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|
| 30.10.2019,
14:10 (Wednesday) |
Orientation meeting for students
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|
| Location: |
Schreiber bldg., room 309, Tel-Aviv University |
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 |
|
|
06.11.2019, 14:10 (Wednesday)
|
Cheuk
Yu Mak (Cambridge, UK)
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|
|
Title:
|
Non-displaceable
Lagrangian links in four-manifolds |
| Location: |
Schreiber bldg., room 309, Tel-Aviv University |
|
|
| Abstract: |
One
of the earliest fundamental applications of Lagrangian Floer
theory is detecting the non-displaceablity of a Lagrangian submanifold.
Many progress and generalisations have been made since then but little
is known when the Lagrangian submanifold is disconnected. In this talk,
we describe a new idea to address this problem. Subsequently, we
explain
how to use Fukaya-Oh-Ohta-Ono and Cho-Poddar theory to show that for
every S2 × S2 with a non-monotone product
symplectic form, there
is a continuum of disconnected, non-displaceable Lagrangian
submanifolds
such that each connected component is displaceable. This is a joint
work
with Ivan Smith.
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SPECIAL ANNOUNCEMENT -
MATHEMATICAL PHYSICS SEMINAR |
|
| 07.11.2019, 16:10 (Thursday) |
Victor
Ivrii (U. Toronto)
|
|
|
| Title: |
Etudes
of Spectral Theory |
| Location: |
Schreiber bldg., room 309, Tel-Aviv University |
|
|
| Abstract: |
I
briefly describe five old but still actively explored problems of the
Spectral Theory of Partial Differential Equations
1. How eigenvalues are distributed (where eigenvalues often mean
squares
of the frequencies in the mechanical or
electromagnetic problems or
energy levels in the quantum mechanics models)
and the relation to
the behaviour of the billiard trajectories.
2. Equidistribution of eigenfunctions and connection to
ergodicity of
billiard trajectories (a quantum chaos
and a classical chaos).
3. Can one hear the shape of the drum?
4. Nodal lines and Chladni plates.
5. Strange spectra of quantum systems. |
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| 13.11.2019,
14:10 (Wednesday) |
NO SEMINAR THIS WEEK!
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20.11.2019,
14:10-15:00 (Wednesday)
|
Sara
Tukachinsky (IAS Princeton) |
|
|
| Title: |
Counts
of pseudoholomorphic curves: Definition, calculations, and more |
| Location: |
Schreiber bldg., room 309, Tel-Aviv University |
|
|
| Abstract: |
Genus
zero open Gromov-Witten (OGW) invariants should count
pseudoholomorphic disks in a symplectic manifold with boundary
conditions
in a Lagrangian submanifold, satisfying various constraints at boundary
and
interior marked points. In a joint work with Jake Solomon we developed
an
approach for defining OGW invariants using machinery from Fukaya
A∞
algebras. In a recent work, also joint with Solomon, we find that the
generating function of OGW satisfies a system of PDE called open WDVV
equations. This PDE translates to an associativity relation for a
quantum
product on the relative cohomology. For projective spaces, open WDVV
give
rise to recursions that, together with other properties, allow the
computation of all OGW invariants. |
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|
|
| 20.11.2019,
15:10-16:00 (Wednesday) |
Yakov Pesin (Penn State University) |
|
|
| Title: |
Reflections on Viana's conjecture: SRB
measures for systems with non-zero Lyapunov exponents |
| Location: |
Schreiber bldg., room 309, Tel-Aviv University |
|
|
| Abstract: |
In his plenary lecture at the ICM in Berlin
in 1998, Viana proposed a
conjecture that a diffeomorphism with non-zero Lyapunov exponents at
Lebesgue almost every point should admit an SRB measure. In this talk I
will describe some recent breakthrough results on existence of SRB
measures for some general hyperbolic systems. |
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SPECIAL ANNOUNCEMENT -
ANALYSIS SEMINAR
|
|
| 19.11.2019,
14:10 (Tuesday) |
Misha Bialy (Tel Aviv University) |
|
|
| Title: |
Magnetic billiards |
| Location: |
Schreiber bldg., room 209, Tel-Aviv University |
|
|
| Abstract: |
I will explain two recent results on
magnetic billiards. First result
is on integrable magnetic billiards in a strong magnetic field.
