectic structure is related to a certain completion of the canonical
BV symplectic structure. A presymplectic gauge PDE gives rise to a BV form
ulation of the underlying system through an appropriate generalization of
the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) sigma-model constructi
on followed by taking the symplectic quotient. The construction is illustr
ated on the standard examples of gauge theories with particular emphasis o
n the Einstein gravity\, where this naturally leads to an elegant presympl
ectic AKSZ representation of the BV extension of the Cartan-Weyl formulati
on of gravity.\n
LOCATION:https://researchseminars.org/talk/GDEq/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petr Vojčák
DTSTART:20211208T162000Z
DTEND:20211208T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/55
DESCRIPTION:Title: On
the algebras of nonlocal symmetries for the (modified) 4D Martı́nez Alo
nso-Shabat equation\nby Petr Vojčák as part of Geometry of different
ial equations seminar\n\n\nAbstract\nWe consider two four-dimensional Lax-
integrable equations known as the 4D Martı́nez Alonso-Shabat equation an
d the modified Martı́nez Alonso-Shabat equation\, respectively. We const
ruct two differential coverings for both of them and describe the algebras
of nonlocal symmetries in these coverings. We also analyze the actions of
the known recursion operators on these nonlocal symmetries.\n\nPartially
based on a joint work with Joseph Krasil'shchik.\n
LOCATION:https://researchseminars.org/talk/GDEq/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Agafonov
DTSTART:20220209T162000Z
DTEND:20220209T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/56
DESCRIPTION:Title: Da
rboux integrability for diagonal systems of hydrodynamic type\nby Serg
ey Agafonov as part of Geometry of differential equations seminar\n\nLectu
re held in room 303 of the Independent University of Moscow.\n\nAbstract\n
We prove that diagonal systems of hydrodynamic type are Darboux integrable
if and only if the Laplace transformation sequences of the system for com
muting flows terminate\, give geometric interpretation for Darboux integra
bility of such systems in terms of congruences of lines and in terms of so
lution orbits with respect to symmetry subalgebras\, show that Darboux int
egrable systems are necessarily semihamiltonian\, and discuss known and ne
w examples.\n
LOCATION:https://researchseminars.org/talk/GDEq/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Kruglikov (UiT the Arctic University of Norway)
DTSTART:20220223T162000Z
DTEND:20220223T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/57
DESCRIPTION:Title: OD
Es with essential contact or point symmetries\nby Boris Kruglikov (UiT
the Arctic University of Norway) as part of Geometry of differential equa
tions seminar\n\nLecture held in room 303 of the Independent University of
Moscow.\n\nAbstract\n(joint work with Eivind Schneider)\n\nWe observe tha
t\, up to conjugation\, a majority of higher order ODEs and ODE systems ha
ve only point fiber-preserving symmetries (surprisingly this is also true
for "most interesting" ODEs). We describe all the exceptions in the case o
f scal ar ODEs and systems of pairs of ODEs on a pair of functions. We exp
loit classifications of Lie algebras of vector fields in 2 and 3 dimension
s.\n\nWhile we can express scalar ODEs with essentially contact or point s
ymmetry algebras via absolute and relative differential invariants\, we ha
ve to invoke also conditional differential invariants in the case of ODE s
ystems to deal with singular orbits of the action. In the scalar case the
result is partially due to Lie\, but we consider the global classification
and discuss the algebra of relative invariants. For systems the result is
new.\n\nInvestigating prolongations of the actions\, we observe some inte
resting relations between different realizations of Lie algebras. We also
note that prolongation of a finite-dimensional Lie algebra acting on a dif
ferential equation may not eventually become free. An example of underdete
rmined ODE with this phenomenon shows limitations of the method of moving
frames.\n
LOCATION:https://researchseminars.org/talk/GDEq/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitri Alekseevsky
DTSTART:20220316T162000Z
DTEND:20220316T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/59
DESCRIPTION:Title: Sp
ecial Vinberg cones and their applications\nby Dmitri Alekseevsky as p
art of Geometry of differential equations seminar\n\nLecture held in room
303 of the Independent University of Moscow.\n\nAbstract\nThe talk is base
d on joint works with Vicete Cortes\, Andrea Spiro and Alessio Marrani.\n\
nA short survey of the Vinberg theory of convex cones (including its infor
mational geometric interpretation) and homogeneous convex cones will be pr
esented. Then we concentrate on the theory of rank 3 special Vinberg cones
\, associated to metric Clifford $Cl({\\mathbb R}^n)$ modules.\n\nA genera
lization of the theory to the indefinite special Vinberg cones\, associate
d to indefinite metric Clifford $Cl({\\mathbb R}^{p\,q})$ modules is indic
ated. An application of special Vinberg cones to $N=2 \, \\\, d=5\,4\,3$ S
upergravity will be considered.\n\nWe will discuss also applications of th
eory of homogeneous convex cones to convex programming\, information geome
try and Frobenius manifolds.\n
LOCATION:https://researchseminars.org/talk/GDEq/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20220309T162000Z
DTEND:20220309T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/60
DESCRIPTION:Title: Mu
ltiplicative kernels\, Non-abelian Abel theorem\, Kontsevich polynomials a
nd around. Part 1\nby Vladimir Rubtsov (Université d'Angers) as part
of Geometry of differential equations seminar\n\n\nAbstract\nWe discuss re
cent progress (published and unpublished yet ) in studies of multiplicativ
e kernels\, initiated by M. Konstevich. We will try to explain various lin
ks and applications of this notion in geometry\, differential equations an
d integrable systems. My talk is based on the paper arXiv:2102.09511 and on two ongoing projects with
I. Gaiur and D. Van Straten.\n
LOCATION:https://researchseminars.org/talk/GDEq/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20220413T162000Z
DTEND:20220413T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/61
DESCRIPTION:Title: Na
tural invariants and classification of quasilinear second-order differenti
al operators\nby Valentin Lychagin as part of Geometry of differential
equations seminar\n\nLecture held in room 303 of the Independent Universi
ty of Moscow.\n\nAbstract\nThis talk is based on joint research with Valer
y Yumaguzhin.\n\nIn the first part\, we outline the method of finding rati
onal natural differential invariants of a class of quasilinear second-orde
r differential operators\, and then we show how these invariants could be
used to get local as well as global classification of such type operators
with respect to the diffeomorphism group.\n
LOCATION:https://researchseminars.org/talk/GDEq/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitri Gurevich
DTSTART:20220330T162000Z
DTEND:20220330T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/62
DESCRIPTION:Title: Qu
antum Matrix Algebras and their applications\nby Dimitri Gurevich as p
art of Geometry of differential equations seminar\n\n\nAbstract\nQuantum M
atrix Algebras are very interesting objects from algebraic viewpoint. Part
icular examples of these algebras are related to Drinfeld-Jimbo Quantum Gr
oups. Some of these QMA admit defining analogs of partial derivatives. In
a limit it is possible to develop a new calculus on the enveloping algebra
s $U(gl(N))$.\n\nOther applications will be discussed.\n
LOCATION:https://researchseminars.org/talk/GDEq/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20220406T162000Z
DTEND:20220406T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/63
DESCRIPTION:Title: Mu
ltiplicative kernels\, Non-abelian Abel theorem\, Kontsevich polynomials a
nd around. Part 2\nby Vladimir Rubtsov (Université d'Angers) as part
of Geometry of differential equations seminar\n\n\nAbstract\nA continuatio
n of the talk on 9 Ma
rch.\n\nWe discuss recent progress (published and unpublished yet) in
studies of multiplicative kernels\, initiated by M. Konstevich. We will tr
y to explain various links and applications of this notion in geometry\, d
ifferential equations and integrable systems. My talk is based on the pape
r arXiv:2102.09511 and on t
wo ongoing projects with I. Gaiur and D. Van Straten.\n
LOCATION:https://researchseminars.org/talk/GDEq/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Marshall
DTSTART:20220504T162000Z
DTEND:20220504T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/64
DESCRIPTION:Title: On
action-angle duality\nby Ian Marshall as part of Geometry of differen
tial equations seminar\n\nLecture held in room 303 of the Independent Univ
ersity of Moscow.\n\nAbstract\nAction-angle duality is a property enjoyed
by systems of Ruijsenaars type - many body systems\; relativistic analogue
s of Calogero-Moser-Sutherland systems - whereby families of integrable sy
stems come in natural pairs: the canonical coordinates of one system are t
he action-angle variables of the other\, and together they generate the wh
ole phase space. I will explain this property\, and why it is special. Whe
n transported to quantum systems\, the action-angle duality property is re
presented in the form of bispectral operators.\n\nI hope also to describe
results obtained with László Fehér in which Hamiltonian reduction is us
ed to obtain systems in action-angle duality relation with one an other.\n
LOCATION:https://researchseminars.org/talk/GDEq/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilia Gaiur (University of Birmingham)
DTSTART:20220511T162000Z
DTEND:20220511T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/65
DESCRIPTION:Title: Ha
miltonian reduction and rational Calogero system\nby Ilia Gaiur (Unive
rsity of Birmingham) as part of Geometry of differential equations seminar
\n\n\nAbstract\nIn my talk I am going to give an introduction to the theor
y of the moment map for the Hamiltonian group action on the symplectic man
ifolds with the focus on Hamiltonian reduction and integrable systems. In
particular\, I will show how to translate symmetries of the Hamiltonian sy
stem to the first integrals using the moment map and what kind of systems
we may obtain by performing such reduction. As the main example\, I will d
emonstrate how to obtain a rational Calogero system from the free particle
system on the cotangent bundle to the Lie algebra $su(n)$.\n
LOCATION:https://researchseminars.org/talk/GDEq/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Smilga
DTSTART:20220518T162000Z
DTEND:20220518T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/66
DESCRIPTION:Title: No
ncommutative quantum mechanical systems associated with Lie algebras\n
by Andrei Smilga as part of Geometry of differential equations seminar\n\n
\nAbstract\nWe consider quantum mechanics on the noncommutative spaces cha
racterized by the commutation relations\n$$ [x_a\, x_b] \\ =\\ i\\theta f_
{abc} x_c\\\,\, $$\nwhere $f_{abc}$ are the structure constants of a Lie a
lgebra. We note that this problem can be reformulated as an ordinary quant
um problem in a commuting {\\it momentum} space. The coordinates are then
represented as linear differential operators $\\hat x_a = -i \\hat D_a = -
iR_{ab} (p)\\\, \\partial /\\partial p_b $. Generically\, the matrix $R_{a
b}(p)$ represents a certain infinite series over the deformation parameter
$\\theta$: $R_{ab} = \\delta_{ab} + \\ldots$. The deformed Hamiltonian\,
$\\hat H \\ =\\ - \\frac 12 \\hat D_a^2\\\,\, $ describes the motion alon
g the corresponding group manifolds with the characteristic size of order
$\\theta^{-1}$. Their metrics are also expressed into certain infinite se
ries in $\\theta$.\n\nFor the algebras $su(2)$ and $u(2)$\, it has been po
ssible to represent the operators $\\hat x_a$ in a simple finite form. A b
yproduct of our study are new nonstandard formulas for the metrics on $SU(
2) \\equiv S^3$ and on $SO(3)$.\n
LOCATION:https://researchseminars.org/talk/GDEq/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Talalaev (Moscow State University)
DTSTART:20221005T162000Z
DTEND:20221005T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/67
DESCRIPTION:Title: Ca
tegory of braided sets\, extensions and 2-analogues\nby Dmitry Talalae
v (Moscow State University) as part of Geometry of differential equations
seminar\n\nLecture held in room 303 of the Independent University of Mosco
w.\n\nAbstract\nA braided set is the same thing as a solution of the set-t
heoretic Yang-Baxter equation. It is important to rephrase this in a categ
orical language from the point of view of natural questions of morphisms\,
extensions and simple objects in this family. I will tell about several r
esults in the problem of constructing extensions of braided sets and how t
his problem can be generalized to 2-braided categories\, how to build exte
nsions of sets with solutions of the Zamolodchikov tetrahedron equation.\n
LOCATION:https://researchseminars.org/talk/GDEq/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Kushner
DTSTART:20221109T162000Z
DTEND:20221109T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/68
DESCRIPTION:Title: On
the integration of suspension filtration equations and thrombus formation
\nby Alexei Kushner as part of Geometry of differential equations semi
nar\n\nLecture held in room 303 of the Independent University of Moscow.\n
\nAbstract\nThe problem of one-dimensional filtration of a suspension in a
porous medium is considered. The process is described by a hyperbolic sys
tem of two first-order differential equations. This system is reduced by a
change of variables to the symplectic equation of the Monge-Ampère type.
