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BEGIN:VEVENT
SUMMARY:Prof Semyon Dyatlov (MIT)
DTSTART:20210129T150000Z
DTEND:20210129T160000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/1
DESCRIPTION:Title: Control of eigenfunctions on negatively curved surfaces\nby Prof Semy
on Dyatlov (MIT) as part of Open PDE and analysis seminar and lectures\n\n
\nAbstract\nGiven an $L^2$-normalized eigenfunction with eigenvalue $\\lam
bda^2$ on a compact Riemannian manifold $(M\,g)$ and a non-empty open subs
et $\\Omega$ of $M$\, what lower bound can we prove on the $L^2$-mass of t
he eigenfunction on $\\Omega$? The unique continuation principle gives a b
ound for any $\\Omega$ which is exponentially small as $\\lambda$ goes to
infinity. On the other hand\, microlocal analysis gives a $\\lambda$-indep
endent lower bound if $\\Omega$ is large enough\, i.e. it satisfies the ge
ometric control condition. This talk presents a $\\lambda$-independent low
er bound for any set $\\Omega$ in the case when $M$ is a negatively curved
surface\, or more generally a surface with Anosov geodesic flow. The pro
of uses microlocal analysis\, the chaotic behaviour of the geodesic flow\,
and a new ingredient from harmonic analysis called the Fractal Uncertaint
y Principle. Applications include control for Schrödinger equation and ex
ponential decay of damped waves. Joint work with Jean Bourgain\, Long Jin
\, and Stéphane Nonnenmacher.\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Prof Eugenia Malinnikova (Stanford)
DTSTART:20210212T150000Z
DTEND:20210212T160000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/2
DESCRIPTION:Title: On Yau’s conjecture for the Dirichlet Laplacian in C^1 domains\nby
Prof Eugenia Malinnikova (Stanford) as part of Open PDE and analysis semin
ar and lectures\n\n\nAbstract\nLet D be a bounded domain in R^n with C^1 b
oundary and let u be a Dirichlet Laplace eigenfunction in D with eigenvalu
e λ. We show that the (n − 1)-dimensional Hausdorff measure of the zero
set of u does not exceed C√λ. The opposite estimate follows from the w
ork of Donnelly and Fefferman. The talk is based on a joint work with A. L
ogunov\, N. Nadirashvili\, and F. Nazarov..\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claude Zuily (Universite Paris-Saclay)
DTSTART:20210205T140000Z
DTEND:20210205T160000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/3
DESCRIPTION:Title: (LECTURE) Quantitative unique continuation: an introduction. After A.Logu
nov and E. Malinnikova.\nby Claude Zuily (Universite Paris-Saclay) as
part of Open PDE and analysis seminar and lectures\n\n\nAbstract\nThe ques
tion of the unique continuation from open sets for solutions of elliptic e
quations with Lipschitz coefficients as well as its quantitative version h
ave been positively answered a long time ago mainly using the technique of
Carleman estimates. The same question where the open set is replaced by a
set of positive measure is more recent. In 2017 A. Logunov and E. Malinni
kova introduced new ideas to face this problem. The present lecture is an
introduction to their techniques which appear to have applications to the
study of the size of the nodal sets of eigenfunctions as well as to contro
l theory.\n\nThis meeting starts at 3pm in France time (9pm ET). Here is t
he zoom link:\nhttps://univ-cotedazur.zoom.us/j/82084903423?pwd=UUJIS0ZSUm
5lR1FZcDFoWTd4Wis3dz09\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tarek Elgindi (Duke University)
DTSTART:20210423T130000Z
DTEND:20210423T150000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/4
DESCRIPTION:Title: (LECTURE) Singularity formation in incompressible fluids\nby Tarek El
gindi (Duke University) as part of Open PDE and analysis seminar and lectu
res\n\n\nAbstract\nI will discuss various aspects of singularity formation
in the incompressible Euler equation in two and three dimensions. In two
dimensions\, important questions relate to the infinite time growth of smo
oth solutions\, filamentation of the vorticity\, and the genericity of thi
s phenomenon. In three dimensions\, we will discuss two methods that have
been used to rigorously construct finite time singularities. In both conte
xts\, an important theme is the identification of stable growth mechanisms
.\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maciej Zworski (UC Berkeley)
DTSTART:20210305T150000Z
DTEND:20210305T170000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/5
DESCRIPTION:Title: (LECTURE) Introduction to non-self-adjoint operators: a case study using
a model of twisted bilayer graphene.\nby Maciej Zworski (UC Berkeley
) as part of Open PDE and analysis seminar and lectures\n\n\nAbstract\nI w