The second result is on the so called Gutkin property for magnetic
billiards, which turns out to be related to the so called Wegner
curves.
These curves give a partial solution to 19th problem by Ulam of the
Scottish cafe book. I will try discuss geometric and analytic aspects
of these two results. Joint work with Andrey Mironov and Lior Shalom. |
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|
| 27.11.2019,
14:10 (Wednesday) |
Gal Binyamini (Weizmann
Institute of Science) |
|
|
| Title: |
The Yomdin-Gromov lemma: a complex-analytic
perspective |
| Location: |
Schreiber bldg., room 309, Tel-Aviv University |
|
|
| Abstract: |
In '87 Yomdin established Shub's entropy
conjecture for
C^\infty maps. A principal ingredient in the proof is a
lemma on C^r-smooth parametrizations of semialgebraic sets,
which was later improved by Gromov and is now known as the
Yomdin-Gromov algebraic lemma. A couple of decades later
the same lemma has played a crucial role in the proof of
the Pila-Wilkie counting theorem, a result on the border
between model theory and diophantine approximation. The
counting theorem was later used in the resolution of several
outstanding conjectures in arithmetic geometry including
the Andre-Oort conjecture and some cases of the Zilber-Pink
conjecture.
Yomdin, and later Burguet, have suggested a conjectural
strengthening of the Yomdin-Gromov lemma. In the topological
direction, this strong form would imply a sharp bound for
the tail entropy of \emph{analytic} maps, conjectured by
Yomdin in '91. In the diophantine direction it would provide
a key step toward an improvement of the counting theorem
conjectured by Wilkie in '06.
In this talk I will discuss a recent proof of the strong
form of the Yomdin-Gromov lemma. The proof involves
complexifying the key building blocks of semialgebraic
geometry. Some classical ideas from complex analysis,
notably the Schwarz-Pick lemma, have surprisingly strong
implications in this complexified setup and provide the
fundamental new ideas that allow for this proof to be
carried out.
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|
| 04.12.2019,
14:10 (Wednesday) |
Shira Tanny (Tel Aviv University)
|
|
|
| Title: |
Floer theory of disjointly supported
Hamiltonians: isolation and interaction |
| Location: |
Schreiber bldg., room 309,
Tel-Aviv University |
|
|
| Abstract: |
The Floer-theoretic interaction between
disjointly supported
Hamiltonians was studied by Polterovich, Seyfaddini, Ishikawa
and Humilière-Le Roux-Seyfaddini, mostly through the relation
between invariants of the sum of Hamiltonians and invariants
of each one. We study this Floer-theoretic interaction on the
level of Floer trajectories, in aspherical manifolds, and
derive some applications. In particular, we prove that the
spectral invariants, with respect to the fundamental and point
classes, of Hamiltonians supported in some domains, are
determined locally. This is a joint work in progress with
Yaniv Ganor. |
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|
|
| 11.12.2019,
14:10 (Wednesday) |
Yoel
Groman (Hebrew University of Jerusalem)
|
|
|
| Title: |
Floer homology for embeddings and mirror
symmetry |
| Location: |
Schreiber bldg., room 309,
Tel-Aviv University |
|
|
| Abstract: |
Floer homology of a compact subset of
closed or geometrically
bounded symplectic manifold was introduced in the 90's, in the
works of Floer, Hofer and Cieliebak, and applied in various ways
to study quantitative questions is symplectic topology. In recent
years there has been some renewed interest and versions of it
have been found to have a close connection to mirror symmetry.
Roughly speaking, the mirror symmetry conjecture states that
symplectic geometry of a given symplectic manifold is encoded
in the analytic geometry of another. In certain cases, considering
Floer homology of a compact set mirrors studying analytic geometry
of subsets in the mirror. I will discuss this idea and some
develpoing methods of computation inspired by it. Based on joint
works in progress with Mohammed Abouzaid and Umut Varolgunes.