It is noteworthy that this symplectic equation cannot be reduced to a lin
ear wave equation by a symplectic transformation (the Lychagin-Rubtsov the
orem works here)\, but it can be done by a contact transformation. This ma
de it possible to find its exact general solution and exact solutions of t
he original system. The solution of the initial-boundary value problem and
the Cauchy problem are constructed.\n\nJoint work with Svetlana Mukhina.\
n
LOCATION:https://researchseminars.org/talk/GDEq/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20221012T162000Z
DTEND:20221012T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/69
DESCRIPTION:Title: On
interplay between jet and information geometries\nby Valentin Lychagi
n as part of Geometry of differential equations seminar\n\nLecture held in
room 303 of the Independent University of Moscow.\n\nAbstract\nWe will co
nsider the procedure of measurement of random vectors\, operators and tens
ors from the double point of view: pure probabilistic and geometrical. Usi
ng the principle of minimum information gain\, we reformulate the probabil
istic approach as studies in the geometry of jet spaces over the manifolds
of extreme measures. Moreover\, the procedure of a measurement itself bec
omes equivalent to study various geometrical structures on integral manifo
lds of the Cartan distribution. We will illustrate all of this for the cas
e of thermodynamics of real gases and phase transitions of the first and s
econd orders.\n
LOCATION:https://researchseminars.org/talk/GDEq/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20221019T162000Z
DTEND:20221019T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/70
DESCRIPTION:Title: Ho
mogeneous Hamiltonian operators\, projective geometry and integrable syste
ms\nby Raffaele Vitolo (Università del Salento) as part of Geometry o
f differential equations seminar\n\n\nAbstract\nFirst-order homogeneous Ha
miltonian operators play a central role in the Hamiltonian formulation of
quasilinear systems of PDEs. They have well-known differential-geometric i
nvariance properties which find application in the theory of Frobenius man
ifolds. In this talk we will show that second and third order homogeneous
Hamiltonian operators are invariant under reciprocal transformations of pr
ojective type\, thus allowing for a projective classification of the opera
tors. Then\, we will describe how the above operators generate known and n
ew integrable systems\, and discuss the invariance properties of the syste
ms under projective transformations.\n
LOCATION:https://researchseminars.org/talk/GDEq/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roberto D'Onofrio
DTSTART:20221116T162000Z
DTEND:20221116T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/73
DESCRIPTION:Title: Mo
nge-Ampère geometry and semigeostrophic equations\nby Roberto D'Onofr
io as part of Geometry of differential equations seminar\n\n\nAbstract\nSe
migeostrophic equations are a central model in geophysical fluid dynamics
designed to represent large-scale atmospheric flows. Their remarkable dual
ity structure allows for a geometric approach through Lychagin's theory of
Monge-Ampère equations. We extend seminal earlier work on the subject by
studying the properties of an induced metric on solutions\, understood as
Lagrangian submanifolds of the phase space. We show the interplay between
singularities\, elliptic-hyperbolic transitions\, and the metric signatur
e through a few visual examples.\n
LOCATION:https://researchseminars.org/talk/GDEq/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Talalaev
DTSTART:20221130T162000Z
DTEND:20221130T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/74
DESCRIPTION:Title: Za
molodchikov Tetrahedron equation\nby Dmitry Talalaev as part of Geomet
ry of differential equations seminar\n\nLecture held in room 303 of the In
dependent University of Moscow.\n\nAbstract\nThe main subject of the talk
is the Zamolodchikov tetrahedron equation\, which is the next n-simplex eq
uation after the Yang-Baxter equation. This equation finds its embodiments
in the theory of cluster manifolds\, exactly-solvable models of statistic
al physics in dimension 3\, as well as the theory of invariants of 2-knots
\, that is\, classes of isotopies of embeddings of a two-dimensional surfa
ce in a 4-dimensional space.\n\nThe main focus of the report will be on th
e definition of this class of equations in terms of the hypercube face col
oring problem\, the cohomology complex associated with each solution of th
e n-simplex equation. We will discuss how these definitions are realized i
n the case of n=3\, that is\, in the case of the tetrahedron equation\, an
d some interesting classes of solutions to this equation arising in modern
mathematics.\n
LOCATION:https://researchseminars.org/talk/GDEq/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Pierre Magnot
DTSTART:20221123T162000Z
DTEND:20221123T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/75
DESCRIPTION:Title: Ne
w perspectives for generalized Kadomtsev-Petviashvili hierarchies\nby
Jean-Pierre Magnot as part of Geometry of differential equations seminar\n
\n\nAbstract\nIn the setting of diffeological differential algebras\, we f
irst expose step by step how the classical algebraic construction of the s
olution of the (classical) Kadomtsev-Petviashvili hierarchy can be extende
d in order to get well-posedness for Kadomtsev-Petviashvili hierarchies in
this generalized setting. Of course\, we give a short exposition of the n
ecessary notions in diffeologies for non-specialists of this topic.\n\nThe
n\, we discuss the Hamiltonian formulation in a refreshed way. Finally\, w
e deduce the corresponding Kadomtsev-Petviashvili equations\, first in an
abstract formulation\, and in a series of examples.\n\nReferences:\n<
a href="https://arxiv.org/abs/1007.3543">arXiv:1007.3543\nhttps://dx.doi.org/10.
1080/14029251.2017.1418057\narXiv:1608.03994\narXiv:2101.04523\, Mi tm
f10046\narXiv:2203.070
62\narXiv:2212.07583\n
LOCATION:https://researchseminars.org/talk/GDEq/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20221221T162000Z
DTEND:20221221T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/76
DESCRIPTION:Title: Ch
opping integrals of the full symmetric Toda system\, a new approach\nb
y Georgy Sharygin as part of Geometry of differential equations seminar\n\
nLecture held in room 303 of the Independent University of Moscow.\n\nAbst
ract\nIn my talk I will try to answer the questions that has been causing
my anxiety for a rather long time: where do the additional integrals of th
e full symmetric Toda system come from\, why they are rational and what do
es all this have to do with "chopping". Even if we can use the AKS method
there remains the question\, why do the initial functions actually commute
(and whether it is possible to find other with the same property). The kn
own answers were concerned either with rather hard straightforward computa
tions\, or with the properties of a Gaudin system\; they look pretty compl
icated. In my talk I will show how one can obtain these integrals with the
help of some simple differential operators (in the manner of the argument
shift method). Besides this\, we will discuss some other possible integra
ls as well as the method to solve the corresponding flows by QR decomposit
ion.\n\nThe talk is based on a common work with Yu. Chernyakov and D. Tala
laev.\n
LOCATION:https://researchseminars.org/talk/GDEq/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20221207T162000Z
DTEND:20221207T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/77
DESCRIPTION:Title: On
normal forms of differential operators\nby Valentin Lychagin as part
of Geometry of differential equations seminar\n\nLecture held in room 303
of the Independent University of Moscow.\n\nAbstract\nIn this talk\, we cl
assify linear (as well as some special nonlinear) scalar diff\nerential op
erators of order $k$ on $n$-dimensional manifolds with respect to the diff
eomorphism pseudogroup.\n
Cases\, when $k = 2$\, $\\forall n$\, and $
k = 3$\, $n = 2$\, were discussed before\, and now we consider cases $k\\g
e5$\, $n = 2$ and $k\\ge4$\, $n = 3$ and $k\\ge3$\, $n\\ge4$. In all these
cases\, the fields of rational differential invariants are generated by t
he 0-order invariants of symbols.\n\nThus\, at first\, we consider the cla
ssical problem of Gl-invariants of $n$-ary forms. We'll illustrate here th
e power of the differential algebra approach to this problem and show how
to find the rational Gl-invariants of $n$-are forms in a constructive way.
\n\nAfter all\, we apply the $n$ invariants principle in order to get (loc
al as well as global) normal forms of linear operators with respect to the
diffeomorphism pseudogroup.\n\nDepending on available time\, we show how
to extend all these results to some classes of nonlinear operators.\n
LOCATION:https://researchseminars.org/talk/GDEq/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Tsarev\, Folkert Müller-Hoissen\, Dmitry Millionschikov\,
Boris Konopelchenko
DTSTART:20221214T140000Z
DTEND:20221214T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/78
DESCRIPTION:Title: On
e day workshop in honor of Maxim Pavlov's 60th birthday\nby Sergey Tsa
rev\, Folkert Müller-Hoissen\, Dmitry Millionschikov\, Boris Konopelchenk
o as part of Geometry of differential equations seminar\n\nLecture held in
room 303 of the Independent University of Moscow.\n\nAbstract\n\n\nSpeaker: Sergey Tsarev (Krasnoyarsk)\n\nTitle
: Hydrodynamic type systems and beyond: a long way towards integr
ability with Maxim Pavlov\n\nSpeaker: Folkert Müller-Hoi
ssen (Göttingen)\n\nTitle: A relative of the NLS equatio
n revisited\n\nSpeaker: Dmitry Millionschikov (Moscow)\n\
nTitle: Growth of Lie algebras and integrability\n\nSpeaker: Boris Konopelchenko\n\nTitle: Multi-
dimensional MAS-Pavlov-Jordan chain and its reduct\nions\n\nThe abstracts\
, slides\, and videos can be found on the page https://gdeq.org/Pavlov60\n
LOCATION:https://researchseminars.org/talk/GDEq/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Orlov
DTSTART:20230208T162000Z
DTEND:20230208T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/79
DESCRIPTION:Title: Hu
rwitz numbers\, matrix models\, commuting operators\nby Alexander Orlo
v as part of Geometry of differential equations seminar\n\nLecture held in
room 303 of the Independent University of Moscow.\n\nAbstract\nWe will an
alyze how matrix models are related to arbitrary Hurwitz numbers. There ar
e equivalent descriptions using\n\n(a) differential operators\n\n(b) oscil
latory algebra and bosonic Fock space.\n\nCommuting sets of such operators
will be presented. This is a modification of Calogero's quantum Hamiltoni
ans at a special point.\n
LOCATION:https://researchseminars.org/talk/GDEq/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Samokhin
DTSTART:20230222T162000Z
DTEND:20230222T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/80
DESCRIPTION:Title: On
perturbations retaining conservation laws of differential equations\n
by Alexey Samokhin as part of Geometry of differential equations seminar\n
\n\nAbstract\nThe talk deals with perturbations of the equation that have
a number of conservation laws. When a small term is added to the equation
its conserved quantities usually decay at individual rates\, a phenomenon
known as a selective decay. These rates are described by the simple law us
ing the conservation laws' generating functions and the added term. Yet so
me perturbation may retain a specific quantity(s)\, such as energy\, momen
tum and other physically important characteristics of solutions. We introd
uce a procedure for finding such perturbations and demonstrate it by examp
les including the KdV-Burgers equation and a system from magnetodynamics.\
n
LOCATION:https://researchseminars.org/talk/GDEq/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20230215T162000Z
DTEND:20230215T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/81
DESCRIPTION:Title: On
invariants and equivalence differential operators under algebraic Lie pse
udogroups actions\nby Valentin Lychagin as part of Geometry of differe
ntial equations seminar\n\nLecture held in room 303 of the Independent Uni
versity of Moscow.\n\nAbstract\nIt is the concluding talk on invariants an
d the equivalence of differential operators under actions of Lie pseudogro
ups. We'll show\, that under some natural algebraic restrictions on Lie ps
eudogroups and nonlinearities of differential operators under consideratio
n\, there is a reasonable description of their orbits under the Lie pseudo
groups\, as well as local model forms. Then\, the general approach will be
applied to the Cartan list of primitive Lie pseudogroups.\n
LOCATION:https://researchseminars.org/talk/GDEq/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henrik Winther
DTSTART:20230301T162000Z
DTEND:20230301T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/82
DESCRIPTION:Title: Je
t functors in noncommutative geometry\nby Henrik Winther as part of Ge
ometry of differential equations seminar\n\n\nAbstract\nWe construct an in
finite family of endofunctors $J_d^n$ on the category of left $A$-modules\
, where $A$ is a unital associative algebra over a commutative ring $k$\,
equipped with an exterior algebra $\\Omega^\\bullet_d$. We prove that thes
e functors generalize the corresponding classical notion of jet functors.
The functor $J_d^n$ comes equipped with a natural transformation from the
identity functor to itself\, which plays the rôle of the classical prolon
gation map. This allows us to define the notion of linear differential ope
rator with respect to $\\Omega^{\\bullet}_d$. These retain most classical
properties of differential operators\, and operators such as partial deriv
atives and connections belong to this class. Moreover\, we construct a fun
ctor of quantum symmetric forms $S^n_d$ associated to $\\Omega^\\bullet_d$
\, and proceed to introduce the corresponding noncommutative analogue of t
he Spencer $\\delta$-complex. We give necessary and sufficient conditions
under which the jet functor $J_d^n$ satisfies the jet exact sequence\, $0\
\rightarrow S^n_d \\rightarrow J_d^n \\rightarrow J_d^{n-1} \\rightarrow 0
$. This involves imposing mild homological conditions on the exterior alge
bra\, in particular on the Spencer cohomology $H^{\\bullet\,2}$.\n\nThis i
s a joint work with K. Flood and M. Mantegazza.\n
LOCATION:https://researchseminars.org/talk/GDEq/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugene Ferapontov (Loughborough University)
DTSTART:20230322T162000Z
DTEND:20230322T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/83
DESCRIPTION:Title: Qu
asilinear systems of Jordan block type\nby Eugene Ferapontov (Loughbor
ough University) as part of Geometry of differential equations seminar\n\n
\nAbstract\nI will discuss integrability aspects of quasilinear systems wh
ose velocity matrix has a nontrivial Jordan block structure. I plan to cov
er the following topics:\n\n- Integrable systems of Jordan block typ
e and modified KP hierarchy\;
\n- Hamiltonian aspects of quasilinear
systems of Jordan block type\;
\n- Example: delta-functional reduct
ions of the soliton gas equation.
