\n** Theorem.** For \\(d\\geq 3\\)
the graph of a connected triangulated \\((d-1)\\)-manifold is generically
globally rigid in \\(\\mathbb R^{d}\\) if and only if the graph is \\((d+1
)\\)-connected and\, if \\(d=3\\)\, not planar.\n

\nThis proves and generalises a conjecture of Connelly. I will also discuss some applic ations of this result and of the techniques we use in the proof. We prove the generic case of a conjecture of Kalai on the reconstructability of a p olytope from its space of stresses. We also use our methods to generalise parts of the Lower Bound Theorem to a larger class of simplicial complexes .\n

\n\n\n** Some context for a general audience.**\n
\nA graph is said to be globally rigid in \\(\\mathbb R^d\\) if a generic
embedding of the vertex set in \\(\\mathbb R^d\\) is determined\, up to is
ometry of \\(\\mathbb R^d\\)\, by the distances between adjacent vertices.
There is a weaker local version of rigidity in which the embedding is on
ly determined within some neighbourhood. More detail\, and examples\, will
be given in the talk. \n

\nJackson and Jordán\, following e arlier work of Connelly\, have characterised graphs that are globally rigi d in \\(\\mathbb R^2\\) in terms of the 2-dimensional rigidity matroid. H owever extending this characterisation to higher dimensions is a very chal lenging open problem. Indeed there are very few examples known of natural ly interesting infinite families of graphs for which the global rigidity p roblem in \\(\\mathbb R^d\\)\, for \\(d\\geq 3\\)\, has been settled. \n < /p>\n

\nOne interesting family of graphs in this context are those aris
ing as graphs of triangulations of manifolds. Fogelsanger showed that the
graph of a triangulated \\(d\\)-manifold is locally rigid in \\(\\mathbb
R^{d+1}\\)\, but the global rigidity problem for such graphs remained open
. Connelly conjectured that \\(4\\)-connected triangulations of non-spheri
cal surfaces are globally rigid in \\(\\mathbb R^3\\). In higher dimensio
ns\, even the global rigidity of graphs of simplicial polytopes remained a
n open question. Kalai\, Tay and others have used the local rigidity theo
ry of graphs to prove important results concerning face numbers of pseudom
anifolds\, but global rigidity theory remains relatively unexplored in tha
t context.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Potapov (University of Liverpool)
DTSTART;VALUE=DATE-TIME:20221026T150000Z
DTEND;VALUE=DATE-TIME:20221026T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/24
DESCRIPTION:Title: Reachability Problems in Matrix Semigroups\nby Igor
Potapov (University of Liverpool) as part of Selected Topics in Mathematic
s - Online Edition\n\n\nAbstract\nMatrices and matrix products play a cruc
ial role in the representation and analysis of various computational proce
sses\, The central decision problem in matrix semigroups is the membership
problem: "Decide whether a given matrix $M$ belongs to a finitely generat
ed matrix semigroup". By restricting $M$ to be the identity (zero) matrix
the problem is known as the identity (mortality) problem. \n\nUnfortunatel
y\, many simply formulated and elementary problems for matrices are inhere
ntly difficult to solve even in dimension two\, and most of these problems
become undecidable in general starting from dimension three or four. For
example\, the identity problem for $3\\times 3$ matrices of integers is th
e long-standing open problem.\n\nIn this talk I will provide an overview a
bout various decision problems in matrix semigroups such as membership\, v
ector reachability\, freeness\, scalar reachability\, etc. Also\, I will f
ocus on the number of state-of-the-art theoretical computer science techni
ques as well as decidability\, undecidability and computational complexity
results.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anton Lukyanenko (George Mason University)
DTSTART;VALUE=DATE-TIME:20221102T160000Z
DTEND;VALUE=DATE-TIME:20221102T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/25
DESCRIPTION:Title: Heisenberg continued fractions: overview and recent resu
lts\nby Anton Lukyanenko (George Mason University) as part of Selected
Topics in Mathematics - Online Edition\n\n\nAbstract\nContinued fraction
theory over the real numbers has a long connection to real hyperbolic geom
etry.\nAbout 10 years ago\, Joseph Vandehey and I proposed a new CF algori
thm over the non-commutative\nHeisenberg group\, which is designed to take
advantage of complex hyperbolic theory\,\nand connects directly to the wo
rk of Falbel-Francsics-Lax-Parker\, Hersonsky-Paulin\, Series\, \nKatok-Ug
arkovici\, Nakada\, Hensley\, and others.\n\nWe have since connected the t
heory to Diophantine approximation\, established ergodicity\nof the Gauss-
type map (for a folded variant of the CF)\, and developed a broader framew
ork of\nIwasawa CFs\, which include many real\, higher-dimensional\, and n
on-commutative CF algorithms.\nMore recently\, we returned to the Euclidea
n setting to explore the dynamics of CFs over the complex \nnumbers\, quat
ernions\, octonions\, as well as defining new CF algorithms in \\(\\mathbb
{R}^3\\).\n\nIn this talk\, I will start by discussing the A. Hurwitz comp
lex CFs as a motivating higher-dimensional algorithm\,\nthen discuss the H
eisenberg group and Heisenberg CFs\, and then provide an overview of my wo
rk with\nVandehey\, finishing with this year's results in Euclidean space.