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|
18.12.2019,
14:10 (Wednesday)
|
Liat
Kessler (University of Haifa at Oranim)
|
|
|
| Title: |
Relating the algebraic and combinatorial
structures of Hamiltonian circle-actions, and finiteness corollaries |
| Location: |
Rosenberg bldg., room 212,
Tel-Aviv University (PLEASE NOTE CHANGE IN LOCATION)
|
|
|
| Abstract: |
We give a soft proof for the finiteness of
maximal Hamiltonian
circle actions on a closed symplectic four-manifold, analogous
to the proof of McDuff and Borisov for the finiteness of toric
actions. Our first step is obtaining a generators and relations
description of the equivariant cohomology module from the decorated
graph associated to the circle action. We further use this
description to prove that the even parts of the equivariant
cohomology modules are weakly isomorphic (and the odd groups
have the same ranks) if and only if the labelled graphs obtained
from the decorated graphs by forgetting the height and area labels
are isomorphic. This leads us to an example of an isomorphism of
equivariant cohomology modules that cannot be induced by an
equivariant diffeomorphism of spaces. |
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|
| 25.12.2019,
14:10 (Wednesday) |
Sergei
Tabachnikov (Penn State University) - Blumenthal Lecture in Geometry
|
|
|
| Title: |
Introducing
symplectic billiards |
| Location: |
Dan David bldg., room 110,
Tel-Aviv University (PLEASE NOTE CHANGE IN LOCATION) |
|
|
| Abstract: |
I shall introduce a new dynamical system
called symplectic
billiards. As opposed to the usual (Birkhoff) billiards, where
length is the generating function, for symplectic billiards the
area is the generating function. I shall describe basic properties
and exhibit several similarities, but also differences, of
symplectic billiards with Birkhoff billiards. Symplectic billiards
can be defined not only in the plane, but also in linear symplectic
spaces, with the symplectic area as the generating function. In
this multi-dimensional setting, I shall discuss the existence of
periodic trajectories and describe the integrable dynamics of
symplectic billiards in ellipsoids. I shall also explore polygonal
symplectic billiards and present such polygonal billiards in which
all orbits are periodic. This later work, still in progress, is
mostly experimental, using specially developed computer programs. |
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SPECIAL ANNOUNCEMENT -
COLLOQUIUM TALK: |
|
| 23.12.2019,
12:15 (Monday) |
Sergei
Tabachnikov (Penn State University) - Blumenthal Lecture in Geometry |
|
|
| Title: |
Four
equivalent properties of integrable billiards |
| Location: |
Schreiber bldg., room 006,
Tel-Aviv University |
|
|
| Abstract: |
Optical properties of conics have been
known since the classical
antiquity (and, according to the legend, put to use by Archimedes
by destroying enemy ships with fire). The reflection in an ideal
mirror is also known as the billiard reflection and, in modern terms,
the billiard inside in ellipse is completely integrable. The interior
of an ellipse is foliated by confocal ellipses that are its caustics:
a ray of light tangent to a caustic remains tangent to it after
reflection ("caustic" means burning).
I shall explain these classic results and some of their geometric
consequences, including the Ivory lemma asserting that the diagonals
of a curvilinear quadrilateral made by arcs of confocal ellipses and
hyperbolas are equal. This lemma is in the heart of Ivory's
calculation of the gravitational potential of a homogeneous ellipsoid.
I shall also describe the string construction that reconstructs a
billiard table from its caustic; in particular, the string construction
on an ellipse yields a confocal ellipse (a theorem of Graves). Then I
will describe four equivalent properties of integrable billiards on
Riemannian surfaces, connecting together the Ivory lemma, the string
construction, billiard caustics, and Liouville metrics. The latter
are generalizations of the metric of an ellipsoid in Euclidean space
whose geodesic flows are completely integrable.