\n
\nThe talk will be based on j
oint work with Lingling Xue\, Maxim Pavlov and Pierandrea Vergallo.\n
LOCATION:https://researchseminars.org/talk/GDEq/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Dobrokhotov and Vladimir Nazaikinskii
DTSTART:20230329T162000Z
DTEND:20230329T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/84
DESCRIPTION:Title: Ex
act and asymptotic solutions of a system of nonlinear shallow water equati
ons in basins with gentle shores\nby Sergey Dobrokhotov and Vladimir N
azaikinskii as part of Geometry of differential equations seminar\n\nLectu
re held in room 303 of the Independent University of Moscow.\n\nAbstract\n
We suggest an effective approximate method for constructing solutions to p
roblems with a free boundary for 1-D and 2-D-systems of nonlinear shallow
water equations in basins with gentle shores. The method is a modification
(and pragmatic simplification) of the Carrier-Greenspan transformation in
the theory of 1-D shallow water over a flat sloping bottom. The result is
as follows: approximate solutions of nonlinear equations are expressed th
rough solutions of naively linearized equations via parametrically defined
functions. This allows us to describe the effects of waves run-up on a sh
ore and their splash. Among the applications we can mention tsunami waves\
, seiches and coastal waves. We also present a comparison of the obtained
formulas with the V.A. Kalinichenko (Institute for Problems in Mechanics R
AS) experiment with standing Faraday waves in an extended basin with gentl
y sloping shores.\n\nJoint work with D. Minenkov.\n
LOCATION:https://researchseminars.org/talk/GDEq/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20230315T162000Z
DTEND:20230315T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/85
DESCRIPTION:Title: Qu
asiderivations and commutative subalgebras of the algebra $U\\mathfrak{gl}
_n$\nby Georgy Sharygin as part of Geometry of differential equations
seminar\n\nLecture held in room 303 of the Independent University of Mosco
w.\n\nAbstract\nLet $\\mathfrak{gl}_n$ be the Lie algebra of $n\\times n$
matrices over a characteristic zero field $\\Bbbk$ (one can take $\\Bbbk=\
\mathbb R$ or $\\mathbb C$)\; let $S(\\mathfrak{gl}_n)$ be the Poisson alg
ebra of polynomial functions on $\\mathfrak{gl}_n^*$\, and $U\\mathfrak{gl
}_n$ the universal enveloping algebra of $\\mathfrak{gl}_n$. By Poincaré-
Birkhoff-Witt theorem $S(\\mathfrak{gl}_n)$ is isomorphic to the graded al
gebra $gr(U\\mathfrak{gl}_n)$\, associated with the order filtration on $U
\\mathfrak{gl}_n$. Let $A\\subseteq S(\\mathfrak{gl}_n)$ be a Poisson-comm
utative subalgebra\; one says that a commutative subalgebra $\\hat A\\subs
eteq U\\mathfrak{gl}_n$ is a $\\textit{quantisation}$ of $A$\, if its imag
e under the natural projection $U\\mathfrak{gl}_n\\to gr(U\\mathfrak{gl}_n
)\\cong S(\\mathfrak{gl}_n)$ is equal to $A$.\n\nIn my talk I will speak a
bout the so-called "argument shift" subalgebras $A=A_\\xi$ in $S(\\mathfra
k{gl}_n)$\, generated by the iterated derivations of central elements in $
S(\\mathfrak{gl}_n)$ by a constant vector field $\\xi$. There exist severa
l ways to define a quantisation of $A_\\xi$\, most of them are related wit
h the considerations of some infinite-dimensional Lie algebras. In my talk
I will explain\, how one can construct such quantisation of $A_\\xi$ usin
g as its generators iterated $\\textit{quasi-derivations}$ $\\hat\\xi$ of
$U\\mathfrak{gl}_n$. These operations are "quantisations" of the derivatio
ns on $S(\\mathfrak{gl}_n)$ and verify an analog of the Leibniz rule. In f
act\, I will show that iterated quasiderivation of certain generating elem
ents in $U\\mathfrak{gl}_n$ are equal to the linear combinations of the el
ements\, earlier constructed by Tarasov.\n
LOCATION:https://researchseminars.org/talk/GDEq/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20230405T162000Z
DTEND:20230405T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/86
DESCRIPTION:Title: La
grangian formalism and the intrinsic geometry of PDEs\nby Konstantin D
ruzhkov as part of Geometry of differential equations seminar\n\n\nAbstrac
t\nThis report is an attempt to answer the following question. Where exact
ly does a differential equation contain information about its variational
nature? Apparently\, in the general case\, the concept of a presymplectic
structure as a closed variational 2-form may not be sufficient to describe
variational principles in terms of intrinsic geometry. I will introduce t
he concept of an internal Lagrangian and relate it to the Vinogradov C-spe
ctral sequence.\n
LOCATION:https://researchseminars.org/talk/GDEq/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Kushner
DTSTART:20230426T162000Z
DTEND:20230426T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/87
DESCRIPTION:Title: Fi
nite-dimensional dynamics of systems of evolutionary differential equation
s with many spatial variables\nby Alexei Kushner as part of Geometry o
f differential equations seminar\n\n\nAbstract\nThe main ideas of the theo
ry of finite-dimensional dynamics were formulated in the 2000s in the work
s of B.S. Kruglikov\, V.V. Lychagin and O.V. Lychagina. These papers also
found finite-dimensional dynamics of the Kolmogorov-Petrovsky-Piskunov and
Korteweg-de Vries equations. This theory is a natural development of the
theory of dynamical systems. Finite-dimensional dynamics make it possible
to find families of solutions depending on a finite number of parameters a
mong all solutions of evolutionary differential equations. Namely\, finite
-dimensional submanifolds are constructed in the space of smooth functions
that are invariant under the flow given by the evolution equation. This r
emoves the question of the existence of solutions\, since such submanifold
s consist of solutions to ordinary differential equations\, and\, moreover
\, gives a constructive method for finding them. Note that finite-dimensio
nal dynamics can exist for equations that do not have symmetries. The talk
will present the results obtained by us for systems of evolutionary equat
ions\, including those with many spatial variables.\n
LOCATION:https://researchseminars.org/talk/GDEq/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Fels
DTSTART:20230510T162000Z
DTEND:20230510T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/88
DESCRIPTION:Title: Va
riational/Symplectic and Hamiltonian Operators\nby Mark Fels as part o
f Geometry of differential equations seminar\n\n\nAbstract\nGiven a differ
ential equation (or system) $\\Delta$ = 0 the inverse problem in the calcu
lus of variations asks if there is a multiplier function $Q$ such that\n\\
[Q\\Delta=E(L)\,\\]\nwhere $E(L)=0$ is the Euler-Lagrange equation for a L
agrangian $L$. A solution to this problem can be found in principle and ex
pressed in terms of invariants of the equation $\\Delta$. The variational
operator problem asks the same question but $Q$ can now be a differential
operator as the following simple example demonstrates for the evolution eq
uation $u_t - u_{xxx} = 0$\,\n\\[D_x(u_t - u_{xxx}) = u_{tx}-u_{xxxx}=E(-\
\frac12(u_tu_x+u_{xx}^2)).\\]\nHere $D_x$ is a variational operator for $u
_t - u_{xxx} = 0$.\n\nThis talk will discuss how the variational operator
problem can be solved in the case of scalar evolution equations and how va
riational operators are related to symplectic and Hamiltonian operators.\n
LOCATION:https://researchseminars.org/talk/GDEq/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Svetlana Mukhina
DTSTART:20230607T162000Z
DTEND:20230607T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/89
DESCRIPTION:Title: Co
ntact vs symplectic geometry\nby Svetlana Mukhina as part of Geometry
of differential equations seminar\n\nLecture held in room 303 of the Indep
endent University of Moscow.\n\nAbstract\nThe report will show how some sy
mplectic Monge-Ampère type equations can be solved by applying contact tr
ansformations to them.\n\nAs is known\, symplectic Monge-Ampère equations
with two independent variables are locally symplectic equivalent to linea
r equations with constant coefficients if and only if the corresponding Ni
jenhuis bracket is zero (the Lychagin-Rubtsov theorem). Necessary and suff
icient conditions for the contact equivalence of the general (not necessar
ily symplectic) Monge-Ampère linear equations were found by Kushner.\n\nU
sing these results\, we consider the problem of constructing exact solutio
ns to some equations arising in filtration theory. We will consider a mode
l of unsteady displacement of oil by a solution of active reagents. This m
odel describes the process of oil extraction from hard-to-recover deposits
. This model is described by a hyperbolic system of partial differential e
quations of the first order of the Jacobi type. Unknown functions are the
water saturation and concentration of reagents in an aqueous solution\, an
d independent variables are time and linear coordinate.\n\nWith the help o
f symplectic and contact transformations\, it is possible to reduce the mo
del equations to a linear wave equation. The exact solution of this system
is obtained and the Cauchy problem is solved.\n
LOCATION:https://researchseminars.org/talk/GDEq/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Doubrov
DTSTART:20230517T162000Z
DTEND:20230517T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/90
DESCRIPTION:Title: Ov
erdetermined systems of PDEs related to representations of semi-simple Lie
algebras\nby Boris Doubrov as part of Geometry of differential equati
ons seminar\n\n\nAbstract\nWe explore a class of finite-type systems of PD
Es whose symbol is determined by an (arbitrary) irreducible representation
of a graded semisimple Lie algebra.\n\nWe show that trivial equations wit
h such symbol correspond to rational homogeneous varieties\, non-trivial l
inear equations define symbol-preserving deformations of such varieties. I
n particular\, we determine when such deformations exist. In terms of the
corresponding PDE system this corresponds to the question when compatibili
ty conditions imply that the system is equivalent to trivial. The answer t
o this question is given in terms of certain Lie algebra cohomology\, whic
h can be effectively computed using the results for the theory of semisimp
le Lie algebras.\n\nWe solve local equivalence problem for such systems un
der fiber+symbol preserving transformations and show how this is related t
o the projective geometry of submanifolds. Finally\, we discuss the case o
f non-linear systems with the same symbol and show that under certain addi
tional conditions their solution spaces admit remarkable geometric structu
res.\n
LOCATION:https://researchseminars.org/talk/GDEq/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maksim Gadzhiev and Alexander Kuleshov
DTSTART:20230531T162000Z
DTEND:20230531T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/91
DESCRIPTION:Title: In
tegrability of the problem of motion of a body with a fixed point in a flo
w of particles\nby Maksim Gadzhiev and Alexander Kuleshov as part of G
eometry of differential equations seminar\n\nLecture held in room 303 of t
he Independent University of Moscow.\n\nAbstract\nThe problem of the motio
n\, in the free molecular flow of particles\, of a rigid body with a fixed
point is considered. The molecular flow is assumed to be sufficiently spa
rse\, there is no interaction between the particles. Based on the approach
proposed by V.V. Beletsky\, an expression is obtained for the moment of f
orces acting on a body with a fixed point. It is shown that the equations
of motion of a body are similar to the classical Euler-Poisson equations o
f motion of a heavy rigid body with a fixed point and are presented in the
form of classical Euler-Poisson equations in the case when the surface of
a body is a sphere. The existence of the first integrals is discussed. Co
nstraints on the system parameters are obtained under which there are inte
grable cases corresponding to the classical Euler-Poinsot\, Lagrange and H
ess cases of integrability of the equations of motion of a heavy rigid bod
y with a fixed point. The case when the surface of the body is an ellipsoi
d is considered. Using the methods developed in the works of V.V. Kozlov\,
proved the absence of an integrable case in this problem\, similar to the
Kovalevskaya case.\n
LOCATION:https://researchseminars.org/talk/GDEq/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20230913T162000Z
DTEND:20230913T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/92
DESCRIPTION:Title: On
equivalence of planar webs\nby Valentin Lychagin as part of Geometry
of differential equations seminar\n\nLecture held in room 303 of the Indep
endent University of Moscow.\n\nAbstract\nIn this talk\, I'll discuss the
equivalence problem for planar d-webs.\n\nTo this end\, the fields of rati
onal differential invariants will be found\, and natural geometric objects
related to planar webs will be discussed.\n\nThe cases of d-webs with d<6
will be discussed in detail.\n\nPlease download the formula file pl.pdf and keep it hand
y during the talk so that the speaker can refer to it.\n
LOCATION:https://researchseminars.org/talk/GDEq/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20230920T162000Z
DTEND:20230920T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/93
DESCRIPTION:Title: In
ternal Lagrangians as variational principles\nby Konstantin Druzhkov a
s part of Geometry of differential equations seminar\n\n\nAbstract\nThe pr
inciple of stationary action deals with Lagrangians defined on jets. Howev
er\, for some reason\, the intrinsic geometry of the corresponding equatio
ns knows about their variational nature. It turns out that the explanation
is quite simple: each stationary-action principle reproduces itself in te
rms of the intrinsic geometry. More precisely\, each admissible Lagrangian
gives rise to a unique integral functional defined on some particular cla
ss of submanifolds of the corresponding equations. Such submanifolds can b
e treated as almost solutions since (informally speaking) they are compose
d of initial-boundary conditions lifted to infinitely prolonged equations.