\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Ovsienko (University of Reims Champagne-Ardenne)
DTSTART;VALUE=DATE-TIME:20221116T160000Z
DTEND;VALUE=DATE-TIME:20221116T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/26
DESCRIPTION:Title: Shadows of numbers: supergeometry with a human face\
nby Valentin Ovsienko (University of Reims Champagne-Ardenne) as part of S
elected Topics in Mathematics - Online Edition\n\n\nAbstract\nIn this elem
entary and accessible to everybody talk\, I will explain an attempt to app
ly supersymmetry and supergeometry to arithmetic. The following general id
ea looks crazy. What if every integer sequence has another integer sequenc
e that follows it like a shadow? I will demonstrate that this is indeed th
e case\, though perhaps not for every integer sequence\, but for many of t
hem. The main examples are those of the Markov numbers and Somos sequences
.\n\nIn the second part of the talk\, I will discuss the notions of supers
ymmetric continued fractions and the modular group\, and arrive at yet a m
ore crazy idea that every rational and every irrational has its own shadow
. The second part of the talk is a joint work with Charles Conley.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karin Baur (University of Leeds)
DTSTART;VALUE=DATE-TIME:20221123T160000Z
DTEND;VALUE=DATE-TIME:20221123T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/27
DESCRIPTION:Title: Frieze patterns and cluster theory\nby Karin Baur (U
niversity of Leeds) as part of Selected Topics in Mathematics - Online Edi
tion\n\n\nAbstract\nCluster categories and cluster algebras can be describ
ed via triangulations of surfaces or via Postnikov diagrams. \nIn type A\
, such triangulations lead to frieze patterns or SL\\(_2\\)-friezes in the
sense of Conway and Coxeter. \nWe explain how infinite frieze patterns a
rise and how Grassmannians or Plücker coordinates give rise to SL\\(_k
\\)-friezes.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Raffaelli (TU Wien)
DTSTART;VALUE=DATE-TIME:20221207T160000Z
DTEND;VALUE=DATE-TIME:20221207T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/28
DESCRIPTION:Title: Curvature-adapted submanifolds of semi-Riemannian groups
\nby Matteo Raffaelli (TU Wien) as part of Selected Topics in Mathemat
ics - Online Edition\n\n\nAbstract\nGiven a semi-Riemannian hypersurface $
M$ of a semi-Riemannian manifold $Q$\, one says that $M$ is $\\textit{curv
ature-adapted}$ if\, for each $p \\in M$\, the normal Jacobi operator and
the shape operator of $M$ at $p$ commute. The first operator measures the
curvature of the ambient manifold along the normal vector of $M$\, whereas
the second describes the curvature of $M$ as a submanifold of $Q$. This c
ondition can be generalized to submanifolds of arbitrary codimension.\n\nI
n this talk I will present joint work with Margarida Camarinha addressing
the case where the ambient manifold is a Lie group equipped with a bi-inva
riant metric. In particular\, we will see that\, if the normal bundle of $
M$ is $\\textit{closed under the Lie bracket}$ (i.e.\, if each normal spac
e corresponds\, under the group's left action\, to a Lie subalgebra)\, the
n curvature adaptedness can be understood geometrically\, in terms of left
translations. Incidentally\, our analysis offers a new case-independent p
roof of a well-known fact: every three-dimensional Lie group equipped with
a bi-invariant semi-Riemannian metric has constant curvature.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Blackman (University of Liverpool)
DTSTART;VALUE=DATE-TIME:20221214T160000Z
DTEND;VALUE=DATE-TIME:20221214T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/29