I shall conclude with a generalization of the famous Birkhoff's
conjecture: If an interior neighborhood of a closed geodesically
convex curve on a Riemannian surface is foliated by billiard caustics,
then the metric in the neighborhood is Liouville, and the curve is a
coordinate line. |
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|
| 01.01.2020,
14:10 (Wednesday) |
NO SEMINAR THIS WEEK! |
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|
| 08.01.2020,
14:10 (Wednesday) |
Louis Ioos (Tel Aviv University) |
|
|
| Title: |
Donaldson's iterations towards canonical
Kähler metrics |
| Location: |
Dan David bldg., room 103, Tel-Aviv University (PLEASE NOTE CHANGE IN LOCATION) |
|
|
| Abstract: |
I will present Donaldson's program on
the approximation
of Kähler metrics of constant scalar curvature by Kähler metrics
associated with a canonical projective embedding, and give estimates
on the speed of convergence of this approximation process. This is
partly based on joint work with Victoria Kaminker, Leonid Polterovich
and Dor Shmoish. |
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|
| 15.01.2020,
14:10 (Wednesday) |
Ood Shabtai (Tel Aviv University) |
|
|
| Title: |
Commutators of spectral projections of quantum observables |
| Location: |
Dan David bldg., room 103, Tel-Aviv University (PLEASE NOTE CHANGE IN LOCATION) |
|
|
| Abstract: |
We consider spectral projections associated with a pair of
non-commuting classical observables, and show that in a number of
cases, the norm of their commutator exhibits a surprising behaviour
in the semiclassical limit. |
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|
|
22.01.2020,
14:00 (Wednesday)
|
Daniel
Peralta-Salas (ICMAT) - MINT Distinguished Lecture |
|
|
| Title: |
Selected topics on the dynamics of the steady Euler flows |
| Location: |
Dan David bldg., room 201, Tel-Aviv University (PLEASE NOTE CHANGE IN LOCATION) |
|
|
| Abstract: |
An inviscid and incompressible fluid in equilibrium on a
Riemannian manifold is described by (an autonomous) vector
field X that satisfies the stationary Euler equations. In this
talk I will focus on the dynamics of steady Euler flows on
compact manifolds, emphasizing the geometric aspects. The
topics I will cover include a characterization of the
volume-preserving vector fields that are Eulerizable and
some rigidity results for steady Euler flows on the round sphere. |
|
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|
SPECIAL ANNOUNCEMENT -
COLLOQUIUM TALK: |
|
| 20.01.2020,
12:15 (Monday) |
Daniel
Peralta-Salas (ICMAT) - MINT Distinguished Lecture
|
|
|
| Title: |
Emergence of topological structures in elliptic PDE |
| Location: |
Schreiber bldg., room 006,
Tel-Aviv University |
|
|
| Abstract: |
In recent years, new experimental methods and designs have
enabled the observation and study of topological structures
emerging from a wide range of physical phenomena, from fluid
mechanics to electromagnetic theory. These structures take
the form, e.g., of knotted vortex tubes in fluid flows, or
knotted topological dislocations in optics and condensed matter
theory. They provide a powerful visual tool to gain understanding
of complex physical systems.
Mathematically, these physical processes are described by a
vector or scalar field which satisfies a system of partial
differential equations, and the emerging topological structures
are instances of invariant manifolds of these fields.