ill use a simple model from physics \n(Tarnopolsky--Kruchkov--Vishwanath\,
2019) to illustrate the wealth of \nstrange phenomena possible for non-se
lf-adjoint (or rather non-normal) \noperators. The model\, which is a simp
le operator on the torus\, explains \nthe origin of ``magic angles" in twi
sted bilayer graphene\, a hot topic \nin physics going by the name of twis
tronics: when two sheets of graphene \nare twisted at a special angle\, th
e material becomes a superconductor. \nBut please do not be scared by the
physics: the talk will be an \nelementary blend of spectral theory\, semic
lassical version of \nHörmander's commutator condition\, representation t
heory of the finite \nHeisenberg group\, and theta functions. Easy to stat
e open problems will \nalso be presented and the results will be illustrat
ed by colorful \nnumerics. Based on joint work with S Becker\, J Wittsten
and M Embree.\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Theodore Drivas (Stony Brooks University)
DTSTART:20210416T130000Z
DTEND:20210416T140000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/6
DESCRIPTION:Title: (SEMINAR) Some remarks on the long-time dynamics of 2D Euler.\nby The
odore Drivas (Stony Brooks University) as part of Open PDE and analysis se
minar and lectures\n\n\nAbstract\nWe describe some known results and open
questions regarding properties of steady solutions of the two-dimensional
incompressible Euler equations\, as well as properties of nearby trajector
ies. Specifically\, we focus on whether steady states can be isolated\, wh
ether\, for solutions starting nearby steady states\, recurrence can occur
or whether singularities must form at long times\, and finally some resul
ts on the infinite-time limit near and far from equilibrium.\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charles Collot (CY Universite)
DTSTART:20210409T140000Z
DTEND:20210409T150000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/7
DESCRIPTION:Title: Singularities\, separation\, and generic self-similar behaviour for the i
nviscid unsteady Prandtl boundary layer\nby Charles Collot (CY Univers
ite) as part of Open PDE and analysis seminar and lectures\n\n\nAbstract\n
The inviscid unsteady Prandtl system in two dimensions describes an incomp
ressible non viscous fluid close to a boundary. First\, we will prove that
the boundary layer separates off the wall if and only if the solution bec
omes singular away from it. Second\, we will present a method to find expl
icitly backward self-similar solutions forming finite time singularities.
Finally\, we will show that one of such self-similar solution is the attra
ctor for singular solutions near blow-up time\, in a generic sense (for a
dense an open set). This explains the generic appearance of the so-called
Van Dommelen and Shen singularity\, and describes completely and rigorousl
y the associated separating structure. The talk will combine ideas for tra
nsport equations\, such as Lagrangian coordinates and incompressibility\,
and for singularity formation\, such as self-similarity and renormalisatio
n. This is joint work with T.-E. Ghoul and N. Masmoudi.\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francois Golse (Ecole Polytechnique)
DTSTART:20210528T130000Z
DTEND:20210528T150000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/8
DESCRIPTION:Title: (Lecture) Optimal Transport Distances in Quantum Mechanics\nby Franco
is Golse (Ecole Polytechnique) as part of Open PDE and analysis seminar an
d lectures\n\n\nAbstract\nThe first part of this talk is focussed on the d
efinition of \nan extension of the Monge-Kantorovich-Wasserstein distance
of exponent 2 \nto the set density operators\, which correspond to probabi
lity measures \nin quantum mechanics. We shall mostly explore the metric p
roperties of \nthis extension\, in particular compare it with the Wasserst
ein metric \nitself\, and discuss variants of the triangle inequality.\n\n
The second part of the talk presents some applications of this notion of \
nquantum Wasserstein distances\, to the uniform convergence of \ntime-spli
tting schemes in the Planck constant for quantum dynamics\, to \neffective
observation inequalities for the Heisenberg or the Schrödinger \nequatio
ns\, and to the uniformity in the Planck constant of convergence \nrates f
or the mean-field limit in quantum mechanics.\n(Based on a series of works
with E. Caglioti\, C. Mouhot and T. Paul)\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thierry Paul (CNRS & Ecole Polytechnique)
DTSTART:20210611T140000Z
DTEND:20210611T150000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/9
DESCRIPTION:Title: (Seminar) Optimal transport and quantum mechanics: more facts and applica