Intrinsic integral functionals produced by variational principles are rel
ated to so-called internal Lagrangians. This relation allows us to introdu
ce the notion of stationary point of an internal Lagrangian\, formulate th
e corresponding intrinsic version of Noether's theorem\, and discuss the n
ondegeneracy of presymplectic structures of differential equations. Despit
e the generality of the approach\, its application to gauge theories prove
s to be challenging. Perhaps the construction needs some modification in t
his case.\n
LOCATION:https://researchseminars.org/talk/GDEq/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andronick Arutyunov
DTSTART:20230927T162000Z
DTEND:20230927T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/94
DESCRIPTION:Title: De
rivations in group algebra bimodules\nby Andronick Arutyunov as part
of Geometry of differential equations seminar\n\nLecture held in room 303
of the Independent University of Moscow.\n\nAbstract\nIf one introduces a
norm in a group algebra which is understood as a vector space and consider
s a closure over this norm\, a natural structure of a free bimodule over a
group ring arises. The most natural example is $\\ell_p(G)$\, for $p \\ge
q 1$. This structure makes it natural to consider the problem of describin
g derivations with values in such bimodules\, which I will talk about. A "
character" approach will be used\, which consists in identifying the deriv
ations with characters on a suitable category (in our case\, the groupoid
of adjoint action of a group on itself)\, and further study is already und
erway with the active use of combinatorial methods.\n
LOCATION:https://researchseminars.org/talk/GDEq/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Doubrov
DTSTART:20231011T162000Z
DTEND:20231011T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/95
DESCRIPTION:Title: Ex
trinsic geometry and linear differential equations of SL(3)-type\nby B
oris Doubrov as part of Geometry of differential equations seminar\n\n\nAb
stract\nAs an application of the general theory on extrinsic geometry\, we
investigate extrinsic geometry of submanifolds in flag varieties and syst
ems of linear PDEs for a class of special interest associated with the adj
oint representation of SL(3). It may be seen as a contact generalization o
f the classical description of surfaces in P^3 in terms of two linear PDEs
of second order.\n\nWe carry out a complete local classification of the h
omogeneous structures in this class. As a result\, we find 7 kinds of new
systems of linear PDE's of second order on a 3-dimensional contact manifol
d each of which has a solution space of dimension 8. Among them there are
included a system of PDE's called contact Cayley's surface and one which h
as SL(2) symmetry.\n\nJoint work with Tohru Morimoto.\n
LOCATION:https://researchseminars.org/talk/GDEq/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuri Sachkov
DTSTART:20231025T162000Z
DTEND:20231025T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/96
DESCRIPTION:Title: Lo
rentzian geometry in the Lobachevsky plane\nby Yuri Sachkov as part of
Geometry of differential equations seminar\n\n\nAbstract\nWe consider lef
t-invariant Lorentzian problems on the group of proper affine functions on
the line. These problems have constant sectional curvature\, thus are loc
ally isometric to standard constant curvature Lorentzian manifolds (Minkow
ski space\, de Sitter space\, and anti-de Sitter space).\n\nFor these prob
lems\, the attainability set is described\, existence of optimal trajector
ies is studied\, a parameterization of Lorentzian length maximizers is obt
ained\, and Lorentzian distance and spheres are described.\n\nFor zero cur
vature problem a global isometry into a half-plane of Minkowski plane is c
onstructed.\n
LOCATION:https://researchseminars.org/talk/GDEq/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Pavlov
DTSTART:20231018T162000Z
DTEND:20231018T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/97
DESCRIPTION:Title: Du
brovin paradigm and beyond\nby Maxim Pavlov as part of Geometry of dif
ferential equations seminar\n\nLecture held in room 303 of the Independent
University of Moscow.\n\nAbstract\nThe paradigm proposed by Boris Dubrovi
n\, consisted of two parts: description of Frobenius manifolds + "recovery
" of an infinite set of dispersion corrections with the requirement of pre
servation of integrability in the sense of existence of the Lax representa
tion.\n\nThe talk will propose infinitely many alternatives to the Frobeni
us manifolds.\n
LOCATION:https://researchseminars.org/talk/GDEq/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Kruglikov (UiT the Arctic University of Norway)
DTSTART:20231108T162000Z
DTEND:20231108T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/98
DESCRIPTION:Title: Di
spersionless integrable systems in dimension 5\nby Boris Kruglikov (Ui
T the Arctic University of Norway) as part of Geometry of differential equ
ations seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GDEq/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Morozov
DTSTART:20231101T162000Z
DTEND:20231101T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/99
DESCRIPTION:Title: Ex
tensions of Lie algebras and integrability of some equations of fluid dyna
mics and magnetohydrodynamics.\nby Oleg Morozov as part of Geometry of
differential equations seminar\n\nLecture held in room 303 of the Indepen
dent University of Moscow.\n\nAbstract\nWe find the twisted extension of t
he symmetry algebra of the 2D Euler equation in the vorticity form and use
it to construct new Lax representation for this equation. Then we conside
r the transformation Lie-Rinehart algebras generated by finite-dimensional
subalgebras of the symmetry algebra and employ them to derive a family of
Lax representations for the Euler equation. The family depends on functio
nal parameters and contains a non-removable spectral parameter. Furthermor
e we exhibit Lax representations for the reduced magnetohydrodynamics equa
tions (or the Kadomtsev-Pogutse equations)\, the ideal magnetohydrodynamic
s equations\, the quasigeostrophic two-layer model equations\, and the Cha
rney-Obukhov equation for the ocean.\n
LOCATION:https://researchseminars.org/talk/GDEq/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Agafonov
DTSTART:20231206T162000Z
DTEND:20231206T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/100
DESCRIPTION:Title: H
exagonal circular 3-webs with polar curves of degree three\nby Sergey
Agafonov as part of Geometry of differential equations seminar\n\nLecture
held in room 303 of the Independent University of Moscow.\n\nAbstract\nLie
sphere geometry describes circles on the unit sphere by polar points of t
hese circles. Therefore a one parameter family of circles corresponds to a
curve and a 3-web of circles\, i.e.\, 3 foliations by circles\, is fixed
by 3 curves. We call the union of these curves the polar curve and show ho
w analysis of the singular set of hexagonal 3-webs helps to classify circu
lar hexagonal 3-webs with polar curves of degree 3. Many of the found webs
are new. The presented results mark the progress in the Blaschke-Bol prob
lem posed almost one hundred years ago. More detail in arXiv:2306.11707.\n
LOCATION:https://researchseminars.org/talk/GDEq/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irina Bobrova
DTSTART:20231122T162000Z
DTEND:20231122T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/101
DESCRIPTION:Title: O
n a classification of non-Abelian Painlevé equations\nby Irina Bobrov
a as part of Geometry of differential equations seminar\n\n\nAbstract\nThe
famous Painlevé equations define the most general special functions and
appear ubiquitously in integrable models. Since the latter have been inten
sively studied in the matrix or\, more general\, non-Abelian case\, exampl
es of non-Abelian Painlevé equations arise.\n\nWe will discuss the proble
m of classifying such equations. This talk is based on a series of papers
joint with Vladimir Sokolov and an ongoing project with Vladimir Retakh\,
Vladimir Rubtsov\, and Georgy Sharygin.\n
LOCATION:https://researchseminars.org/talk/GDEq/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev
DTSTART:20231129T162000Z
DTEND:20231129T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/102
DESCRIPTION:Title: P
resymplectic minimal models of local gauge theories\nby Maxim Grigorie
v as part of Geometry of differential equations seminar\n\nLecture held in
room 303 of the Independent University of Moscow.\n\nAbstract\nWe describ
e how the BV-AKSZ construction (or\, more generally\, finite dimensional s
ymplectic gauge PDE) can be extended to generic local gauge field theories
including non-topological and non-diffeomorphism-invariant ones. The mini
mal formulation of this sort has a finite-dimensional target space which i
s a pre Q-manifold equipped with a compatible presymplectic structure. The
nilpotency condition for the homological vector field is replaced with a
presymplectic version of the classical BV master equation. Given such a pr
esymplectic BV-AKSZ formulation\, it defines a standard jet-bundle BV form
ulation by taking a symplectic quotient of the respective super jet-bundle
. In other words all the information about the underlying PDE\, its Lagran
gian\, and the corresponding BV formulation turns out to be encoded in the
finite dimensional graded geometrical object. Standard examples include Y
ang-Mills\, Einstein gravity\, conformal gravity etc.\n
LOCATION:https://researchseminars.org/talk/GDEq/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yagub Aliyev
DTSTART:20231213T162000Z
DTEND:20231213T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/103
DESCRIPTION:Title: A
pollonius problem and caustics of an ellipsoid\nby Yagub Aliyev as par
t of Geometry of differential equations seminar\n\n\nAbstract\nIn the talk
we discuss Apollonius Problem on the number of normals of an ellipse pass
ing through a given point. It is known that the number is dependent on the
position of the given point with respect to a certain astroida. The inter
section points of the astroida and the ellipse are used to study the case
when the given point is on the ellipse. The problem is then generalized fo
r 3-dimensional space\, namely for Ellipsoids. The number of concurrent no
rmals in this case is known to be dependent on the position of the given p
oint with respect to caustics of the ellipsoid. If the given point is on t
he ellipsoid then the number of normals is dependent on position of the po
int with respect to the intersections of the ellipsoid with its caustics.
The main motivation of this talk is to find parametrizations and classify
all possible cases of these intersections.\n
LOCATION:https://researchseminars.org/talk/GDEq/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20240214T162000Z
DTEND:20240214T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/104
DESCRIPTION:Title: O
n flows and filtration in the presence of thermodynamic processes: general
ized Navier-Stokes equations\nby Valentin Lychagin as part of Geometry
of differential equations seminar\n\nLecture held in room 303 of the Inde
pendent University of Moscow.\n\nAbstract\nWe plan to present a generaliza
tion of the Navier-Stokes equations that describes the flows of homogeneou
s multicomponent media in the presence of various thermodynamic processes\
, especially chemical reactions. To achieve this\, we discuss the classica
l thermodynamics of homogeneous multicomponent media and related thermodyn
amic processes (especially chemical reactions) from the contact geometry p
erspective.\n\nIt makes it possible to work with thermodynamic processes a
s contact vector fields on a contact manifold and easily include in the st
andard scheme of continuous mechanics. At the end\, we outline methods of
solving resulting equations and discuss possible singularities arising in
solutions.\n\nPlease download the formula file fl.pdf and keep it handy during the talk so that the spea
ker can refer to it.\n
LOCATION:https://researchseminars.org/talk/GDEq/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20240221T162000Z
DTEND:20240221T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/105
DESCRIPTION:Title: B
i-Hamiltonian systems and projective geometry\nby Raffaele Vitolo (Uni
versità del Salento) as part of Geometry of differential equations semina
r\n\n\nAbstract\nWe introduce the problem of classification of bi-Hamilton
ian structures of KdV type under projective reciprocal transformations. Th
is problem leads naturally to studying the compatibility of a first order
localizable homogeneous Hamiltonian operator with a higher order homogeneo
us Hamiltonian operator. We study the simplest second-order and third-orde
r case where the orbit contains a constant operator. Computations with wea
kly non local Hamiltonian operators have been made by techniques developed
in a previous paper.\n\nJoint work with P. Lorenzoni.\n
LOCATION:https://researchseminars.org/talk/GDEq/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20240306T162000Z
DTEND:20240306T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/106
DESCRIPTION:Title: D
eformation quantisation of the argument shift on $U\\mathfrak{gl}(n)$\
nby Georgy Sharygin as part of Geometry of differential equations seminar\
n\nLecture held in room 303 of the Independent University of Moscow.\n\nAb
stract\nArgument shift algebras are the commutative subalgebras in the sym
metric algebras of a Lie algebra\, generated by the iterated derivations (
in direction of a constant vector field) of Casimir elements in $S\\mathfr
ak{gl}(n)$. In particular all these quasiderivations do mutually commute.