DESCRIPTION:Title: Cutting Sequences and the p-adic Littlewood Conjecture\nby John Blackman (University of Liverpool) as part of Selected Topics
in Mathematics - Online Edition\n\n\nAbstract\nOne of the main themes of D
iophantine approximation is the study of how well real numbers can be appr
oximated by rational numbers. Classically\, a real number is defined to be
well-approximable if the Markov constant is 0\, i.e. \\(M(x):=\\lim \\inf
{q||qx||}=0\\). Otherwise\, the number is badly approximable\, with large
r values of \\(M(x)\\) indicating worse rates of approximation. As a sligh
t twist on this notion of approximability\, the \\(p\\)-adic Littlewood Co
njecture asks if -- given a prime \\(p\\) and a badly approximable number
\\(x\\) -- one can always find a subsequence of \\(xp^k\\) such that the M
arkov constant of this sequence tends to \\(0\\)\, i.e. if \\(\\lim \\inf
{M(xp^k)} =0\\).\n\nIn this talk\, I will outline a brief history of the \
\(p\\)-adic Littlewood Conjecture and discuss how hyperbolic geometry can
be used to help understand the problem further. In particular\, I will dis
cuss how one can represent integer multiplication of continued fractions b
y replacing one triangulation of the hyperbolic plane with an alternative
triangulation. Finally\, I will give a reformulation of pLC using infinite
loops -- a family of objects that arise from this setting.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andy Hone (University of Kent)
DTSTART;VALUE=DATE-TIME:20230208T160000Z
DTEND;VALUE=DATE-TIME:20230208T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/30
DESCRIPTION:Title: Continued fractions from hyperelliptic curves\nby An
dy Hone (University of Kent) as part of Selected Topics in Mathematics - O
nline Edition\n\n\nAbstract\nWe consider a family of nonlinear maps that a
re generated from the continued fraction expansion of a function on a hype
relliptic curve of genus \\(g\\)\, as originally described by van der Poor
ten. Using the connection with the classical theory of \\(J\\)-fractions a
nd orthogonal polynomials\, we show that in the simplest case \\(g=1\\) th
is provides a straightforward derivation of Hankel determinant formulae fo
r the terms of a general Somos-\\(4\\) sequence\, which were found in part
icular cases by Chang\, Hu\, and Xin. We extend these formulae to the high
er genus case\, and prove that generic Hankel determinants in genus \\(2\\
) satisfy a Somos-\\(8\\) relation. Moreover\, for all \\(g\\) we show tha
t the iteration for the continued fraction expansion is equivalent to a di
screte Lax pair with a natural Poisson structure\, and the associated nonl
inear map is a discrete integrable system\, connected with solutions of th
e infinite Toda lattice. If time permits\, we will also mention the link t
o (Stieltjes) \\(S\\)-fractions via contraction\, and a family of maps ass
ociated with the Volterra lattice\, described in current joint work with J
ohn Roberts and Pol Vanhaecke.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Iskander Aliev (Cardiff University)
DTSTART;VALUE=DATE-TIME:20230301T160000Z
DTEND;VALUE=DATE-TIME:20230301T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/31
DESCRIPTION:Title: Sparse integer points in rational polyhedra: bounds for
the integer Caratheodory rank\nby Iskander Aliev (Cardiff University)
as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\n
We will give an overview of the recent results on sparse integer points (t
hat is\, the integer points with a relatively large number of zero coordin
ates) in the rational polyhedra of the form \\(\\{x: Ax=b\, x\\geq 0\\}\\)
\, where \\(A\\) is an integer matrix\, and \\(b\\) is an integer vector.
