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CALSCALE:GREGORIAN
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BEGIN:VEVENT
SUMMARY:Sergey Avvakumov (University of Copenhagen\, Denmark)
DTSTART:20201202T131000Z
DTEND:20201202T135000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/1
DESCRIPTION:Title: A subexponential size $\\mathbb{R}P^N$\nby Sergey Av
vakumov (University of Copenhagen\, Denmark) as part of Combinatorics and
Geometry Days III\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pablo Soberon (Baruch College\, City University of New York\, US)
DTSTART:20201202T135500Z
DTEND:20201202T143500Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/2
DESCRIPTION:by Pablo Soberon (Baruch College\, City University of New York
\, US) as part of Combinatorics and Geometry Days III\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Garber (University of Texas Rio Grande Valley\, US)
DTSTART:20201202T144000Z
DTEND:20201202T152000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/3
DESCRIPTION:Title: Helly numbers for crystals and cut-and-project sets\
nby Alexey Garber (University of Texas Rio Grande Valley\, US) as part of
Combinatorics and Geometry Days III\n\n\nAbstract\nIn this talk I'll intro
duce Helly numbers for (discrete) point sets and explain why finite Helly
numbers exist for periodic and certain quasiperiodic sets in Euclidean spa
ce of any dimension though the bounds in the latter case seem to be extrem
ely non-optimal. I'll also show that for a wider class of Meyer sets Helly
numbers could be infinite.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Attila Jung (Institute of Mathematics\, ELTE Eötvös Loránd Univ
ersity\, Hungary)
DTSTART:20201202T160000Z
DTEND:20201202T164000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/4
DESCRIPTION:Title: Quantitative Fractional Helly and (p\,q)-Theorems\nb
y Attila Jung (Institute of Mathematics\, ELTE Eötvös Loránd University
\, Hungary) as part of Combinatorics and Geometry Days III\n\nAbstract: TB
A\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:József Solymosi (University of British Columbia\, Canada)
DTSTART:20201203T130000Z
DTEND:20201203T134500Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/5
DESCRIPTION:Title: Planar point sets with many similar triangles\nby J
ózsef Solymosi (University of British Columbia\, Canada) as part of Combi
natorics and Geometry Days III\n\n\nAbstract\nElekes and Erdős proved tha
t for any triangle $T$\, there are $n$-element planar point sets with $\\O
mega(n^2)$ triangles similar to $T$. It was proved shortly after that if t
he number of equilateral triangles is at least $(1/6 + \\epsilon)n^2$ then
the pointset should contain large parts of a triangular lattice. On the o
ther hand\, no lattice is guaranteed for $cn^2$ similar copies if $c < 1/6
$. We will show that one can still expect some structural results for tria
ngles\, even if the number of similar copies is as low as $n^{11/6 + \\eps
ilon}$.\n\n\nJoint work with Dhruv Mubayi.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivér Janzer (ETH Zürich\, Switzerland)
DTSTART:20201203T135000Z
DTEND:20201203T143500Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/6
DESCRIPTION:Title: On the Zarankiewicz problem for graphs with bounded VC-d
imension\nby Olivér Janzer (ETH Zürich\, Switzerland) as part of Com
binatorics and Geometry Days III\n\n\nAbstract\nThe problem of Zarankiewic
z asks for the maximum number of edges in a bipartite graph on $n$ vertice
s which does not contain the complete bipartite graph $K_{k\,k}$ as a subg
raph. A classical theorem due to Kővári\, Sós and Turán says that this
number of edges is $O(n^{2 - 1/k})$. An important variant of this problem
is the analogous question in bipartite graphs with VC-dimension at most $
d$\, where $d$ is a fixed integer such that $k\\ge d\\ge 1$. A remarkable
result of Fox\, Pach\, Sheffer\, Suk and Zahl with multiple applications i
n incidence geometry shows that\, under this additional hypothesis\, the n
umber of edges in a bipartite graph on $n$ vertices and with no copy of $K
_{k\,k}$ as a subgraph must be $O(n^{2 - 1/d})$. This theorem is sharp whe
n $d=2$\, because any $K_{2\,2}$-free graph has VC-dimension at most $2$\,
and there are well-known examples of such graphs with $\\Omega(n^{3/2})$