In my talk I will show that a similar statement holds for the algebra $U\\
mathfrak{gl}(n)$ and its quasiderivations: namely\, I will show that itera
ted quasiderivations of the central elements of $U\\mathfrak{gl}(n)$ with
respect to a constant quasiderivation do mutually commute. Our proof is ba
sed on the existence and properties of "Quantum Mischenko-Fomenko" algebra
s\, and (which is worse) cannot be extended to other Lie algebras\, but we
believe that the fact that the "shift operator" can be raised to $U\\math
frak{gl}(n)$ is an interesting fact.\n
LOCATION:https://researchseminars.org/talk/GDEq/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20240320T162000Z
DTEND:20240320T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/107
DESCRIPTION:Title: B
esselland: autour and beyond. Part 1\nby Vladimir Rubtsov (Université
d'Angers) as part of Geometry of differential equations seminar\n\n\nAbst
ract\nI shall try to explain – why it is interesting to study and to gen
eralize analytic solutions of modified Bessel equation. My talk is based o
n ongoing projects in progress with V. Buchstaber\, I. Gaiur and D. Van St
raten.\n
LOCATION:https://researchseminars.org/talk/GDEq/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20240327T162000Z
DTEND:20240327T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/108
DESCRIPTION:Title: B
esselland: autour and beyond. Part 2\nby Vladimir Rubtsov (Université
d'Angers) as part of Geometry of differential equations seminar\n\n\nAbst
ract\nI shall try to explain – why it is interesting to study and to gen
eralize analytic solutions of modified Bessel equation. My talk is based o
n ongoing projects in progress with V. Buchstaber\, I. Gaiur and D. Van St
raten.\n
LOCATION:https://researchseminars.org/talk/GDEq/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gerard Helminck
DTSTART:20240417T162000Z
DTEND:20240417T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/109
DESCRIPTION:Title: A
construction of solutions of an integrable deformation of a commutative L
ie algebra of skew Hermitian $\\mathbb{Z}\\times\\mathbb{Z}$-matrices\
nby Gerard Helminck as part of Geometry of differential equations seminar\
n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GDEq/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20240403T162000Z
DTEND:20240403T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/110
DESCRIPTION:Title: B
esselland: autour and beyond. Part 3\nby Vladimir Rubtsov (Université
d'Angers) as part of Geometry of differential equations seminar\n\n\nAbst
ract\nContinuation of the talks held on 20 and 27 March. \n\nI shall try t
o explain – why it is interesting to study and to generalize analytic so
lutions of modified Bessel equation. My talk is based on ongoing projects
in progress with V. Buchstaber\, I. Gaiur and D. Van Straten.\n
LOCATION:https://researchseminars.org/talk/GDEq/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20240501T162000Z
DTEND:20240501T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/111
DESCRIPTION:Title: I
nternal Lagrangians and gauge systems\nby Konstantin Druzhkov as part
of Geometry of differential equations seminar\n\n\nAbstract\nIn classical
mechanics\, the Hamiltonian formalism is given in terms of instantaneous p
hase spaces of mechanical systems. This explains why it can be interpreted
as an encapsulation of the Lagrangian formalism into the intrinsic geomet
ry of equations of motion. This observation can be generalized to the case
of arbitrary variational equations. To do this\, we describe instantaneou
s phase spaces using the intrinsic geometry of PDEs. The description is gi
ven by the lifts of involutive codim-1 distributions from the base of a di
fferential equation viewed as a bundle with a flat connection (Cartan dist
ribution). Such lifts can be considered differential equations\, which one
can regard as gauge systems. They encode instantaneous phase spaces. In a
ddition\, each Lagrangian of a variational system generates a unique eleme
nt of a certain cohomology of the system. We call such elements internal L
agrangians. Internal Lagrangians can be varied within classes of paths in
the instantaneous phase spaces. This fact yields a direct (non-covariant)
reformulation of the Hamiltonian formalism in terms of the intrinsic geome
try of PDEs. Finally\, the non-covariant internal variational principle gi
ves rise to its covariant child.\n
LOCATION:https://researchseminars.org/talk/GDEq/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eivind Schneider
DTSTART:20240515T162000Z
DTEND:20240515T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/112
DESCRIPTION:Title: I
nvariant divisors and equivariant line bundles\nby Eivind Schneider as
part of Geometry of differential equations seminar\n\n\nAbstract\nScalar
relative invariants play an important role in the theory of group actions
on a manifold as their zero sets are invariant hypersurfaces. Relative inv
ariants are central in many applications\, where they often are treated lo
cally\, since an invariant hypersurface is not necessarily the locus of a
single function. Our aim is to outline a global theory of relative invaria
nts in the complex analytic setting. For a Lie algebra $\\mathfrak{g}$ of
holomorphic vector fields on a complex manifold $M$\, any holomorphic $\\m
athfrak{g}$-invariant hypersurface is given in terms of a $\\mathfrak{g}$-
invariant divisor. This generalizes the classical notion of scalar relativ
e $\\mathfrak{g}$-invariant. Since any $\\mathfrak{g}$-invariant divisor g
ives rise to a $\\mathfrak{g}$-equivariant line bundle\, we investigate th
e group $\\mathrm{Pic}_{\\mathfrak{g}}(M)$ of $\\mathfrak{g}$-equivariant
line bundles. A cohomological description of $\\mathrm{Pic}_{\\mathfrak{g}
}(M)$ is given in terms of a double complex interpolating the Chevalley-Ei
lenberg complex for $\\mathfrak{g}$ with the Čech complex of the sheaf of
holomorphic functions on $M$. In the end we will discuss applications of
the theory to jet spaces and differential invariants.\n\nThe talk is based
on joint work with Boris Kruglikov (arXiv:2404.19439).\n
LOCATION:https://researchseminars.org/talk/GDEq/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev
DTSTART:20240522T162000Z
DTEND:20240522T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/113
DESCRIPTION:Title: G
auge PDEs on manifolds with boundaries and asymptotic symmetries\nby M
axim Grigoriev as part of Geometry of differential equations seminar\n\n\n
Abstract\nWe propose a framework to study local gauge theories on manifold
s with boundaries and their asymptotic symmetries\, which is based on repr
esenting them as so-called gauge PDEs. These objects extend the convention
al BV-AKSZ sigma-models to the case of not necessarily topological and dif
feomorphism invariant systems and are known to behave well when restricted
to submanifolds and boundaries. We introduce the notion of gauge PDE with
boundaries\, which takes into account generic boundary conditions\, and a
pply the framework to asymptotically flat gravity. In so doing\, we start
with a suitable representation of gravity as a gauge PDE with boundaries\,
which implements the Penrose description of asymptotically simple spaceti
mes. We then derive the minimal model of the gauge PDE induced on the boun
dary and observe that it provides the Cartan (frame-like) description of a
(curved) conformal Carollian structure on the boundary. Furthermore\, imp
osing a version of the familiar boundary conditions in the induced boundar
y gauge PDE\, leads immediately to the conventional BMS algebra of asympto
tic symmetries.\n\nMore references: arXiv:2212.11350\;\narX
iv:1207.3439\, arXiv:1305.01
62\, arXiv:1903.02820\,
arXiv:1009.0190\n
LOCATION:https://researchseminars.org/talk/GDEq/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20240911T162000Z
DTEND:20240911T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/114
DESCRIPTION:Title: O
n a structure of the first order differential operators\nby Valentin L
ychagin as part of Geometry of differential equations seminar\n\nLecture h
eld in room 303 of the Independent University of Moscow.\n\nAbstract\nThe
various geometrical structures associated with differential operators of t
he first order shall be discussed as well as notions of singular and regul
ar points. At the end normal forms of operators at regular points will be
presented.\n
LOCATION:https://researchseminars.org/talk/GDEq/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pavel Bedrikovetsky (University of Adelaide)
DTSTART:20240918T162000Z
DTEND:20240918T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/115
DESCRIPTION:Title: E
xact solutions and upscaling in conservation law systems\nby Pavel Bed
rikovetsky (University of Adelaide) as part of Geometry of differential eq
uations seminar\n\n\nAbstract\nNumerous transport processes in nature and
industry are described by $n\\times n$ conservation law systems $u_t+f(u)_
x=0$\, $u=(u^1\,\\dots\,u^n)$. This corresponds to upper scale\, like rock
or core scale in porous media\, column length in chemical engineering\, o
r multi-block scale in city transport. The micro heterogeneity at lower sc
ales introduces $x$- or $t$-dependencies into the large-scale conservation
law system\, like $f=f(u\,x)$ or $f(u\,t)$. Often\, numerical micro-scale
modelling highly exceeds the available computational facilities in terms
of calculation time or memory. The problem is a proper upscaling: how to "
average" the micro-scale $x$-dependent $f(u\,x)$ to calculate the upper-sc
ale flux $f(u)$?\n\nWe present general case for $n=1$ and several systems
for $n=2$ and $3$. The key is that the Riemann invariant at the microscale
is the "flux" rather than "density". It allows for exact solutions of sev
eral 1D problems: "smoothing" of shocks and "sharpening" of rarefaction wa
ves into shocks due to microscale $x$- and $t$-dependencies\, flows in pie
cewise homogeneous media. It also allows formulating an upscaling algorith
m based on the analytical solutions and its invariant properties.\n
LOCATION:https://researchseminars.org/talk/GDEq/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Morozov
DTSTART:20241009T162000Z
DTEND:20241009T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/116
DESCRIPTION:Title: L
ax representations for Euler equations of ideal hydrodynamics\nby Oleg
Morozov as part of Geometry of differential equations seminar\n\n\nAbstra
ct\nI will discuss Lax representations with non-removable parameters for t
he Euler equations of ideal hydrodynamics on a 2D Riemannian manifold and
for the 3D Euler-Helmholtz equations.\n
LOCATION:https://researchseminars.org/talk/GDEq/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20241106T162000Z
DTEND:20241106T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/117
DESCRIPTION:Title: R
emarkable properties of the full symmetric Toda system\nby Georgy Shar
ygin as part of Geometry of differential equations seminar\n\nLecture held
in room 303 of the Independent University of Moscow.\n\nAbstract\nA full
symmetric Toda system is a Hamiltonian dynamical system on the space of sy
mmetric real matrices with zero trace\, generalizing the usual open Toda c
hain. This system is given by the Lax equation $\\dot L=[L\,M(L)]$\, where
$M(L)$ is the (naive) antisymmetrization of the symmetric matrix $L$: the
difference of its super and subdiagonal parts (with zeros on the diagonal
). The Hamiltonianity of this system comes from the identification of the
space of symmetric matrices with the space dual to the algebra of upper tr
iangular matrices\, with the Hamilton function being $1/2Tr(L^2)$. This sy
stem can be further generalized to obtain systems on the spaces of "genera
lized symmetric matrices"\, the symmetric components of the Cartan expansi
on of the semi-simple real Lie algebras. In a somewhat unexpected way\, al
l these systems turn out to be integrable (in the sense of having a suffic
iently large commutative algebra of first integrals) and possess a number
of remarkable properties which I will discuss: their trajectories always c
onnect fixed points corresponding to the elements of the Weyl group of the
original Lie algebra\, and two such points are connected if and only if t
he elements of the Weyl group are comparable in Bruhat order\; in the case
of a system on spaces of generalized symmetric matrices\, this property a
llows one to describe the intersections of the real Bruhat cells\; this sy
stem has a large set of symmetries (sufficient for it to be Lie-Bianchi in
tegrable)\; its additional first integrals can be obtained by a "cut" proc
edure\, and the trajectories of the corresponding Hamiltonian fields can b
e obtained by the QR decomposition\; if time permits\, I will describe alt
ernative families of first integrals (commutative and non-commutative)\; f
inally\, I will describe a way to lift the extra first integrals of the "c
ut" into the universal enveloping algebra with commutativity preserved.\n\
nThe talk is based on a series of works by the author jointly with Yu. Che
rnyakov\, A. Sorin and D. Talalaev.\n
LOCATION:https://researchseminars.org/talk/GDEq/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Zheglov
DTSTART:20241113T162000Z
DTEND:20241113T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/118
DESCRIPTION:Title: N
ormal forms for differential operators\nby Alexander Zheglov as part o
f Geometry of differential equations seminar\n\nLecture held in room 303 o
f the Independent University of Moscow.\n\nAbstract\nIn my talk I'll give
an overview of the results obtained by me\, as well as jointly with co-aut
hors\, related to the problem of classifying commuting (scalar) differen
tial\, or more generally\, differential-difference or integral-different
ial operators in several variables.\n\nConsidering such rings as subrings
of a certain complete non-commutative ring $\\hat{D}_n^{sym}$ (not the kno
wn ring of formal pseudo-differential operators!)\, the normal forms of
differential operators mentioned in the title are obtained after conjugati
on by some invertible operator ("Schur operator")\, calculated with the he
lp of one of the operators in a ring. Normal forms of commuting op
erators are polynomials with constant coefficients in the differentiatio
n\, integration and shift operators\, which have a finite order in each va
riable\, and can be effectively calculated for any given commuting operato
rs.\n\nI'll talk about some recent applications of the theory of normal fo
rms: an effective parametrisation of torsion free sheaves with vanishing
cohomologies on a projective curve\, and a correspondence between solutio
ns to the string equation and pairs of commuting ordinary differential op
erators of rank one.\n
LOCATION:https://researchseminars.org/talk/GDEq/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yasushi Ikeda
DTSTART:20241120T130000Z
DTEND:20241120T144000Z
DTSTAMP:20260422T225928Z
UID:GDEq/119
DESCRIPTION:Title: Q
uantum argument shifts in general linear Lie algebras\nby Yasushi Iked
a as part of Geometry of differential equations seminar\n\n\nAbstract\nArg
ument shift algebras in $S(g)$ (where $g$ is a Lie algebra) are Poisson co
mmutative subalgebras (with respect to the Lie-Poisson bracket)\, generate
d by iterated argument shifts of Poisson central elements. Inspired by the
