Geometry & Dynamics Seminar 2017-18


The seminar will take place in Schreiber Building room 309, on Wednesdays at 14:10.

Please check each announcement since this is sometimes changed.

 

Upcoming Talks        Previous Talks        Previous Years











25.10.2017, 14:10 (Wednesday) Orientation meeting for students




Location: Schreiber bldg., room 309, Tel-Aviv University




31.10.2017,12:10 (Tuesday)

SPECIAL LECTURE - PLEASE NOTE THE DATE,  PLACE & TIME

Jean-Michel Bismut, Université Paris-Sud (Orsay)




Title:
Hypoelliptic Laplacian, index theory and the trace formula
Location: Schreiber bldg., room 209, Tel-Aviv University



Abstract: The hypoelliptic Laplacian is a family of operators, indexed by $b\in
\mathbf{R}_{+}^{*}$, acting on the
total space of the tangent  bundle of a Riemannian manifold, that
interpolates between the ordinary Laplacian as $b\to 0$ and the
generator of the geodesic flow as $b\to + \infty $.  These operators
are not elliptic, they are not self-adjoint, they are hypoelliptic.

The hypoelliptic deformation preserves subtle invariants of the
Laplacian. In the case of locally symmetric spaces, the deformation
is essentially isospectral.

In a first part of the talk, I will describe the geometric
construction of the hypoelliptic Laplacian in the context of de Rham
theory. In a second part, I will explain applications to the trace
formula.









01.11.2017, 14:10 (Wednesday) Daniel Rosen (TAU) 



Title: Duality of Caustics in Minkowski Billiards
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: Mathematical billiards are a classical and well-studied class of dynamical systems,
"a mathematician’s playground". Convex caustics, which are curves to which billiard
trajectories remain forever tangent, play an important role in the study of billiards.
In this talk we will discuss convex caustic in Minkowski billiards, which is the
generalization of classical billiards no non-Euclidean normed planes. In this case a
natural duality arises from, roughly speaking,  interchanging the roles of the billiard
table and the unit ball of the (dual) norm. This leads to duality of caustics in Minkowski
billiards. Such a pair of caustics is dual in a strong sense, and in particular they have
equal perimeters and other classical parameters. We will show that, when the norm is
Euclidean, every caustic possesses a dual caustic, but in general this phenomenon fails.
Based on joint work with S. Artstein-Avidan, D. Florentin, and Y. Ostrover .








08.11.2017, 14:10 (Wednesday)
Mads Bisgaard (ETH)




Title: Topology of small Lagrangian cobordisms
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: I will discuss how one can study Lagrangian cobordisms from the point
of view of quantitative symplectic topology: It turns out that if a Lagrangian
cobordism is sufficiently small (in a sense which can be made precise), then
its topology is to a large extend determined by its boundary. I will show how
this principle allows one to derive several homological uniqueness results for
small Lagrangian cobordisms. In particular (under the smallness assumption)
I will prove homological uniqueness of the class of Lagrangian cobordisms which,
by Biran-Cornea’s Lagrangian cobordism theory, induces operations on a version
of the derived Fukaya category. If time permits it, I will indicate a link from these
ideas to Vassilyev’s theory of Lagrange characteristic classes and the classification
of caustics.








15.11.2017, 14:10 (Wednesday) Dmitry Novikov (Weizmann Institute)



Title: Complex cellular parameterization
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: We introduce the notion of a complex cell, a complex analog
of the cell decompositions used in real algebraic and analytic
geometry. Complex cells defined using holomorphic data admit a
natural notion of analytic continuation called $\delta$-extension,
which gives rise to a rich hyperbolic geometric structure absent in
the real case. We use this structure to prove that complex cellular
decompositions share some interesting features with the classical
constructions in the theory of resolution of singularities. Restriction
of a complex cellular decomposition to the reals recovers the preparation
theorem for subanalytic functions, and can be viewed as an analytic
continuation thereof.

A key difference in comparison to the classical resolution of
singularities is that the cellular decompositions are intrinsically
uniform over (sub)analytic families. We deduce a subanalytic version
of the Yomdin-Gromov theorem where $C^k$-smooth
maps are replaced by mild maps.