edges. However\, it turns out this phenomenon no longer carries through fo
r any larger $d$.\n\n\nWe show the following improved result: the maximum
number of edges in bipartite graphs with no copies of $K_{k\, k}$\nand VC-
dimension at most $d$ is $o(n^{2 - 1/d})$\, for every $k \\ge~d~\\ge~3.$ \
n\n\nJoint work with Cosmin Pohoata.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Géza Tóth (Rényi Institute of Mathematics\, Hungary)
DTSTART:20201203T144500Z
DTEND:20201203T153000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/7
DESCRIPTION:Title: Crossings between non-homotopic edges\nby Géza Tót
h (Rényi Institute of Mathematics\, Hungary) as part of Combinatorics and
Geometry Days III\n\n\nAbstract\nWe call a multigraph non-homotopic if it
can be drawn in the plane in such a way that no two edges connecting the
same pair of vertices can be continuously transformed into each other with
out passing through a vertex\, and no loop can be shrunk to its end-vertex
in the same way. It is easy to see that a non-homotopic multigraph on $n
> 1$ vertices can have arbitrarily many edges. We prove that the number of
crossings between the edges of a non-homotopic multigraph with $n$ vertic
es and $m > 4n$ edges is larger than $c \\frac{m^2}{n}$\nfor some constant
$C > 0$\, and that this bound is tight up to a polylogarithmic factor. We
also show that the lower bound is not asymptotically sharp as $n$ is fixe
d and $m \\rightarrow \\infty$.\n\n\nJoint work with János Pach and Gábo
r Tardos.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Tsiutsiurupa (MIPT\, Russia)
DTSTART:20201202T164500Z
DTEND:20201202T172500Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/8
DESCRIPTION:Title: Functional Lowner Ellipsoid\nby Igor Tsiutsiurupa (M
IPT\, Russia) as part of Combinatorics and Geometry Days III\n\n\nAbstract
\nWe extend the notion of the smallest volume ellipsoid containing a conve
x body in to the setting of logarithmically concave functions. We consider
a vast class of logarithmically concave functions whose superlevel sets a
re concentric ellipsoids. For a fixed function from this class\, we consid
er the set of all its "affine" positions. For any log-concave function f o
n R^d\, we consider functions belonging to this set of "affine" positions\
, and find the one with the smallest integral under the condition that it
is pointwise greater than or equal to f. We study the properties of existe
nce and uniqueness of the solution to this problem. Finally\, extending th
e notion of the outer volume ratio\, we define the outer integral ratio of
a log-concave function and give an asymptotically tight bound on it. \n\n
Joint work with Grigory Ivanov.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elisabeth Werner (Case Western Reserve University\, US)
DTSTART:20201202T173000Z
DTEND:20201202T181000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/9
DESCRIPTION:Title: Convex Floating Bodies of Equilibrium\nby Elisabeth
Werner (Case Western Reserve University\, US) as part of Combinatorics and
Geometry Days III\n\n\nAbstract\nWe study a long standing open problem by
Ulam\, which is whether the Euclidean ball is the unique body of uniform
density which will float in equilibrium in any direction. We answer this p
roblem in the class of origin symmetric n-dimensional convex bodies whose
relative density to water is 1/2. For n=3\, this result is due to Falconer
.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rom Pinchasi (Technion\, Israel)
DTSTART:20201203T153500Z
DTEND:20201203T162000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/10
DESCRIPTION:Title: Conics and Small Number of Distinct Directions\nby
Rom Pinchasi (Technion\, Israel) as part of Combinatorics and Geometry Day
s III\n\n\nAbstract\nGiven a set $P$ of $n$ points in general position in
the plane\, Jamison conjectured in 1986 that if $P$ determines at most $m
\\le 2n - C$ distinct directions\, then $P$ is contained in the set of ver
tices of a regular $m$-gon. Jamison verified his conjecture for $m = n$ an
d recently the case $m = n + 1$ was proved by Pilatte. We will discuss thi
s conjecture and prove it in the case $m \\le n + O(\\sqrt{n})$. \n\n\nBas
ed on joint works with Mehdi Makhul and with Alexandr Polyanskii.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Litvak (University of Alberta\, Canada)