quantum partial derivatives on $U(gl_d)$ proposed by Gurevich\, Pyatov\,
and Saponov\, I and Georgy Sharygin showed that the quantum argument shift
algebras are generated by iterated quantum argument shifts of central ele
ments in $U(gl_d)$. In this talk\, I will introduce a formula for calculat
ing iterated quantum argument shifts and generators of the quantum argumen
t shift algebras up to the second order\, recalling the main theorem.\n\nN
ote the non-standard start time!\n
LOCATION:https://researchseminars.org/talk/GDEq/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ekaterina Shemyakova
DTSTART:20241127T162000Z
DTEND:20241127T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/120
DESCRIPTION:Title: O
n differential operators generating higher brackets\nby Ekaterina Shem
yakova as part of Geometry of differential equations seminar\n\n\nAbstract
\nOn supermanifolds\, a Poisson structure can be either even\, correspondi
ng to a Poisson bivector\, or odd\, corresponding to an odd Hamiltonian qu
adratic in momenta. An odd Poisson bracket can also be defined by an odd s
econd-order differential operator that squares to zero\, known as a "BV-ty
pe" operator.\n\nA higher analog\, $P_\\infty$ or $S_\\infty$\, is a serie
s of brackets of alternating parities or all odd\, respectively\, that sat
isfy relations that are higher homotopy analogs of the Jacobi identity. Th
ese brackets are generated by arbitrary multivector fields or Hamiltonians
. However\, generating an $S_\\infty$-structure by a higher-order differen
tial operator is not straightforward\, as this would violate the Leibniz i
dentities. Kravchenko and others studied these structures\, and Voronov ad
dressed the Leibniz identity issue by introducing formal $\\hbar$-differen
tial operators.\n\nIn this talk\, we revisit the construction of an $\\hba
r$-differential operator that generates higher Koszul brackets on differen
tial forms on a $P_\\infty$-manifold.\n\nIt is well known that a chain map
between the de Rham and Poisson complexes on a Poisson manifold at the sa
me time maps the Koszul bracket of differential forms to the Schouten brac
ket of multivector fields. In the $P_\\infty$-case\, however\, the chain m
ap is also known\, but it does not connect the corresponding bracket struc
tures. An $L_\\infty$-morphism from the higher Koszul brackets to the Scho
uten bracket has been constructed recently\, using Voronov's thick morphis
m technique. In this talk\, we will show how to lift this morphism to the
level of operators.\n\nThe talk is partly based on joint work with Yagmur
Yilmaz.\n
LOCATION:https://researchseminars.org/talk/GDEq/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20241218T162000Z
DTEND:20241218T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/122
DESCRIPTION:Title: T
owards a theory of homotopy structures for differential equations: First d
efinitions and examples. Part 1\nby Vladimir Rubtsov (Université d'An
gers) as part of Geometry of differential equations seminar\n\n\nAbstract\
nWe define $A_\\infty$-algebra structures on horizontal and vertical coho
mologies of (formally integrable) partial differential equations.\n\nSince
higher order $A_\\infty$-algebra operations are related to Massey product
s\, our observation implies the existence of invariants for differential
equations that go beyond conservation laws.\n\nWe also propose notions of
formality for PDEs\, and we present examples of formal equations.\n
LOCATION:https://researchseminars.org/talk/GDEq/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20241211T162000Z
DTEND:20241211T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/123
DESCRIPTION:Title: I
nvariant reduction for PDEs. I: Conservation laws of 1+1 systems of evolut
ion equations\nby Konstantin Druzhkov as part of Geometry of different
ial equations seminar\n\n\nAbstract\nAmong various methods for constructin
g exact solutions of partial differential equations\, the symmetry-invaria
nt approach is particularly noteworthy. This method is especially effectiv
e in the case of point symmetries\, but when it comes to higher symmetries
\, additional steps are required to obtain invariant solutions. It turns o
ut that systems that describe symmetry-invariant solutions inherit symmetr
y-invariant geometric structures even in the case of higher symmetries. Mo
reover\, the reduction of invariant conservation laws of 1+1 systems of ev
olution equations can be described as an algorithm and implemented in Mapl
e. Starting from invariant conservation laws\, we get constants of invaria
nt motion. They are analogs of first integrals of ODEs\, and one can use t
hem in the same way. In particular\, a sufficient number of independent co
nstants of invariant motion allows one to integrate the corresponding syst
em for invariant solutions.\n
LOCATION:https://researchseminars.org/talk/GDEq/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20250108T162000Z
DTEND:20250108T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/124
DESCRIPTION:Title: T
owards a theory of homotopy structures for differential equations: First d
efinitions and examples. Part 2\nby Vladimir Rubtsov (Université d'An
gers) as part of Geometry of differential equations seminar\n\n\nAbstract\
nWe define $A_\\infty$-algebra structures on horizontal and vertical coho
mologies of (formally integrable) partial differential equations.\n\nSince
higher order $A_\\infty$-algebra operations are related to Massey product
s\, our observation implies the existence of invariants for differential
equations that go beyond conservation laws.\n\nWe also propose notions of
formality for PDEs\, and we present examples of formal equations.\n
LOCATION:https://researchseminars.org/talk/GDEq/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Kryczka
DTSTART:20250219T162000Z
DTEND:20250219T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/125
DESCRIPTION:Title: S
ingularities and Bi-complexes for PDEs\nby Jacob Kryczka as part of Ge
ometry of differential equations seminar\n\n\nAbstract\nMany moduli spaces
in geometry and physics\, like those appearing in symplectic topology\, q
uantum gauge field theory (e.g. in homological mirror symmetry and Donalds
on-Thomas theory) are constructed as parametrizing spaces of solutions to
non-linear partial differential equations modulo symmetries of the underly
ing theory. These spaces are often non-smooth and possess multi non-equidi
mensional components. Moreover\, when they may be written as intersections
of higher dimensional components they typically exhibit singularities due
to non-transverse intersections. To account for symmetries and provide a
suitable geometric model for non-transverse intersection loci\, one should
enhance our mathematical tools to include higher and derived stacks. Seco
ndary Calculus\, due to A. Vinogradov\, is a formal replacement for the di
fferential calculus on the typically infinite dimensional space of solutio
ns to a non-linear partial differential equation and is centered around th
e study of the Variational Bi-complex of a system of equations. In my ta
lk I will discuss a generalization in the setting of (relative) homotopica
l algebraic geometry for possibly singular PDEs.\n\nThis is based on a ser
ies of joint works with Artan Sheshmani and Shing-Tung Yau.\n
LOCATION:https://researchseminars.org/talk/GDEq/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Rudinsky
DTSTART:20250226T162000Z
DTEND:20250226T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/126
DESCRIPTION:Title: W
eak gauge PDEs\nby Dmitry Rudinsky as part of Geometry of differential
equations seminar\n\nLecture held in room 303 of the Independent Universi
ty of Moscow.\n\nAbstract\nGauge PDEs are flexible graded geometrical obje
cts that generalise AKSZ sigma models to the case of local gauge theories.
However\, aside from specific cases - such as PDEs of finite type or topo
logical field theories - gauge PDEs are inherently infinite-dimensional. I
t turns out that these objects can be replaced by finite dimensional objec
ts called weak gauge PDEs. Weak gauge PDEs are equipped with a vertical in
volutive distribution satisfying certain properties\, and the nilpotency c
ondition for the homological vector field is relaxed so that it holds modu
lo this distribution. Moreover\, given a weak gauge PDE\, it induces a sta
ndard jet-bundle BV formulation at the level of equations of motion. In ot
her words\, all the information about PDE and its corresponding BV formula
tion turns out to be encoded in the finite-dimensional graded geometrical
object. Examples include scalar field theory and self-dual Yang-Mills theo
ry.\n
LOCATION:https://researchseminars.org/talk/GDEq/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20250305T162000Z
DTEND:20250305T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/127
DESCRIPTION:Title: I
nvariant reduction for PDEs. II: The general mechanism\nby Konstantin
Druzhkov as part of Geometry of differential equations seminar\n\n\nAbstra
ct\nGiven a local (point\, contact\, or higher) symmetry of a system of pa
rtial differential equations\, one can consider the system that describes
the invariant solutions (the invariant system). It seems natural to expect
that the invariant system inherits symmetry-invariant geometric structure
s in a specific way. We propose a mechanism of reduction of symmetry-invar
iant geometric structures\, which relates them to their counterparts on th
e respective invariant systems. This mechanism is homological and covers t
he stationary action principle and all terms of the first page of the Vino
gradov C-spectral sequence. In particular\, it applies to invariant conser
vation laws\, presymplectic structures\, and internal Lagrangians. A versi
on of Noether's theorem naturally arises for systems that describe invaria
nt solutions. Furthermore\, we explore the relationship between the C-spec
tral sequences of a system of PDEs and systems that are satisfied by its s
ymmetry-invariant solutions. Challenges associated with multi-reduction un
der non-commutative symmetry algebras are also clarified.\n
LOCATION:https://researchseminars.org/talk/GDEq/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20250312T162000Z
DTEND:20250312T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/128
DESCRIPTION:Title: T
urbulence geometry and Navier-Stokes equations\nby Valentin Lychagin a
s part of Geometry of differential equations seminar\n\nLecture held in ro
om 303 of the Independent University of Moscow.\n\nAbstract\nIt is propose
d to consider turbulent media and\, in particular\, random vector fields f
rom a geometric point of view. This leads to a geometry similar to\, but n
ot identical to\, Finsler's.\n\nWe show that a turbulence generates a hori
zontal differential symmetric 2-form on the tangent bundle\, which we call
the Mahalanobis metric.\n\nThus\, vector fields on the underlying manifol
d generate Riemannian structures on it by the restriction of the Mahalanob
is metric on the graphs of vector fields.\n\nIn the case of so-called Gaus
sian turbulences\, these Riemannian structures coincide and generate a uni
que Riemannian structure.\n\nMoreover\, similar to Finsler geometry\, turb
ulence generates a nonlinear connection in the tangent bundle but the Maha
lanobis metric generates an affine connection in the tangent bundle.\n\nTh
is affine connection and the Mahalanobis metric give us everything we need
to construct the Navier-Stokes equations for turbulent media.\n\nWe will
present two implementations of this scheme: for the flow of ideal gases an
d plasma\, where turbulence is described by the Maxwell-Boltzmann distribu
tion law\, and compare them with the standard Navier-Stokes equations.\n
LOCATION:https://researchseminars.org/talk/GDEq/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20250319T162000Z
DTEND:20250319T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/129
DESCRIPTION:Title: G
eometry of the full symmetric Toda system\nby Georgy Sharygin as part
of Geometry of differential equations seminar\n\nLecture held in room 303
of the Independent University of Moscow.\n\nAbstract\nFull symmetric Toda
system is the Lax-type system $\\dot L=[M(L)\,L]$\, where the variable $L$
is a real symmetric $n\\times n$ matrix and $M(L)=L_+-L_-$ denotes its "
naive" anti-symmetrisation\, i.e. the matrix constructed by taking the dif
ference of strictly upper- and lower-triangular parts $L_+$ and $L_-$ of $
L$. This system has lots of interesting properties: it is a Liouville-in
tegrable Hamiltonian system (with rational first integrals)\, it is also
super-integrable (in the sense of Nekhoroshev)\, its singular points and t
rajectories represent the Hasse diagram of Bruhat order on permutations gr
oup. Its generalizations to other semisimple real Lie algebras have simila
r properties. In my talk I will sketch the proof of some of these properti
es and will describe a construction of infinitesimal symmetries of the Tod
a system. It turns out that there are many such symmetries\, their constru
ction depends on representations of $\\mathfrak{sl}_n$. As a byproduct w
e prove that the full symmetric Toda system is integrable in the sense of
Lie-Bianchi criterion.\n\nThe talk is based on a series of papers joint wi
th Yu.Chernyakov\, D.Talalaev and A.Sorin.\n
LOCATION:https://researchseminars.org/talk/GDEq/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20250402T162000Z
DTEND:20250402T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/131
DESCRIPTION:Title: T
he quantum argument shift method on $U\\mathfrak{gl}_n$\nby Georgy Sha
rygin as part of Geometry of differential equations seminar\n\nLecture hel
d in room 303 of the Independent University of Moscow.\n\nAbstract\nThe te
rm "argument shift method" is used for a simple and efficient method to co
nstruct commutative subalgebras in Poisson algebras by deforming the Casi
mir elements in them. This method is primarily used to search for Poisson
commutative subalgebras in symmetric algebras of various Lie algebras\; i
t is closely related with the bi-Hamiltonian induction (Lenard-Magri schem
e). However little is known about the possible extension of this method t
o the quantum algebras associated with given Poisson algebras\; this is tr
ue even for the symmetric algebra of a Lie algebra\, where the quantizati
on is well known (it is equal to the universal enveloping algebra). I will
tell about a particular case\, the algebra $U\\mathfrak{gl}_n$\, for whi
ch one can find a shifting operator raising to this algebra the shift on
$S(\\mathfrak{gl}_n)$\, and prove that this operator verifies the same c
ondition as before: when used to deform the elements in the center of $U\
\mathfrak{gl}_n$\, it yields a set of commuting elements.\n\nThe talk is p
artially based on joint works with Y.Ikeda.\n
LOCATION:https://researchseminars.org/talk/GDEq/131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Kruglikov (UiT the Arctic University of Norway)
DTSTART:20250409T162000Z
DTEND:20250409T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/132
DESCRIPTION:Title: O
n inverse variational problem: examples\nby Boris Kruglikov (UiT the A
rctic University of Norway) as part of Geometry of differential equations
seminar\n\n\nAbstract\nInverse problem of the calculus of variations is a