In particular\, we will discuss the bounds on the Integer Caratheodory ran
k in various settings and proximity/sparsity transference results.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikolay Moshchevitin (Israel Institute of Technology (Technion))
DTSTART;VALUE=DATE-TIME:20230308T160000Z
DTEND;VALUE=DATE-TIME:20230308T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/32
DESCRIPTION:Title: Geometry of Diophantine Approximation\nby Nikolay Mo
shchevitin (Israel Institute of Technology (Technion)) as part of Selected
Topics in Mathematics - Online Edition\n\n\nAbstract\nWe discuss some cla
ssical and modern results related to the geometry of Diophantine Approxima
tion\, in particular some multidimensional generalizations of continued fr
actions algorithm related to patterns of the best approximations. Importan
t tools for the study of the properties of approximations are related to i
rrationality measure functions. We will give a brief introduction into the
theory and explain a recent conjecture by Schmidt and Summerer and its so
lution.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bob Connelly (Cornell University)
DTSTART;VALUE=DATE-TIME:20230315T160000Z
DTEND;VALUE=DATE-TIME:20230315T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/33
DESCRIPTION:Title: Global Rigidity of Braced Convex Polygons\nby Bob Co
nnelly (Cornell University) as part of Selected Topics in Mathematics - On
line Edition\n\n\nAbstract\nA framework in the plane is globally rigid if
any other realization of the framework with corresponding edges the same l
ength is congruent. For example\, a collection of triangles placed end-to
-end without overlap such that a bar connecting the first triangle to the
last\, intersecting the interior of each triangle\, is globally rigid. W
e would like to tell when a convex polygon with braces inside connecting t
he vertices so that for \\(n\\) vertices there are \\(n-2\\) internal bra
ces\, then this framework is always globally rigid. But we can’t do tha
t yet. However\, we have some interesting classes of braced convex polygo
nal frameworks that are always globally rigid.\n\n \n\nThis is joint work
with Bob Connelly\, Bill Jackson\, Shin-ichi Tanagawa\, and Albert Zhen\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Garrity (Williams College)
DTSTART;VALUE=DATE-TIME:20230322T160000Z
DTEND;VALUE=DATE-TIME:20230322T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/34
DESCRIPTION:Title: On Partition Numbers and Multi-dimensional Continued Fra
ctions\nby Thomas Garrity (Williams College) as part of Selected Topic
s in Mathematics - Online Edition\n\n\nAbstract\nThis talk will link parti
tion numbers from combinatorics with a certain multi-dimensional continued
fraction algorithm from number theory and dynamical systems.\n\nAndrew an
d Eriksson's Introduction to Integer Partitions starts with discussing Eul
er's identity\, *Every number has as many integer partitions into odd p
arts as into distinct parts*. As they state\, this is quite surprising
if you have never seen it before. There are\, though\, many other equally
if not more surprising partition identities. For all there are two basic
questions. First\, how to even guess the existence of any potential partit
ion identities. Then\, once a possible potential identity is conjectured\,
how to prove it.\n\nIn joint work with Bonanno\, Del Vigna and Isola\, th
ere was developed a link between traditional continued fractions and the s
low triangle map (a type of multi-dimensional continued fraction algorithm
) with integer partitions of numbers into two or three distinct parts\, wi
th multiplicity. These maps were initially introduced for number theoretic
reasons but have over the years exhibited many interesting dynamical prop
erties. In work with Wael Baalbaki\, we will see that the slow triangle ma
p\, when extended to higher dimensions\, will provide a natural map (an al
most internal symmetry) from the set of integer partitions to itself.\n\nT
hus we will allow us to create a new technique for generating any number o
f partition identities.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pavel Tumarkin (Durham University)
DTSTART;VALUE=DATE-TIME:20230329T150000Z
DTEND;VALUE=DATE-TIME:20230329T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/35
DESCRIPTION:Title: Farey graph and ideal tetrahedra\nby Pavel Tumarkin
(Durham University) as part of Selected Topics in Mathematics - Online Edi
tion\n\n\nAbstract\nWe construct a 3-dimensional analog of the Farey tesse
lation and show that it inherits many properties of the usual 2-dimensiona
l Farey graph. As a by-product\, we get a 3-dimensional counterpart of the
Ptolemy relation. The talk is based on an ongoing work joint with Anna Fe