(joint work with Gal Binyamini)








22.11.2017, 14:10 (Wednesday) Boris Kruglikov (University of Tromsø)




Title: Integrability in Grassmann geometry and twistor theory
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: This reviews a series of works joint with E.Ferapontov, D.Calderbank,
B.Doubrov and V.Novikov. It will be explained that for several important
classes of PDEs the integrability by the method of hydrodynamic reductions
is equivalent to a Lax representation. This includes equations of Hirota type
and also PDE systems encoded by submanifolds in Grassmannians. For the
latter the integrability can be interpreted geometrically. In 3D and 4D the
integrability is also shown to be equivalent to Einstein-Weyl and, respectively,
self-dual geometry on solutions. This relates dispersionless integrability to the
twistor theory.
Ref: J.Diff.Geom.97 (2014), arXiv:1503.02274, arXiv:1612.02753, arXiv:1705.06999.








29.11.2017, 14:10 (Wednesday) Igor Uljarevic




Title: Floer homology and contact Hamiltonians
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: In the setting of symplectic manifolds which are convex at infinity,
we use a version of the Aleksandrov maximum principle to extend the class of
Hamiltonians that one can use in the direct limit when constructing symplectic
homology. As an application, we detect elements of infinite order in the symplectic
mapping class group of a Liouville domain and prove existence results for
translated points.
The talk is based on joint work with W. Merry.








06.12.2017, 14:10 (Wednesday) Dmitry Faifman (University of Toronto)



Title: Contact curvatures and integral geometry of the contact sphere
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: Valuations are finitely additive measures on nice subsets, for example
the Euler characteristic, volume and surface area are valuations. During
the 20th century, valuations have been studied predominantly on convex
bodies and polytopes, in linear spaces and lattices. Valuations on manifolds
were introduced about 15 years ago by S. Alesker, with contributions by
A. Bernig, J. Fu and others, and immediately brought under one umbrella a
range of classical results in Riemannian geometry, notably Weyl's tube
formula and the Chern-Gauss-Bonnet theorem. These results circle around
the real orthogonal group. In the talk, the real symplectic group will be the
central player. Drawing inspiration from the Lipschitz-Killing curvatures in
the Riemannian setting, we will construct some natural valuations on contact
and almost contact manifolds, which generalized the Gaussian curvature.
We will also construct symplectic-invariant distributions on the grassmannian,
leading to Crofton-type formulas on the contact sphere and symplectic space.








13.12.2017, 14:10 (Wednesday) Sergey Fomin (University of Michigan) - MINT distinguished lecture




Title: Morsifications and mutations
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: I will discuss a surprising connection between singularity theory and cluster algebras, more specifically
between (1) the topology of isolated singularities of plane curves and (2) the mutation equivalence of the
quivers associated with their morsifications. Joint work with Pavlo Pylyavskyy and Eugenii Shustin.








20.12.2017, 14:10 (Wednesday) NO SEMINAR THIS WEEK










27.12.2017, 14:10 - 15:00 (Wednesday) Kei Irie (Kyoto University and Simons Center for Geometry and Physics)




Title: Denseness of minimal hypersurfaces for generic metrics
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: We prove that, on a smooth closed manifold of dimension $3 \le d \le 7$
with a $C^\infty$-generic Riemannian metric, the union of closed embedded
minimal hypersurfaces is dense. This is joint work with F.Marques and A.Neves.

The proof is based on min-max theory for the volume functional on the space of
codimension 1 (flat) cycles,  which was originally developed by Almgren and Pitts.
The key ingredient of the proof is the ``Weyl law''(proved by Liokumovich, Marques and Neves),
which says that the asymptotic of min-max values in this theory recovers the volume of a
Riemannian manifold.







27.12.2017, 15:10 - 16:00 (Wednesday) Iosif Polterovich (Université de Montréal) 



Title: Isoperimetric inequalities for Laplace eigenvalues on surfaces: some recent developments
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: Isoperimetric inequalities for Laplace eigenvalues have a long history,
going back to the celebrated Rayleigh-Faber-Krahn inequality for the fundamental tone.
Still, many basic questions remain unanswered, particularly, for higher eigenvalues.
In the talk I will give an overview of some recent developments in the study of
isoperimetric inequalities for eigenvalues on compact surfaces with a Riemannian metric.
In particular, I will discuss a solution of a conjecture posed by N. Nadirashvili in 2002
regarding the maximization of higher Laplace-Beltrami eigenvalues on the sphere
(joint with M. Karpukhin, N. Nadirashvili and A. Penskoi).