DTSTART:20201203T163000Z
DTEND:20201203T171000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/11
DESCRIPTION:Title: A remark on the minimal dispersion\nby Alexander Li
tvak (University of Alberta\, Canada) as part of Combinatorics and Geometr
y Days III\n\n\nAbstract\nWe improve known upper bounds for the minimal di
spersion of a point set in the unit cube and its inverse. Some of our boun
ds are sharp up to logarithmic factors.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philippe Moustrou (UiT – The Arctic University of Norway)
DTSTART:20201204T120000Z
DTEND:20201204T124000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/12
DESCRIPTION:Title: Exact semidefinite programming bounds for packing probl
ems\nby Philippe Moustrou (UiT – The Arctic University of Norway) as
part of Combinatorics and Geometry Days III\n\n\nAbstract\nIn the first p
art of the talk\, we present how semidefinite programming methods can prov
ide upper bounds for various geometric packing problems\, such as kissing
numbers\, spherical codes\, or packings of spheres into a larger sphere. W
hen these bounds are sharp\, they give additional information on optimal c
onfigurations\, that may lead to prove the uniqueness of such packings. Fo
r example\, we show that the lattice E8 is the unique solution for the kis
sing number problem on the hemisphere in dimension 8.\n\nHowever\, semidef
inite programming solvers provide approximate solutions\, and some additio
nal work is required to turn them into an exact solution\, giving a certif
icate that the bound is sharp. In the second part of the talk\, we explain
how\, via our rounding procedure\, we can obtain an exact rational soluti
on of semidefinite program from an approximate solution in floating point
given by the solver.\n\n\nJoint work with Maria Dostert and David de Laat.
\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Dostert (KTH Royal Institute of Technology\, Stockholm)
DTSTART:20201204T124500Z
DTEND:20201204T132500Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/13
DESCRIPTION:Title: Semidefinite programming bounds for the average kissing
number\nby Maria Dostert (KTH Royal Institute of Technology\, Stockho
lm) as part of Combinatorics and Geometry Days III\n\n\nAbstract\nThe aver
age kissing number of $\\mathbb{R}^n$ is the supremum of the average degre
es of contact graphs of packings of finitely many balls (of any radii) in
$\\mathbb{R}^n$. In this talk I will provide an upper bound for the averag
e kissing number based on semidefinite programming that improves previous
bounds in dimensions 3\, ...\, 9. A very simple upper bound for the averag
e kissing number is twice the kissing number\; in dimensions 6\, ...\, 9 o
ur new bound is the first to improve on this simple upper bound. \n\n\nJoi
nt work with Alexander Kolpakov and Fernando Mário de Oliveira Filho.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Krupp (University of Cologne\, Germany)
DTSTART:20201204T133000Z
DTEND:20201204T141000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/14
DESCRIPTION:Title: Calculating the EHZ Capacity of Polytopes\nby Stefa
n Krupp (University of Cologne\, Germany) as part of Combinatorics and Geo
metry Days III\n\n\nAbstract\nConsider the Euclidean space $R^n$ with an e
ven dimension n. Equipped with a nondegenerate and alternating (sometimes
called skew-symmetric) bilinear form this is referred to as a symplectic s
pace. Symplectic spaces appear for instance if we express classical mechan
ics in a general way. The interest in the study of symplectic spaces arose
in the 1980s due to the celebrated non-squeezing theorem by Gromov. In pa
rticular\, Gromov's result required the existence of symplectic invariants
\, so called symplectic capacities. A symplectic capacity maps a nonnegati
ve number to each subset of $R^n$ while fulfilling certain properties. By
now\, several families of such invariants have been found. However\, they
are notoriously hard to compute. In my talk I will introduce a specific sy
mplectic capacity\, i.e. the Ekeland-Hofer-Zehnder (EHZ) capacity\, restri
cted to polytopes. More precisely\, I will state a result by Abbondandolo
and Majer which formulates the EHZ capacity as an optimization problem. Af