The goal of this lecture is to introduce the theory we have
recently developed in collaboration with A. Enciso to address
the study of these problems. Some of the questions that I will
consider are: does there exist a steady Euler flow having stream
lines of all knot and link types? Can the nodal components of the
eigenfunctions of a Schrodinger operator exhibit arbitrary
topology? I will focus on the aspects of the theory for
elliptic (time independent) PDE. |
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|
|
| 11.03.2020,
14:10 (Wednesday) |
Lev Buhovsky (Tel Aviv University)
|
|
|
| Title: |
The Arnold conjecture, spectral invariants and C^0 symplectic topology |
| Location: |
Schreiber bldg., room 309,
Tel-Aviv University |
|
|
| Abstract: |
The Arnold conjecture about fixed points of Hamiltonian diffeomorphisms
was partly motivated by the celebrated Poincare-Birkhoff fixed point
theorem for an area-preserving homeomorphism of an annulus in the
plane. Despite the fact that the Arnold conjecture was formulated in
the smooth setting, several attempts to return to the continuous
setting of homeomorphisms and to study the conjecture in this setting
has been made afterwards. In this talk I will describe some old and
more recent results on the subject. Based on a joint work (partly in
progress) with V. Humiliere and S. Seyfaddini. |
|
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|
|
| 18.03.2020,
14:10 (Wednesday) |
NO SEMINAR THIS WEEK! |
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|
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|
| 25.03.2020,
14:10 (Wednesday) |
NO SEMINAR THIS WEEK! |
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|
|
| 01.04.2020,
14:10 (Wednesday) |
Semyon Alesker (Tel Aviv University)
|
|
|
| Title: |
Multiplicative structure on valuations on convex sets and its
complex analogue. |
Location: (VIRTUAL SEMINAR!)
|
Zoom session, the link is available upon request by email
|
|
|
| Abstract: |
Valuation on convex sets is a classical object in convexity with
traditionally strong relations to integral geometry. Valuation is a
finitely additive measure on the class of all convex compact sets in R^n.
Translation invariant valuations continuous in the Hausdorff metric are
particularly well studied objects. During the last 25 year there was a
considerable progress in their study and in its integral geometric
applications.
Some years ago the speaker has introduced a canonical multiplicative
structure on them. First I will review some of its properties (e.g.
versions of Poincare duality and hard Lefschetz theorem) and, if time
permits, applications to integral geometry.
That will be a motivation to introduce a complex analogue of the algebra of
even translation invariant valuations. While at the moment it lacks a
geometric interpretation, it has non-trivial algebraic properties. Thus it
satisfies again Poincare duality and hard Lefschetz theorem. |
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|
| 22.04.2020,
14:10 (Wednesday) |
Michael Entov (Technion) |
|
|
| Title: |
Rigidity of Lagrangian tori in K3 surfaces |
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email |
|
|
| Abstract: |
A Kahler-type form is a symplectic form compatible with an integrable
complex structure. Sheridan and Smith previously proved, using deep
methods of homological mirror symmetry, that for any Maslov-zero
Lagrangian torus L in a K3 surface M equipped with a Kahler-type
form of a *particular kind*, the integral homology class of L has
to be non-zero and primitive. I will discuss how to extend this
result to *arbitrary* Kahler-type forms on M using dynamical
properties of the action of the diffeomorphism group of M on the
space of such forms. These dynamical properties are obtained using
a version of Ratner's theorem. This is a joint work in progress
with M.Verbitsky. |
|
|
|
|
|
| 06.05.2020,
14:10 (Wednesday) |
Boris Vertman (University of Oldenburg) |
|
|
| Title: |
Mean curvature flow in Lorentzian space times |
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email |
|
|
| Abstract: |
Hypersurfaces of zero or constant mean curvature play a central
role in the proof of the Positive Mass Theorem and also in the
analysis of the Cauchy problem for asymptotically flat space-times.
Mean curvature flow can be a tool to construct such hypersurfaces.