03.01.2018, 14:10 (Wednesday) Julian Chaidez (University of California, Berkeley) 



Title: The Conley-Zehnder Index In Singular Contact Geometry
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: The Conley-Zehnder (CZ) index is an important invariant of closed orbits of smooth
Hamiltonian flows and Reeb flows. In this talk, I will discuss a version of the CZ index
for various "singular" contact geometry problems, such as Reeb dynamics on polytopes
and dynamical billiards. We will show how this CZ index can be applied to convert some
results in smooth contact geometry into results about singular contact geometry using a
limiting argument.








10.01.2018, 14:00 (Wednesday)
Pazit Haim-Kislev (TAU)



Title: The EHZ capacity of convex polytopes
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: The EHZ capacity is a well-studied symplectic invariant that measures the "symplectic size"
of convex sets, by taking the minimal action of a closed characteristic on the boundary.
We introduce a simplification to the problem of finding a closed characteristic with minimal
action for the case of convex polytopes. We use this to give a combinatorial formula for the
EHZ capacity of convex polytopes, and to prove a certain subadditivity property of the capacity
of a general convex body.








17.01.2018, 14:10 (Wednesday) Daniel Alvarez-Gavela (Stanford University)




Title: Singularities of fronts and their simplification
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: We will present a full h-principle for the simplification of
singularities of Lagrangian and Legendrian fronts. We give several
applications to symplectic and contact topology, including relations to
pseudo-isotopy theory and to Nadler's program for the arborealization of
Lagrangian skeleta.








07.03.2018, 14:10 (Wednesday) Vukasin Stojisavljevic (TAU) 



Title: Persistence barcodes and Laplace eigenfunctions on surfaces
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: Using barcodes, we will define a family of functionals on the space of continuous
functions on a smooth manifold. These functionals behave well with respect to
C^0-distance between functions. Furthermore, we will show that on the space of
linear combinations of Laplace eigenfunctions on surfaces these functionals have
certain upper bounds. As an application of all of these properties we obtain results
about C^0-approximations of a function by linear combinations of Laplace
eigenfunctions. Based on a joint work with I. Polterovich and L. Polterovich.








14.03.2018, 14:10 (Wednesday) Matthias Meiwes (TAU)




Title: Dynamically exotic contact spheres
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: Topological entropy is a dynamical invariant that codifies in a single number
the exponential instability of a dynamical system. In this talk I'll discuss some
results on the existence of contact structures on manifolds such that every Reeb
flow has positive topological entropy. In dimensions greater than 5, even spheres,
in contrast to their simple topology, carry such contact structures.  I'll discuss how,
in many cases, algebraic growth properties of wrapped Floer homology play a
crucial role in detecting positive entropy. Finally, I will describe a more recent
approach to the construction of such contact manifolds, using the growth of
Rabinowitz Floer homology. This is joint work with Marcelo Alves.








21.03.2018, 14:10 (Wednesday) Peter Ozsvath (Princeton University) - Blumenthal lectures in Geometry




Title: An algebraic construction of knot Floer homology
Location: Schreiber bldg., room 309, Tel-Aviv University 



Abstract: Bordered Floer homology is an invariant for three-manifolds with
boundary.  I will discuss an algebraic approach to computing knot
Floer homology, based on decomposing knot diagrams.  This is joint
work with Zoltan Szabo, influenced by earlier joint work with Ciprian
Manolescu and Sucharit Sarkar; and Robert Lipshitz and Dylan Thurston.








28.03.2018, 14:10 (Wednesday) Yaniv Ganor (TAU)




Title: A homotopical viewpoint at the Poisson bracket invariants for tuples of sets
Location: Schreiber bldg., room 309, Tel-Aviv University




Abstract: We suggest a homotopical description of the Poisson
bracket invariants for tuples of closed sets in symplectic manifolds.
It implies that these invariants depend only on the union of the sets
along with topological data.








11.04.2018, 14:10 (Wednesday) Jarek Kedra (University of Aberdeen)




Title: Boundedness properties of SL(n,Z)
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: A group G is called bounded if any conjugation-invariant norm on G has
finite diameter. When looking at conjugation-invariant word norms then the
diameter depends on the choice of a generating set. I will discuss the
subtleties of this dependence, provide more details on the behaviour of
SL(3,Z) and present some applications.