likson\, Oleg Karpenkov and Khrystyna Serhiyenko.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Egon Schulte (Northeastern University)
DTSTART;VALUE=DATE-TIME:20230405T150000Z
DTEND;VALUE=DATE-TIME:20230405T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/36
DESCRIPTION:Title: Skeletal Polyhedral Geometry and Symmetry\nby Egon S
chulte (Northeastern University) as part of Selected Topics in Mathematics
- Online Edition\n\n\nAbstract\nThe study of highly symmetric structures
in Euclidean \\(3\\)-space has a long and fascinating history tracing back
to the early days of geometry. With the passage of time\, various notions
of polyhedral structures have attracted attention and have brought to lig
ht new exciting figures intimately related to finite or infinite groups of
isometries. A radically different\, skeletal approach to polyhedra was pi
oneered by Grunbaum in the 1970's building on Coxeter's work. A polyhedron
is viewed not as a solid but rather as a finite or infinite periodic geom
etric edge graph in space equipped with additional polyhedral super-struct
ure imposed by the faces. Since the mid 1970's there has been a lot of act
ivity in this area. Much work has focused on classifying skeletal polyhedr
a and complexes by symmetry\, with the degree of symmetry defined via dist
inguished transitivity properties of the geometric symmetry groups. These
skeletal figures exhibit fascinating geometric\, combinatorial\, and algeb
raic properties and include many new finite and infinite structures.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valérie Berthé (IRIF)
DTSTART;VALUE=DATE-TIME:20230412T150000Z
DTEND;VALUE=DATE-TIME:20230412T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/37
DESCRIPTION:Title: Balanced words and symbolic dynamical systems\nby Va
lérie Berthé (IRIF) as part of Selected Topics in Mathematics - Online E
dition\n\n\nAbstract\nThe chairman assignment problem can be stated as fol
lows: \\(k\\) states are assumed to form a union and each year a union cha
irman must be selected so that at any time the cumulative number of chairm
en of each state is proportional to its weight.\nIt is closely related to
the (discrete) apportionment problem\, which has its origins in the questi
on of allocating seats in the house of representatives in the United State
s\, in a proportional way to the population of each state.\nThe richness o
f this problem lies in the fact that it can be reformulated both as a sequ
encing problem in operations research for optimal routing and scheduling\,
and as a symbolic discrepancy problem\, in the field of word combinatoric
s\, where the discrepancy measures the difference between the number of oc
currences of a letter in a prefix of an infinite word and the expected val
ue in terms of frequency of occurrence of this letter. \nWe will see in th
is lecture how to construct infinite words with values in a finite alphabe
t having the smallest possible discrepancy\, by revisiting a construction
due to R. Tijdeman in terms of dynamical systems.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jörg Thuswaldner (Montanuniversität Leoben)
DTSTART;VALUE=DATE-TIME:20230419T150000Z
DTEND;VALUE=DATE-TIME:20230419T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/38
DESCRIPTION:Title: Multidimensional continued fractions and symbolic coding
s of toral translations\nby Jörg Thuswaldner (Montanuniversität Leob
en) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstra
ct\nThe aim of this lecture is to find symbolic codings for translations o
n the $d$-dimensional torus that enjoy many of the well-known properties o
f Sturmian sequences (like low complexity\, balance of factors\, bounded r
emainder sets of any scale). Inspired by the approach of G. Rauzy we const
ruct such codings by the use of multidimensional continued fraction algori
thms that are realized by sequences of substitutions. This is joint work w
ith V. Berthé and W. Steiner.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Gekhtman (University of Notre Dame)
DTSTART;VALUE=DATE-TIME:20230510T150000Z
DTEND;VALUE=DATE-TIME:20230510T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/40
DESCRIPTION:Title: Unified approach to exotic cluster structures in simple
Lie groups\nby Michael Gekhtman (University of Notre Dame) as part of
Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nWe present
a construction for cluster charts in simple Lie groups compatible with Poi
sson structures in the Belavin-Drinfeld classification. The key ingredient
is a birational Poisson map from the group to itself that transform a Poi
sson bracket associated with a nontrivial Belavin-Drinfeld data into the s
tandard one. It allows us to obtain a cluster chart as a pull-back of the
Berenstein-Fomin-Zelevinsky cluster coordinates on the open double Bruhat