We discuss local existence of the flow for non-compact space-like
hypersurfaces in Robertson-Walker space-times. This is a joint project
with Giuseppe Gentile. |
|
|
|
|
|
| 13.05.2020,
14:10 (Wednesday) |
Louis Ioos (Tel Aviv University) |
|
|
| Title: |
Almost-representations of the Lie algebra of SU(2) and quantization of the sphere |
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email |
|
|
| Abstract: |
I will show that almost-representations of the Lie algebra of SU(2)
are well approximated by actual representations. The motivation for
this result comes from the study of quantizations of the two-dimensional
sphere, and I will show how it applies to the classification of such
quantizations. This is a joint work in progress with David Kazhdan and
Leonid Polterovich. |
|
|
|
|
|
SPECIAL ANNOUNCEMENT -
Joint TAU Geometry & Dynamics and WIS Midrasha on Groups seminar: |
|
| 20.05.2020,
14:10 (Wednesday) |
Tsachik Gelander (Weizmann Institute of Science) |
|
|
| Title: |
Convergence of normalized Betti numbers in nonpositive curvature |
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email |
|
|
| Abstract: |
I will show that if X is any symmetric space other than 3-dimensional
hyperbolic space and M is any finite volume manifold that is a quotient
of X, then the normalized Betti numbers of M are "testable", i.e. one
can guess their values by sampling the manifold at random places. This
is joint with Abert, Biringer and Bergeron, and extends some of our
older work with Nikolov, Raimbault and Samet. The content of the recent
paper involves a random discretization process that converts the "thick
part" of M into a simplicial complex, together with analysis of the
"thin parts" of M. As a corollary, we obtain that whenever X is a higher
rank irreducible symmetric space and M_i is any sequence of distinct
finite volume quotients of X, the normalized Betti numbers of the M_i
converge to the "L^2-Betti numbers" of X. |
|
|
|
|
|
| 27.05.2020,
14:10 (Wednesday) |
Victor Ivrii (University of Toronto) |
|
|
| Title: |
Heavy atoms and molecules: Thomas-Fermi and Scott approximations |
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email |
|
|
| Abstract: |
The purpose of this talk is to discuss the rigorous mathematical results:
* Thomas-Fermi approximation to the ground state energy, excessive
positive and negative charges, and ionization energy (old results).
* Thomas-Fermi and Scott approximations to the ground state density
(new results). |
|
|
|
|
|
| 03.06.2020,
14:10 (Wednesday) |
Marco Mazzucchelli (ENS de Lyon) |
|
|
| Title: |
Closed geodesics on reversible Finsler 2-spheres |
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email |
|
|
| Abstract: |
In this talk, I will show that two celebrated theorems on closed
geodesics of Riemannian 2-spheres still hold for the larger class
of reversible Finsler 2-spheres: Lusternik-Schnirelmann's theorem
asserting the existence of three simple closed geodesics, and
Bangert-Franks-Hingston's theorem asserting the existence of
infinitely many closed geodesics. I will sketch the proofs of
these statements, employing in particular the Finsler generalization
of Grayson's curve shortening flow developed by Angenent-Oaks.
This is joint work with Guido De Philippis, Michele Marini, and
Stefan Suhr. |
|
|
|
|
|
| 10.06.2020,
14:10 (Wednesday) |
Tali Pinsky (Technion) |
|
|
| Title: |
New Anosov flows from the modular surface |
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email
|
|
|
| Abstract: |
The modular surface is the space of lattices in the plane. It has a
natural structure of a hyperbolic surface and thus one can consider
the geodesic flow on this surface. The geodesic flow is Anosov,
meaning at each point it has a contracting and an expanding direction.
It is well known for its relation to number theory.
In the talk I will explain how one can use cut and paste techniques
to construct new Anosov flows out of geodesic flows, and how this can
be used to solve an open question in dynamics: Does there exist a
three manifold carrying infinitely many different Anosov flows?
This is a joint project with Adam Clay. |
|
|
|
|
|
| 17.06.2020,
14:10 (Wednesday) |
Ran Tessler (Weizmann Institute of Science)
|
|
|
| Title: |
Open r-spin intersection theory |
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email
|
|
|
| Abstract: |
In 1992 Witten conjectured that the intersection theory on the moduli
of r-spin curves gives rise to a Gelfand Dickey tau function, and
proved his conjecture in genus 0.
Recently, in a joint work with Buryak and Clader we made a similar
conjecture/construction in the open setting:
We conjectured that intersection theory on the moduli of r-spin
surfaces with boundaries should give rise to the Gelfand Dickey *wave*
function and proved it in genus 0. In my talk I will describe all this,
in particular, I'll explain what is an r-spin structure, what is the
Gelfand-Dickey hierarchy and what is the motivation. If time permits,
a mirror theorem (based on work with Gross and Kelly) will also be shown. |
|
|
|
|
|
22.06.2020,
14:10 (Monday)
(PLEASE NOTE CHANGE IN DATE!)
|
Emanuel Milman (Technion) |
|
|
| Title: |
Functional Inequalities on sub-Riemannian manifolds via QCD
|
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email
|
|
|
| Abstract: |
We are interested in obtaining Poincare and log-Sobolev inequalities
on domains in sub-Riemannian manifolds (equipped with their natural
sub-Riemannian metric and volume measure).