vast subject with many results. I will discuss some examples related to OD
Es\, making an emphasis on parametrized vs unparametrized problems.\n\nThe
simplest and most studied case is about systems of second order different
ial equations\, also known as path geometries. Here I will mention some re
sults joint with Vladimir Matveev arXiv:2203.15029.\n\nThen I will discuss recent results joint with
Vladimir Matveev and Wijnand Steneker arXiv:2412.04890 about variationality of so-called conformal ge
odesics. This system is given by third order differential equations\, whic
h makes it rather unconventional for traditional approaches. I will also m
ention an on-going project using the invariant variational bicomplex.\n
LOCATION:https://researchseminars.org/talk/GDEq/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20250326T162000Z
DTEND:20250326T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/133
DESCRIPTION:Title: A
non-trivial conservation law with a trivial characteristic\nby Konsta
ntin Druzhkov as part of Geometry of differential equations seminar\n\n\nA
bstract\nAs far as I am aware\, no nontrivial conservation laws surviving
to the second page of Vinogradov's C-spectral sequence have been establish
ed. It turns out that presymplectic structures that cannot be described in
terms of cosymmetries produce such conservation laws for closely related
overdetermined systems. In particular\, the presymplectic structure $D_x$
of the potential mKdV equation gives rise to such a conservation law for t
he overdetermined system $u_t = 4u_x^3 + u_{xxx}$\, $u_y = 0$. While this
example is somewhat degenerate\, it may be one of the simplest systems exh
ibiting this phenomenon.\n
LOCATION:https://researchseminars.org/talk/GDEq/133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evgeny Ferapontov (Loughborough University)
DTSTART:20250416T162000Z
DTEND:20250416T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/134
DESCRIPTION:Title: L
agrangian multiforms and dispersionless integrable systems\nby Evgeny
Ferapontov (Loughborough University) as part of Geometry of differential e
quations seminar\n\nLecture held in room 303 of the Independent University
of Moscow.\n\nAbstract\nWe demonstrate that interesting examples of Lagra
ngian multiforms appear naturally in the theory of multidimensional disper
sionless integrable systems as (a) higher-order conservation laws of linea
rly degenerate PDEs in 3D\, and (b) in the context of Gibbons-Tsarev equat
ions governing hydrodynamic reductions of heavenly type equations in 4D.\n
\nBased on joint work with Mats Vermeeren.\n
LOCATION:https://researchseminars.org/talk/GDEq/134/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20250423T162000Z
DTEND:20250423T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/135
DESCRIPTION:Title: B
i-Hamiltonian structures of WDVV-type\nby Raffaele Vitolo (Università
del Salento) as part of Geometry of differential equations seminar\n\n\nA
bstract\nWe study a class of nonlinear partial differential equations (PDE
s) that admit the same bi-Hamiltonian structure as the Witten-Dijkgraaf-Ve
rlinde-Verlinde (WDVV) equations: a Ferapontov-type first-order Hamiltonia
n operator and a homogeneous third-order Hamiltonian operator in a canonic
al Doyle-Potëmin form\, which are compatible. Properties of these systems
and their classification in low dimension will be discussed.\n\nJoint wor
k with S. Opanasenko.\n
LOCATION:https://researchseminars.org/talk/GDEq/135/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Doubrov
DTSTART:20250514T162000Z
DTEND:20250514T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/136
DESCRIPTION:Title: E
quivalence of scalar ODEs under contact\, point and fiber-preserving trans
formations\nby Boris Doubrov as part of Geometry of differential equat
ions seminar\n\n\nAbstract\nIt is known that the equivalence problem for s
calar ODEs of order 3 and higher can be solved via the construction of a c
anonical Cartan connection. The invariants then appear as part of the curv
ature of this connection. This allows to describe explicitly all scalar OD
Es of order 3 and higher that can be brought to the trivial equation by co
ntact transformations. The goal of this talk is to show how most of this s
tory can be extended to the equivalence of scalar ODEs under point and fib
er-preserving transformations.\n
LOCATION:https://researchseminars.org/talk/GDEq/136/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev
DTSTART:20250521T162000Z
DTEND:20250521T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/137
DESCRIPTION:Title: B
ackground fields and symmetries in the gauge PDE approach\nby Maxim Gr
igoriev as part of Geometry of differential equations seminar\n\nLecture h
eld in room 303 of the Independent University of Moscow.\n\nAbstract\nWe d
evelop an extension of the (presympletic) gauge PDE approach to describe l
ocal gauge theories with background fields. It turns out that such theorie
s correspond to (presymplectic) gauge PDEs whose base spaces are again gau
ge PDEs describing background fields. As such\, the geometric structure is
that of a bundle over a bundle over a given spacetime. Gauge PDEs over ba
ckgrounds arise naturally when studying linearisation\, coupling (gauge) f
ields to background geometry\, gauging global symmetries\, etc. Less obvio
us examples involve parameterised systems\, Fedosov equations\, and the so
-called homogeneous (presymplectic) gauge PDEs. The latter are the gauge-i
nvariant generalisations of the familiar homogeneous PDEs and they provide
a very concise description of gauge fields on homogeneous spaces such as
higher spin gauge fields on Minkowski\, (A)dS\, and conformal spaces. Fina
lly\, we briefly discuss how the higher-form symmetries and their gauging
fit into the framework using the simplest example of the Maxwell field.\n
LOCATION:https://researchseminars.org/talk/GDEq/137/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20250924T162000Z
DTEND:20250924T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/138
DESCRIPTION:Title: I
nformation geometry of turbulent media\nby Valentin Lychagin as part o
f Geometry of differential equations seminar\n\nLecture held in room 303 o
f the Independent University of Moscow.\n\nAbstract\nIn the talk\, we plan
to discuss a method of geometrization of statistics on the example of ran
dom vectors and its application to turbulent media\, by which we understan
d random vector fields on base manifolds.\n\nWe show that this approach gi
ves rise to various geometric structures on the tangent as well as cotange
nt bundles.\n\nAmong these\, the most important is the Mahalanobis metric
on the tangent bundle\, which allows us to obtain all the necessary ingred
ients for the description of flows in turbulent media.\n\nAs an illustrati
on of the method\, we consider its applications to flows of real gases bas
ed on Maxwell-Boltzmann-Gibbs statistics.\n
LOCATION:https://researchseminars.org/talk/GDEq/138/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20251112T162000Z
DTEND:20251112T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/139
DESCRIPTION:Title: I
nvariant reduction for PDEs. III: Poisson brackets\nby Konstantin Druz
hkov as part of Geometry of differential equations seminar\n\n\nAbstract\n
I will show that\, under suitable conditions\, finite-dimensional systems
describing invariant solutions of PDEs inherit local Hamiltonian operators
through the mechanism of invariant reduction\, which applies uniformly to
point\, contact\, and higher symmetries. The inherited operators endow th
e reduced systems with Poisson bivectors that relate constants of invarian
t motion to symmetries. The induced Poisson brackets agree with those of t
he original systems\, up to sign. At the core of this construction lies th
e interpretation of Hamiltonian operators as degree-2 conservation laws of
degree-shifted cotangent equations.\n
LOCATION:https://researchseminars.org/talk/GDEq/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Doubrov
DTSTART:20251001T162000Z
DTEND:20251001T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/140
DESCRIPTION:Title: G
eometry of CR manifolds via finite type systems of PDEs\nby Boris Doub
rov as part of Geometry of differential equations seminar\n\n\nAbstract\nW
e show how complexification of CR manifolds leads to systems of PDEs with
finite-dimensional solution space. Applications of this approach include c
lassification of homogeneous 5D CR manifolds and identification models wit
h large symmetry in other dimensions.\n
LOCATION:https://researchseminars.org/talk/GDEq/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20251015T162000Z
DTEND:20251015T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/141
DESCRIPTION:Title: O
n the geometry of WDVV equations and their Hamiltonian formalism in arbitr
ary dimension\nby Raffaele Vitolo (Università del Salento) as part of
Geometry of differential equations seminar\n\n\nAbstract\nIt is known tha
t in low dimensions WDVV equations can be rewritten as commuting quasiline
ar bi-Hamiltonian systems. We extend some of these results to arbitrary di
mension N and arbitrary scalar product $\\eta$. In particular\, we show th
at WDVV equations can be interpreted as a set of linear line congruences i
n suitable Plücker embeddings. This form leads to their representation as
Hamiltonian systems of conservation laws. Moreover\, we show that in low
dimensions and for an arbitrary $\\eta$ WDVV equations can be reduced to p
assive orthonomic form. This leads to the commutativity of the Hamiltonian
systems of conservation laws. We conjecture that such a result holds in a
ll dimensions.\n
LOCATION:https://researchseminars.org/talk/GDEq/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20251022T162000Z
DTEND:20251022T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/142
DESCRIPTION:Title: K
ällén function and around. Part 1\nby Vladimir Rubtsov (Université
d'Angers) as part of Geometry of differential equations seminar\n\nLecture
held in room 303 of the Independent University of Moscow.\n\nAbstract\nTh
ere are elementary functions that turn out to be ubiquitous. The Källén
function is one of them. Of course\, it is incomparably less well-known t
han the exponential function (although\, in a certain sense\, it is relate
d to it).\n\nOriginally arising in "school-textbooks" mathematics\, the K
ällén function was later "rediscovered" by physicists in the context of
scattering amplitude calculations. It is directly connected with the famou
s combinatorial generating functions (for the Catalan numbers)\, appears i
n the study of solutions of classical ODEs\, and is related to the general
ized hypergeometric functions of Appell and Kampé de Fériet as well as t
o properties of discriminants.\n\nI will try to tell some stories around o
f this remarkable function.\n
LOCATION:https://researchseminars.org/talk/GDEq/142/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20251029T162000Z
DTEND:20251029T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/143
DESCRIPTION:Title: V
an den Berg double Poisson brackets on finite-dimensional algebras\nby
Georgy Sharygin as part of Geometry of differential equations seminar\n\n
Lecture held in room 303 of the Independent University of Moscow.\n\nAbstr
act\nOne of the basic principles of algebraic noncommutative geometry is t
he condition proposed by Kontsevich and Rosenberg that a "geometric" struc
ture on a noncommutative algebra A should generate a similar ordinary\, "c
ommutative" structure on its representation spaces $Rep_d(A)=Hom(A\,Mat_d(
k))$. The concept of "double Poisson brackets" was introduced by van den B
erg (and almost simultaneously\, in a slightly modified form\, by Crowley-
Bovey) in 2008 as an answer to the question of which noncommutative struct
ures correspond to Poisson brackets on representation spaces. The resultin
g construction turned out to be quite rich and interesting\, however\, the
vast majority of examples of such structures now deal with algebras A tha
t are (close to being) free. In my talk\, based on a joint work with my ma
ster's student A. Hernandez-Rodriguez\, I will describe some simple exampl
es of how such structures look on finite-dimensional algebras.\n
LOCATION:https://researchseminars.org/talk/GDEq/143/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Kruglikov (UiT the Arctic University of Norway)
DTSTART:20251008T162000Z
DTEND:20251008T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/144
DESCRIPTION:Title: R
ational integrals of geodesic flows\nby Boris Kruglikov (UiT the Arcti
c University of Norway) as part of Geometry of differential equations semi
nar\n\n\nAbstract\nPolynomial (in momenta) integrals of geodesic flows\, a
lso known as Killing tensors of the metric\, play an important role in fin
ite-dimensional integrable systems. Recently\, rational integrals came in
focus of investigations. (These are natural\, especially for algebraic Ham
iltonian actions.) I will discuss the problem of their computations and co
unt\, relation to relative Killing tensors and show some examples.\n
LOCATION:https://researchseminars.org/talk/GDEq/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev
DTSTART:20251105T162000Z
DTEND:20251105T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/145
DESCRIPTION:Title: G
auge PDEs on spaces with asymptotic boundaries\nby Maxim Grigoriev as
part of Geometry of differential equations seminar\n\n\nAbstract\nI plan
to discuss a general setup for studying the boundary structure of gauge f
ields on spaces with asymptotic boundaries. The main example of this situa
tion is asymptotically-anti-de-Sitter (AdS) or flat gravity and (optionall
y) gauge fields living on such a background. A suitable tool to study syst
ems of this sort in a geometrical way is the so-called gauge PDE on spac
es with (asymptotic) boundaries. When applied to the case of asymptotical
ly-AdS gravity this gives the generalization of the familiar Fefferman-Gr
aham construction that also takes the subleading boundary value into accou
nt. When additional (gauge) fields are present this generalizes the known
gauge PDE approach to boundary values of AdS gauge fields. An interesting
feature is that the gauge PDE induced on the boundary is itself a fibre
bundle of gauge PDEs (also known as gauge PDE over background)\, where th
e base describes the leading (conformal geometry in the case of gravity) w
hile the fiber correspond to the subleading (conserved currents).\n
LOCATION:https://researchseminars.org/talk/GDEq/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evgeny Ferapontov (Loughborough University)
DTSTART:20251217T162000Z
DTEND:20251217T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/146
DESCRIPTION:Title: I
nvolutive scroll structures on solutions of 4D dispersionless integrable h
ierarchies\nby Evgeny Ferapontov (Loughborough University) as part of
Geometry of differential equations seminar\n\n\nAbstract\nA rational norma
l scroll structure on an (n+1)-dimensional manifold M is defined as a fi
eld of rational normal scrolls of degree n-1 in the projectivised cotangen
t bundle $PT^*M$.\n\nWe show that geometry of this kind naturally arises o
n solutions of various 4D dispersionless integrable hierarchies of hea
venly type equations. In this context\, rational normal scrolls coincide