cell. This is a joint work with M. Shapiro and A. Vainshtein.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Uricchio (WPI)
DTSTART;VALUE=DATE-TIME:20230426T150000Z
DTEND;VALUE=DATE-TIME:20230426T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/41
DESCRIPTION:by Nathan Uricchio (WPI) as part of Selected Topics in Mathema
tics - Online Edition\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ekaterina Shemyakova (University of Toledo)
DTSTART;VALUE=DATE-TIME:20231018T150000Z
DTEND;VALUE=DATE-TIME:20231018T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/42
DESCRIPTION:Title: On super cluster algebras based on super Plü\;cker
and super Ptolemy relations\nby Ekaterina Shemyakova (University of To
ledo) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbst
ract\nI will speak about super exterior powers and our results on super an
alogs of Plü\;cker embedding for the Grassmann manifold.\n\nThe probl
em was motivated by the search for the definition of super cluster algebra
s. Based on the obtained super Plü\;cker relations (which we have for
the general case)\, we propose a super cluster structure for super Grass
mannians \\(\\mathrm{Gr}_{2|0}(n|1)\\). The exchange graph structure is n
ow understood. \n\nWe show how to simplify the super Plü\;cker relat
ions for \\(\\mathrm{Gr}_{r|1}(n|1)\\)\, which can be seen as dual to $\\
mathrm{Gr}_{n-r|0}(n|1)$. We also present how super Ptolemy relations of P
enner-Zeitlin for decorated super Teichmü\;ller space --- the basis o
f Musiker-Ovenhouse-Zhang's super cluster algebra definition --- can be re
-written as super Plü\;cker relations.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Derek Kitson (Mary Immaculate College)
DTSTART;VALUE=DATE-TIME:20231206T160000Z
DTEND;VALUE=DATE-TIME:20231206T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/43
DESCRIPTION:Title: Rigid graphs in dimension 3\nby Derek Kitson (Mary I
mmaculate College) as part of Selected Topics in Mathematics - Online Edit
ion\n\n\nAbstract\nA graph is rigid in \\(d\\)-\;dimensional Euclidean
space if there is an embedding of the vertices which admits no non-\;tr
ivial edge-\;length preserving continuous motion. Rigid graphs in dimen
sions \\(1\\) and \\(2\\) are characterised by simple counting rules\, but
currently no such rules are available in higher dimensional Euclidean spa
ces. We will provide a gentle introduction to graph rigidity and report on
recent progress in characterising rigid graphs for a class of cylindrical
normed spaces of dimension \\(3\\). This is joint work with Sean Dewar (U
niversity of Bristol).\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sophie Morier-Genoud (University of Reims)
DTSTART;VALUE=DATE-TIME:20231011T150000Z
DTEND;VALUE=DATE-TIME:20231011T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/44
DESCRIPTION:Title: q-Analogs of rational numbers and the Burau representati
on of the braid group B3.\nby Sophie Morier-Genoud (University of Reim
s) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstrac
t\nThe most popular \\(q\\)-\;analogs of numbers are certainly the \\(q
\\)-\;integers and the \\(q\\)-\;binomial coefficients of Gauss whic
h both appear in various areas of mathematics and physics. Classical seque
nces of integers often have interesting \\(q\\)-\;analogs. With Valenti
n Ovsienko we recently suggested a notion of \\(q\\)-\;analogs for rati
onal numbers. Our approach is based on combinatorial properties and contin
ued fraction expansions of the rationals. The subject can be developed in
connections with various topics such as enumerative combinatorics\, cluste
r algebras\, homological algebra\, knots invariants... I will give an over
view of the theory and present an application to the problem of classific
ation of faithful specialisations of the Burau representation of B3. This
last part is joint work with V. Ovsienko and A. Veselov.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fran Burstall (University of Bath)
DTSTART;VALUE=DATE-TIME:20231129T160000Z
DTEND;VALUE=DATE-TIME:20231129T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/45
DESCRIPTION:Title: Isothermic surfaces and Noether's theorem\nby Fran B
urstall (University of Bath) as part of Selected Topics in Mathematics - O
nline Edition\n\n\nAbstract\nIsothermic surfaces were intensively studied
in the late 19^{th} century and have seen a recent revival of inte
rest due to links with soliton theory. In this talk\, I will describe this
classical theory and the modern integrable systems approach via a pencil
of flat connections. I will explain how this connections arise from a vari
ational characterisation of isothermic surfaces\, due to Bohle-Peters-Pink
all\, together with Noether's theorem. This gives a puzzling link to the c
onservations laws for CMC surfaces discovered by Korevaar-Kusner-Solomon.\