It is well-known that strictly sub-Riemannian manifolds do not satisfy
any type of Curvature-Dimension condition CD(K,N), introduced by
Lott-Sturm-Villani some 15 years ago, so we must follow a different
path. We show that while ideal (strictly) sub-Riemannian manifolds do
not satisfy any type of CD condition, they do satisfy a quasi-convex
relaxation thereof, which we name QCD(Q,K,N). As a consequence, these
spaces satisfy numerous functional inequalities with exactly the same
quantitative dependence (up to a factor of Q) as their CD counterparts.
We achieve this by extending the localization paradigm to completely
general interpolation inequalities, and a one-dimensional comparison
of QCD densities with their "CD upper envelope". We thus obtain the
best known quantitative estimates for (say) the L^p-Poincare and
log-Sobolev inequalities on domains in the ideal sub-Riemannian setting,
which in particular are independent of the topological dimension. For
instance, the classical Li-Yau / Zhong-Yang spectral-gap estimate holds
on all Heisenberg groups of arbitrary dimension up to a factor of 4.
No prior knowledge will be assumed, and we will (hopefully) explain
all of the above notions during the talk.
|
|
|
|
|
| 24.06.2020,
14:10 (Wednesday) |
Uri Bader (Weizmann Institute of Science) |
|
|
| Title: |
Totally geodesic subspaces and arithemeticity phenomena in hyperbolic manifolds |
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email |
|
|
| Abstract: |
In this talk I will survey a well known, still wonderful, connection
between geometry and arithmetics and discuss old and new results in
this topic. The starting point of the story is Cartan's discovery
of the correspondence between semisimple Lie groups and symmetric
spaces. Borel and Harish-Chandra, following Siegel, later realized
a fantastic further relation between arithmetic subgroups of semisimple
Lie groups and locally symmetric space - every arithemtic group gives
a locally symmetric space of finite volume. The best known example
is the modular curve which is associated in this way with the group
SL_2(Z). This relation has a partial converse, going under the name
"arithmeticity theorem", which was proven, under a higher rank
assumption, by Margulis and in some rank one situations by Corlette
and Gromov-Schoen. The rank one setting is related to hyperbolic
geometry - real, complex, quaternionic or octanionic.
There are several open questions regarding arithmeticity of locally
hyperbolic manifolds of finite volume over the real or complex fields
and there are empirical evidences relating these questions to the
geometry of totally geodesic submanifolds.
Recently, some of these questions were solved by Margulis-Mohammadi
(real hyp. 3-dim), Baldi-Ullmo (complex hyp.) and B-Fisher-Miller-Stover.
The techniques involve a mixture of ergodic theory, algebraic groups
theory and hodge theory. After surveying the above story, explaining
all the terms and discussing some open questions, I hope to have a
little time to say something about the proofs. |
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| 01.07.2020,
14:10 (Wednesday) |
Yusuke Kawamoto (Ecole Normale Superieure) |
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| Title: |
Homogeneous quasimorphism, C^0-topology and Lagrangian intersection |
| Location: (VIRTUAL SEMINAR!) |
Zoom session, the link is available upon request by email |
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| Abstract: |
The goal of the talk is to construct a non-trivial homogeneous
quasimorphism on the group of Hamiltonian diffeomorphisms of the
2- and 4-dimensional quadric which is continuous with respect to both
C^0-topology and the Hofer metric. This answers a variant of a question
of Entov-Polterovich-Py. A comparison of spectral invariants for
quantum cohomology rings with different coefficient fields plays a
crucial role in the proof which might be of independent interest.
If time permits, we will see how this comparison can be used to answer
a question of Polterovich-Wu. |
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