terwards\, I will discuss this optimization problem in more detail as well
as strategies to solve it. \n\n\nJoint work with Daniel Rudolf (Ruhr-Univ
ersity Bochum).\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Kolpakov (University of Neuchâtel\, Switzerland)
DTSTART:20201204T141500Z
DTEND:20201204T145500Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/15
DESCRIPTION:Title: Space vectors forming rational angles: on a question of
J.H. Conway\nby Alexander Kolpakov (University of Neuchâtel\, Switze
rland) as part of Combinatorics and Geometry Days III\n\n\nAbstract\nWe cl
assify all sets of nonzero vectors in $\\mathbb{R}^3$ such that the angle
formed by each pair is a rational multiple of $\\pi$. The special case of
four-element subsets lets us classify all tetrahedra whose dihedral angles
are multiples of $\\pi$\, solving a 1976 problem of Conway and Jones: the
re are $2$ one-parameter families and $59$ sporadic tetrahedra\, all but t
hree of which are related to either the icosidodecahedron or the $B_3$ roo
t lattice. The proof requires the solution in roots of unity of a $W(D_6)$
-symmetric polynomial equation with $105$ monomials (the previous record w
as only $12$ monomials).\n\n\nJoint work with Kiran Kedlaya\, Bjorn Poonen
\, and Michael Rubinstein.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Dragnev (Purdue University Fort Wayne\, US)
DTSTART:20201204T153000Z
DTEND:20201204T161000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/16
DESCRIPTION:Title: Mastodon Theorem - 20 Years in the Making\nby Peter
Dragnev (Purdue University Fort Wayne\, US) as part of Combinatorics and
Geometry Days III\n\n\nAbstract\nThe Mastodon theorem (PD.\, D. Legg\, D.
Townsend\, 2002)\, establishes that the regular bi-pyramid (North and Sout
h poles\, and an equilateral triangle on the Equator) is the unique up to
rotation five-point configuration on the sphere that maximizes the product
of all mutual (Euclidean) distances.\n\nIn a joint work with Oleg Musin w
e generalize the Mastodon Theorem to $n+2$\npoints on $S^{n-1}$\, namely w
e characterize all stationary configurations\, and show that all local min
ima occur when a configuration splits in two orthogonal simplexes of $k$ a
nd $l$ vertices\, $k+l=n+2$ \, with global minimum attained when $k = l$ o
r $k = l + 1$ depending on the parity of $n$.\n\n\nJoint work with Oleg Mu
sin.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Musin (University of Texas Rio Grande Valley\, US)
DTSTART:20201204T161500Z
DTEND:20201204T165500Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/17
DESCRIPTION:Title: Optimal spherical configurations\, majorization and f-d
esigns\nby Oleg Musin (University of Texas Rio Grande Valley\, US) as
part of Combinatorics and Geometry Days III\n\n\nAbstract\nWe consider the
majorization (Karamata) inequality and minimums of the majorization (M-se
ts) for f-energy potentials of m-point configurations in a sphere. In part
icular\, we discuss the optimality of regular simplexes\, describe M-sets
with a small number of points\, define and discuss spherical f-designs.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Glazyrin (University of Texas Rio Grande Valley\, US)
DTSTART:20201204T170000Z
DTEND:20201204T174000Z
DTSTAMP:20260404T094700Z
UID:combgeo_days_three/18
DESCRIPTION:Title: Domes over curves\nby Alexey Glazyrin (University o
f Texas Rio Grande Valley\, US) as part of Combinatorics and Geometry Days
III\n\n\nAbstract\nA closed polygonal curve is called integral if it is c
omposed of unit segments. Kenyon's problem asks whether for every integral
curve\, there is a dome over this curve\, i.e. whether the curve is a bou
ndary of a polyhedral surface whose faces are equilateral triangles with u
nit edge lengths. In this talk\, we will give a necessary algebraic condit
ion when the curve is a quadrilateral\, thus giving a negative solution to
Kenyon's problem in full generality. We will then explain why domes exist
over a dense set of integral curves and give an explicit construction of
domes over all regular polygons. Finally\, we will formulate several open
questions related to the initial problem of Kenyon.\n
LOCATION:https://stable.researchseminars.org/talk/combgeo_days_three/18/
END:VEVENT
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