n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladlen Timorin (Higher School of Economics)
DTSTART;VALUE=DATE-TIME:20231213T160000Z
DTEND;VALUE=DATE-TIME:20231213T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/46
DESCRIPTION:Title: Aperiodic points for outer billiards\nby Vladlen Tim
orin (Higher School of Economics) as part of Selected Topics in Mathematic
s - Online Edition\n\n\nAbstract\nThis is a joint project with A. Kanel
5\;Belov\, Ph. Rukhovich\, and V. Zgurskii. A Euclidean outer billiard on
a convex figure in the plane is the map sending a point outside the figure
to the other endpoint of a segment touching the figure at the middle. Ite
rating such a process was suggested by J. Moser as a crude model of planet
ary motion. Polygonal outer billiards are arguably the principal examples
of Euclidean piecewise rotations\, which serve as a natural generalization
of interval exchange maps. They also found applications in electrical eng
ineering. Previously known rigorous results on outer billiards on regular
\\(N\\)-\;polygons are\, apart from trivial

cases of \\(N=3\,4\,
6\\)\, based on dynamical self-\;similarities (this approach was origin
ated by S. Tabachnikov). Dynamical self-\;similarities have been found
so far only for \\(N=5\,7\,8\,9\,10\,12\\). In his ICM 2022 address\, R. S
chwartz asked whether outer billiard on the regular \\(N\\)-\;gon ha
s an aperiodic orbit if \\(N\\) is not \\(3\\)\, \\(4\\)\, \\(6\\)

. We
answer this question in affirmative for \\(N\\) not divisible by \\(4\\).
Our methods are not based on self-\;similarity. Rather\, scissor congr
uence invariants (including that of Sah-\;Arnoux-\;Fathi) play a key
role.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Kleptsyn (Université de Rennes)
DTSTART;VALUE=DATE-TIME:20231108T160000Z
DTEND;VALUE=DATE-TIME:20231108T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/47
DESCRIPTION:Title: From the percolation theory to Fuchsian equations and Ri
emann-\;Hilbert problem\nby Victor Kleptsyn (Université de Rennes)
as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\
nConsider the critical percolation problem on the hexagonal lattice: each
of (tiny) hexagons is independently declared «\; open »\; or &la
quo\; closed »\; with probability (\\(1/2\\)) &mdash\; by a fair coin
tossing. Assume that on the boundary of a simply connected domain four po
ints A\,B\,C\,D are marked. Then either there exists an «\; open &raq
uo\; path\, joining AB and CD\, or there is a «\; closed »\; pat
h\, joining AD and BC (one can recall the famous «\; Hex »\; gam
e here).\n\n\n\n

\n\nCardy's formula\, rigorously proved by S. S mirnov\, gives an explicit value of the limit of such percolation probabil ity\, when the same smooth domain is put onto lattices with smaller and sm aller mesh. Though\, a next natural question is: what if more than four po ints are marked? And thus that there are more possible configurations of o pen/closed paths joining the arcs? \n\n\n

\n\n\nIn our joint work
with M. Khristoforov we obtain the answer as an explicit integral for the
case of six marked points on the boundary\, passing through Fuchsian diffe
rential equations\, Riemann surfaces\, and Riemann-\;Hilbert problem. W
e also obtain a generalisation of this answer to the case when one of the
marked points is inside the domain (and not on the boundary).\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Iván Rasskin (Aix-Marseille University)
DTSTART;VALUE=DATE-TIME:20231115T160000Z
DTEND;VALUE=DATE-TIME:20231115T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/48
DESCRIPTION:Title: On the arithmetic and geometric properties of regular po
lytopal sphere packings and their connection to knot theory\nby Iván
Rasskin (Aix-Marseille University) as part of Selected Topics in Mathemati
cs - Online Edition\n\n\nAbstract\nhe Apollonian Circle Packing (ACP) is a
classic geometric fractal with diverse applications across various domain
s\, particularly in number theory. This is due to its ability to be realiz
ed as an integral packing\, where the curvatures of all the circles are in
tegers. The ACP is constructed iteratively\, beginning with an initial pac
king whose combinatorial structure is encoded by a tetrahedron. By changin
g the initial configuration\, the ACP can be generalized for any polyhedro
n. However\, not every polyhedron is integral in the sense that it can gen
erate an integral packing. Moreover\, in higher dimensions\, not every pol
ytope is crystallographic\, meaning that it can generate an Apollonian-lik
e sphere packing. In this talk\, we will study the case of regular polytop
es in any dimension to determine whether they are integral and crystallogr
aphic. Additionally\, we will explore how the symmetry inherent in the pol
ytope can be leveraged to extract special cross-sections of the packings.