with the characteristic varieties (principal symbols) of the hierarchy. F
urthermore\, such structures automatically satisfy an additional property
of involutivity.\n\nOur main result states that involutive scroll structur
es are themselves governed by a dispersionless integrable hierarchy\, na
mely\, the hierarchy of conformal self-duality equations.\n\nBased on join
t work with Boris Kruglikov.\n
LOCATION:https://researchseminars.org/talk/GDEq/146/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Pavlov
DTSTART:20251203T162000Z
DTEND:20251203T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/147
DESCRIPTION:Title: I
somonodromic deformations and the Tsarev generalized hodograph method\
nby Maxim Pavlov as part of Geometry of differential equations seminar\n\n
Lecture held in room 303 of the Independent University of Moscow.\n\nAbstr
act\nGeneral and particular solutions of the so called semi-Hamiltonian hy
drodynamic type systems can be obtained by the Tsarev Generalized Hodograp
h Method. Here we show that a natural extension of this approach applied t
o dispersive integrable systems is determined by isomonodromic deformation
s.\n
LOCATION:https://researchseminars.org/talk/GDEq/147/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20251210T162000Z
DTEND:20251210T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/148
DESCRIPTION:Title: K
ällén function and around. Part 2\nby Vladimir Rubtsov (Université
d'Angers) as part of Geometry of differential equations seminar\n\n\nAbstr
act\nA continuation of the talk on 22 October.\n\nThere are elementary functions that turn
out to be ubiquitous. The Källén function is one of them. Of course\,
it is incomparably less well-known than the exponential function (although
\, in a certain sense\, it is related to it).\n\nOriginally arising in "sc
hool-textbooks" mathematics\, the Källén function was later "rediscovere
d" by physicists in the context of scattering amplitude calculations. It i
s directly connected with the famous combinatorial generating functions (f
or the Catalan numbers)\, appears in the study of solutions of classical O
DEs\, and is related to the generalized hypergeometric functions of Appell
and Kampé de Fériet as well as to properties of discriminants.\n\nI wil
l try to tell some stories around of this remarkable function.\n
LOCATION:https://researchseminars.org/talk/GDEq/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Markov
DTSTART:20251224T162000Z
DTEND:20251224T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/149
DESCRIPTION:Title: B
oundary calculus for gauge fields on asymptotically AdS spaces\nby Mik
hail Markov as part of Geometry of differential equations seminar\n\n\nAbs
tract\nI plan to discuss the applications of the gauge PDE approach to the
study of the boundary structure of gauge fields on the conformal boundary
of asymptotically AdS (also known as Poincaré-Einstein) manifolds.\n\nTh
e main result is the construction of an efficient calculus for the gauge P
DE induced on the boundary\, which allows one to systematically derive Wey
l-invariant equations induced on the boundary. The so-called obstruction e
quations (e.g. Bach in dimension d=4)\, higher conformal Yang-Mills equati
ons\, and GJMS operators are derived systematically\, as the constraints o
n the leading boundary value of\, respectively\, the metric\, YM field\,
and the critical scalar field. In particular\, the higher conformal Yang-
Mills equation in dimension d=8\, obtained within this framework appears t
o be new. The Weyl-invariant equations on the subleading boundary data for
these fields are also derived.\n\nThe approach is very general and can
be considered as an extension of the Fefferman-Graham construction that is
applicable to generic gauge fields and explicitly takes into account both
the leading and the subleading sector.\n
LOCATION:https://researchseminars.org/talk/GDEq/149/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20260211T162000Z
DTEND:20260211T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/150
DESCRIPTION:Title: O
n the management of thermodynamic processes\nby Valentin Lychagin as p
art of Geometry of differential equations seminar\n\nLecture held in room
303 of the Independent University of Moscow.\n\nAbstract\nAt the beginning
of the talk\, three geometric approaches to thermodynamics will be discus
sed.\n\nThe first approach is the Gibbs energy approach\, which will be re
formulated in terms of contact geometry and where the description of subst
ances (the so-called equations of state) is given in terms of Legendre sub
manifolds in thermodynamic contact phase spaces\, and thermodynamic proces
ses\, as well as controls\, will be given by contact vector fields.\n\nThe
second approach is based on information geometry and follows the principl
e of maximum entropy\, also known as the principle of minimum information
gain or Occam's razor. Both of these approaches lead us to the same model
of thermodynamics\, but they also introduce important new concepts\, such
as the Gibbs-Duhem principle and Riemannian structures on Legendre submani
folds.\n\nThe third approach is based on the geometry of jet spaces (or th
e geometry of differential equations)\, and it provides a more convenient
apparatus for the practical description and calculation of both equations
of state and thermodynamic processes\, taking into account phase transitio
ns.\n\nThermodynamic process control will be understood as a thermodynamic
process that does not destroy the process in question\, but allows it to
be accelerated or slowed down. \n\nThe set of controls forms a Lie algebr
a\, in which the Lie algebra of symmetries is a Lie subalgebra. \n\nWe wi
ll present equations that depend on the equations of state of the medium a
nd allow us to find control processes\, as well as illustrate their applic
ation in the case of adiabatic processes.\n\nIf time permits\, phase trans
itions of the first\, second and higher orders will be considered\, both i
n thermodynamic processes and in controls\, as well as their connection wi
th Arnold's theory on the singularities of projections of Lagrangian manif
olds.\n\nPlease download the formula file td.pdf and keep it handy during the talk so tha
t the speaker can refer to it.\n
LOCATION:https://researchseminars.org/talk/GDEq/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20260218T162000Z
DTEND:20260218T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/151
DESCRIPTION:Title: 2
-Valued algebraic groups\, the Chazy equation\, and quasimodular forms
\nby Vladimir Rubtsov (Université d'Angers) as part of Geometry of differ
ential equations seminar\n\nLecture held in room 303 of the Independent Un
iversity of Moscow.\n\nAbstract\nI will discuss some (un)known relations b
etween the objects in the title.\n\nIn particular\, the celebrated Chazy e
quation emerges as an associativity condition. \n\nThe talk is based on o
ngoing joint work with V. Buchstaber and M. Kornev (Steklov Mathematical Institute\, RAS).\
n
LOCATION:https://researchseminars.org/talk/GDEq/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Duyunova
DTSTART:20260225T162000Z
DTEND:20260225T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/152
DESCRIPTION:Title: D
ifferential invariants and quotient of the Euler equations on a sphere
\nby Anna Duyunova as part of Geometry of differential equations seminar\n
\nLecture held in room 303 of the Independent University of Moscow.\n\nAbs
tract\nWe consider the Euler system on a sphere written in stereographic c
oordinates. Since the system is underdetermined we consider flow of a medi
um taking into account thermodynamic equations of state.\n\nLie algebras o
f symmetries of the Euler system are found and we give their classificatio
n depending on possible equations of state. Among these Lie algebras there
is one that preserves any thermodynamic equation. Such symmetries and the
corresponding rational differential invariants we call kinematic. The fie
ld of kinematic differential invariants is described: basis differential i
nvariants as well as invariant derivations are found. Then we find relatio
ns (syzygies) between the second-order invariants\, from which we find a q
uotient equation for the Euler system on a sphere.\n\nJoint work with Vale
ntin Lychagin and Sergey Tychkov.\n\nPlease download the formula file di.pdf and keep it
handy during the talk so that the speaker can refer to it.\n
LOCATION:https://researchseminars.org/talk/GDEq/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20260318T162000Z
DTEND:20260318T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/153
DESCRIPTION:Title: B
i-Hamiltonian systems from homogeneous operators\nby Raffaele Vitolo (
Università del Salento) as part of Geometry of differential equations sem
inar\n\n\nAbstract\nMany "famous" integrable systems (KdV\, AKNS\, dispers
ive water waves etc.) have a bi-Hamiltonian pair of the following form: $A
_1 = P_1 + R_k$ and $A_2 = P_2$\, where $P_1$\, $P_2$ are homogeneous firs
t-order Hamiltonian operators and $R_k$ is a homogeneous Hamiltonian opera
tor of degree (order) $k$. The Hamiltonian property of $P_1$\, $P_2$ and t
heir compatibility were given an explicit analytic form and geometric inte
rpretation long ago (Dubrovin\, Novikov\, Ferapontov\, Mokhov). The Hamilt
onian property of $R_k$ was studied in the past (Doyle\, Potemin\; $k=2\,3
$) and recently revisited with interesting results.\n\nIn this talk\, we i
llustrate the analytic form and some preliminary geometric interpretation
of the compatibility conditions between $P_i$ and $R_k$\, $k=2\,3$.\n\nSee
the recent papers arXiv:2602.1
4739\, arXiv:2407.17189
\, arXiv:2311.13932.\n\nJoi
nt work with P. Lorenzoni and S. Opanasenko.\n
LOCATION:https://researchseminars.org/talk/GDEq/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Kuleshov
DTSTART:20260325T162000Z
DTEND:20260325T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/154
DESCRIPTION:Title: E
xact solutions of some problems of rigid body dynamics\nby Alexander K
uleshov as part of Geometry of differential equations seminar\n\nLecture h
eld in room 303 of the Independent University of Moscow.\n\nAbstract\nInte
rest in integrable problems in mechanics has never waned. Finding new inte
grable cases of differential equations of motion for various mechanical sy
stems\, as well as finding solutions in quadratures for these cases\, is o
ne of the main problem of theoretical mechanics. The problem of exact inte
gration of differential equations of motion has several aspects. The geome
tric aspect is associated with the qualitative study of the regular behavi
or of the trajectories of integrable systems. The constructive aspect is a
ssociated with finding the conditions under which an algorithm for explici
t solving differential equations using quadratures can be specified. In th
is regard\, another important aspect of the range of issues under consider
ation arises: the explicit solution of systems of differential equations.
For certain classes of differential equations\, relying on their specific
structure\, special methods can be used. An example here is the broad and
important class of linear differential equations. The study of many proble
ms in mechanics and mathematical physics reduces to solving a second-order
linear homogeneous differential equation. If\, by changing the independen
t variable\, it is possible to reduce the corresponding second-order linea
r differential equation to an equation with rational coefficients\, then t
he necessary and sufficient for solvability by quadratures for such an equ
ation are determined by the so-called Kovacic algorithm. In 1986\, the Ame
rican mathematician J. Kovacic presented an algorithm for finding Liouvill
ian solutions of a second-order linear homogeneous differential equation w
ith rational coefficients. If the differential equation has no Liouvillian
solution\, the algorithm also allows one to establish this fact.\n\nThis
talk will discuss the application of the Kovacic algorithm to investigate
the existence of Liouvillian solutions in the problem of motion of a rotat
ionally symmetric rigid body on a perfectly rough plane and on a perfectly
rough sphere. It will also discuss the application of the algorithm to in
vestigate the existence of Liouvillian solutions in the problem of motion
of a heavy homogeneous ball on a fixed perfectly rough surface of revoluti
on. The existence of Liouvillian solutions in the Hess case of the problem
of motion of a heavy rigid body with a fixed point is also analyzed.\n
LOCATION:https://researchseminars.org/talk/GDEq/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wijnand Steneker
DTSTART:20260408T162000Z
DTEND:20260408T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/155
DESCRIPTION:Title: O
n globally invariant Euler-Lagrange equations for curves\nby Wijnand S
teneker as part of Geometry of differential equations seminar\n\n\nAbstrac
t\nInvariant Lagrangians yield invariant Euler-Lagrange equations and loca
l methods for computing these are well-established\, starting with Anderso
n and Griffiths. We focus on global algebraic invariants\, using an invar
iant version of variational bicomplex or\, more generally\, C-spectral seq
uence. One motivation is the question\, posed by Kogan and Olver\, whether
invariant variational problems with only singular extremals can exist. We
show that the example of conformal geodesics answers this question positi
vely and motivates the need for global invariant methods. We then discuss
how to compute invariant Euler-Lagrange equations using global invariants
and how this can be applied in practice\, both as a supplementary tool for
existing local methods\, as well as in a purely global setting. We demons
trate these principles with some examples\, all for systems of ODEs (unpar
ametrized curves). \n\nThis talk is based on joint work with Boris Krugli
kov and Eivind Schneider (Tromsø).\n
LOCATION:https://researchseminars.org/talk/GDEq/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20260415T162000Z
DTEND:20260415T180000Z
DTSTAMP:20260422T225928Z
UID:GDEq/156
DESCRIPTION:Title: I
nvariant reduction for PDEs. IV: Symmetries that rescale geometric structu
res\nby Konstantin Druzhkov as part of Geometry of differential equati
ons seminar\n\n\nAbstract\nFor a system of differential equations and a sy
mmetry\, the framework of invariant reduction systematically computes how
invariant geometric structures are inherited by the subsystem governing i
nvariant solutions. In this setting\, the reduction of structures invarian
t under a two-dimensional Lie algebra requires its commutativity. We exte
nd this mechanism to the case where geometric structures are invariant und
er one symmetry $X$ and are rescaled\, by a factor of $-a$\, by another sy
mmetry $X_s$ satisfying $[X_s\, X] = aX$. As an application\, we describe
a class of exact solutions to systems possessing sufficiently many symmet
ries and conservation laws subject to certain compatibility conditions. Th
ese solutions are invariant under pairs of symmetries and are completely d
etermined by explicitly constructed functions that are constant on them\;
the description is geometric and does not require any integrability-relat
ed structures such as Lax pairs.\n
LOCATION:https://researchseminars.org/talk/GDEq/156/
END:VEVENT
END:VCALENDAR