tions\nby Thierry Paul (CNRS & Ecole Polytechnique) as part of Open PD
E and analysis seminar and lectures\n\n\nAbstract\nAfter showing that the
extension of the Monge-Kantorovich-Wasserstein distance introduced in the
talk by F. Golse is more convenient to separate density matrices than the
usual Schatten topologies usually used in quantum mechanics\, we shall sho
w how (and explain why) they produce a cost for the quantum bipartite matc
hing problem which is cheapper than the corresponding classical one. We sh
all then show that a quantum version of the Kantorovich duality provides a
form of Knott-Smith-Brenier theorem in quantum mechanics\, under technica
l conditions on the density matrices involved\, with a suitable quantum de
finition of the gradient of an observable\, naturally constructed on the c
lassical one. The finite rank case\, always tractable\, will give rise its
elf to a non-gradient «flow » without classical counterpart. Finally\, w
e will study transport associated to a semiquantum analogue of the Wassers
tein distances and show that they involve a generalization the Legendre tr
ansform between classical and quantum densities. (Based on a series of wor
ks with E. Caglioti\, F. Golse and C. Mouhot)\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chiara Saffirio (U. Basel)
DTSTART:20210618T130000Z
DTEND:20210618T140000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/10
DESCRIPTION:Title: (Seminar) Mean-field evolution of fermionic mixed states with singular
interaction potentials\nby Chiara Saffirio (U. Basel) as part of Open
PDE and analysis seminar and lectures\n\n\nAbstract\nWe will consider the
many-body evolution of initially \nconfined fermions interacting through a
singular potential. In a joint \nmean-field and semiclassical scaling and
using second quantization \ntechniques\, we will show that\, for mixed st
ates enjoying a semiclassical \nstructure\, the many-body dynamics can be
approximated in Schatten norms \nby the time-dependent Hartree-Fock equati
on. In particular\, we will \nhighlight the advantages and drawbacks of co
nsidering such strong \ntopology instead of the quantum Wasserstein distan
ce introduced in [F. \nGolse\, C. Mouhot and T. Paul\, Commun. Math. Phys.
343\, 165-205 (2016]).\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cambyze Rouze (U. München)
DTSTART:20210618T141500Z
DTEND:20210618T150000Z
DTSTAMP:20260422T225704Z
UID:OpenPDEA/11
DESCRIPTION:Title: (Seminar) Quantum modified logarithmic Sobolev inequalities\nby Camb
yze Rouze (U. München) as part of Open PDE and analysis seminar and lectu
res\n\n\nAbstract\nFunctional inequalities constitute by now a well-establ
ished \ntheory with many connections to other fields of mathematics such a
s \nconcentration of measure\, mixing times of Markov processes or optimal
\ntransport to name only a few. Among these inequalities\, the modified
\nlogarithmic Sobolev inequality (MLSI) controls the exponential entropic
\nconvergence of a Markov semigroup towards its stationary measure. \nAlth
ough introduced almost simultaneously\, their quantum analogues have \nlon
g suffered from the loss of certain key properties inherent to the \npassa
ge to the non-commutative realm. Perhaps the most important of \nthese is
the tensorization property\, which often allows one to prove a \nfunctiona
l inequality for a Markov process on an uncountable state space \nby reduc
tion to the two-points space.\nDue to the absence of generic tensorization
results for the MLSI in the \nquantum setting\, one is often forced to pr
ove it case by case. However\, \nin the recent years\, a new approach to t
he problem emerged from the \ncommunities of operator algebras and quantum
information theory. Here\, \ninstead of proving the tensorization of MLSI
for a product of \nsemigroups\, one considers a stronger inequality which
naturally \ntensorizes\, namely the complete modified logarithmic Sobolev
inequality \n(CMLSI). The latter consists in proving the inequality for t
he semigroup \ntensorized with the identity semigroup over an arbitrarily
large matrix \nalgebra. The existence of CMLSI for all quantum Markov semi
groups on \nmatrix algebras was however left as an open conjecture.\nIn th
is talk\, I will provide a proof of the conjecture for the class of \nreve
rsible quantum Markov semigroups. This talk is intended to be \nself-conta
ined and does not require previous knowledge about quantum \nmechanics or
quantum information theory. It is based on a joint work \nwith Li Gao\, a
preprint of which is available here: \nhttps://arxiv.org/abs/2102.04146.\n
LOCATION:https://researchseminars.org/talk/OpenPDEA/11/
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