Furthermore\, we will demonstrate how a specific section of an orthoplicia
l/hyperoctahedral Apollonian sphere packing can be utilized as a geometric
framework to establish an upper bound on a knot invariant for rational li
nks.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Herman Servatius (Worcester Polytechnic Institute)
DTSTART;VALUE=DATE-TIME:20231101T160000Z
DTEND;VALUE=DATE-TIME:20231101T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/49
DESCRIPTION:Title: Rigidity and movability of configurations in the project
ive plane\nby Herman Servatius (Worcester Polytechnic Institute) as pa
rt of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nConfi
gurations of points and lines in the plane have a long history. The Theore
m of Pappus from the fourth century begins a classical theory that has bee
n advanced by Desargues\, Pascal\, Cayley\, Steinitz\, Grassman and many o
thers. The study of such objects and their generalizations has deep roots
in algebra\, geometry\, topology and combinatorics.\n\nIn this talk we dis
cuss recent work which is the result of regarding these classical structur
es as geometric constraint systems. The objects of interest then become th
e topology\, geometry\, and parameterizations of the space of realizations
of a configuration. Some of the tools derive from those developed by civi
l and mechanical engineers in the analysis of the statics of structures an
d the kinematics of linkages.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pär Kurlberg (KTH)
DTSTART;VALUE=DATE-TIME:20231004T150000Z
DTEND;VALUE=DATE-TIME:20231004T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/50
DESCRIPTION:Title: Repulsion in number theory and physics\nby Pär Kurl
berg (KTH) as part of Selected Topics in Mathematics - Online Edition\n\n\
nAbstract\nZeros of the Riemann zeta function and eigenvalues of quantized
chaotic Hamiltonians appears to have something in common. Namely\, they
both seem to be ruled by random matrix theory and consequently should exhi
bit "repulsion" in the sense that small gaps between elements are very rar
e. More mysteriously\, while zeros of different L-\;functions (i.e.\,
generalizations of the Riemann zeta function) are "mostly independent" the
y also exhibit subtle repulsion effects on zeros of **other** L-\;fu
nctions.\n\nWe will give a survey of the above phenomena. Time permitting
we will also discuss repulsion between eigenvalues of "arithmetic Seba bi
lliards"\, a certain singular perturbation of the Laplacian on the 3D toru
s \\(R^3/Z^3\\). The perturbation is weak enough to allow for arithmetic
features from the unperturbed system to be brought into play\, yet strong
enough to *probably* induce repulsion.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Shallit (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20231122T160000Z
DTEND;VALUE=DATE-TIME:20231122T170000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/51
DESCRIPTION:Title: Proving results in combinatorics on words and number the
ory using a decidable logic theory\nby Jeffrey Shallit (University of
Waterloo) as part of Selected Topics in Mathematics - Online Edition\n\n\n
Abstract\nDavid Hilbert's dream\, of a deterministic finite procedure that
could decide if a given theorem statement is true or false\, was killed o
ff by Gä\;del and Turing. Nevertheless\, there are some logical theori
es\, such as Presburger arithmetic\, that are decidable.\n\nIn this talk I
will discuss one such theory\, Bü\;chi arithmetic\, and its implement
ation in a computer system called Walnut. With this free software\, one ca
n prove non-trivial theorems in combinatorics on words and number theory\;
it suffices to state the desired result in first-order logic\, type it in
to the system\, and wait. So far\, it has been used in over 70 papers in t
he literature\, and has even detected errors in some published papers.\n\n
My recent book\, *The Logical Approach to Automatic Sequences*\, pu
blished by Cambridge University Press\, has more details.\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Esterov (London Institute for Mathematical Sciences)
DTSTART;VALUE=DATE-TIME:20231025T150000Z
DTEND;VALUE=DATE-TIME:20231025T160000Z
DTSTAMP;VALUE=DATE-TIME:20231130T084410Z
UID:SelectedTopics-Liverpool/52
DESCRIPTION:by Alexander Esterov (London Institute for Mathematical Scienc
es) as part of Selected Topics in Mathematics - Online Edition\n\nAbstract
: TBA\n
LOCATION:https://researchseminars.org/talk/SelectedTopics-Liverpool/52/
END:VEVENT
END:VCALENDAR