Joint work with Assaf Libman and Ben Martin.








25.04.2018, 14:10 (Wednesday) Sheng-Fu Chiu (Academia Sinica Institute of Mathematics, Taiwan)



Title: Microlocal Sheaf Theory and Contact Non-squeezing
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: In this talk we will discuss the contact non-squeezing problem of pre-quantized
balls in the ambient pre-quantization space. We will provide an approach from
the mixed viewpoints of operator valued measurement, categorification of
symplectic/contact morphism, and the homological algebra of microlocal sheaf theory.








Workshop: Topological data analysis meets symplectic topology

NO SEMINAR THIS WEEK!










09.05.2018, 14:10-15:00 (Wednesday) Michael Brandenbursky (BGU)



Title: Entropy, metrics and quasi-morphisms
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: One of the mainstream and modern tools in the study of non abelian groups
are quasi-morphisms. These are functions from a group to the reals which
satisfy homomorphism condition up to a bounded error. Nowadays they are
used in many fields of mathematics. For instance, they are related to bounded
cohomology, stable commutator length, metrics on diffeomorphism groups,
displacement of sets in symplectic topology, dynamics, knot theory, orderability,
and the study of mapping class groups and of concordance group of knots.

Let S be a compact oriented surface. In this talk I will discuss several invariant
metrics and quasi-morphisms on the identity component Diff_0(S, area) of the
group of area preserving diffeomorphisms of S. In particular, I will show that
some quasi-morphisms on Diff_0(S, area) are related to the topological entropy.
More precisely, I will discuss a construction of infinitely many linearly independent
quasi-morphisms on Diff_0(S, area) whose absolute values bound from below the
topological entropy. If time permits, I will define a bi-invariant metric on this group,
called the entropy metric, show that it is unbounded, and will discuss a relation with
Katok's conjecture. Based on a joint work with M. Marcinkowski.







09.05.2018, 15:10-16:00 (Wednesday) Leonid Potyagailo (Université Lille 1)




Title: Martin and Floyd boundaries of finitely generated groups

Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: ABSTRACT








16.05.2018, 14:10 (Wednesday) Egor Shelukhin (University of Montreal)




Title: On the geometry of Lagrangian flux
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: We discuss the space of first cohomology classes of a given Lagrangian submanifold L
that can be obtained as the flux of Lagrangian isotopies starting with it. When the flux
is required to move along a fixed direction as the isotopy evolves, or when the isotopy
stays within a given Weinstein neighborhood of L, we obtain bounds on this space
(and restrictions on neighborhoods), and in some cases, its exact calculation.
This direction of research was initiated by Benci, Sikorav, and Eliashberg, and was recently
studied by Entov-Ganor-Membrez. An important part of our considerations is based on
an invariant produced from the Fukaya algebra of the Lagrangian, and its properties.
This invariant can be considered to be, roughly, the minimal area of a holomorphic disk on
L with a non-trivial algebraic count. This is joint work with Dmitry Tonkonog and Renato Vianna.








23.05.2018, 14:10 (Wednesday) Asaf Kislev (TAU)




Title: Sharp bounds on the boundary depth
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: The plan for this talk is to show that the bottleneck distance between Floer
barcodes associated to Hamiltonian diffeomorphisms \phi and \psi, is Lipschitz
with respect to the spectral norm \gamma(\phi \psi^{-1} ). We use this to show
that Usher's boundary depth of RP^n in CP^n with arbitrary Hamiltonian
perturbations is bounded above by n / (2n+2), while the absolute boundary depth
of any Hamiltonian diffeomorphism of CP^n is bounded above by n/(n+1).
We prove that these bounds are sharp, and give some examples and applications.
Joint work with Egor Shelukhin.








30.05.2018, 14:10-15:00 (Wednesday) Marie-Claude Arnaud (University of Avignon)




Title: Arnol’d-Liouville theorems in low regularity
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: Classical Arnol’d Liouville theorem describes precisely the Dynamics of Hamiltonian systems
that have enough independent C2 integrals. For such Hamiltonians, it is known that there is
an invariant Lagrangian foliation that is symplectically diffeomorphic to the standard one and
that the Dynamics restricted to every invariant leaf is conjugate to a translation flow. Here we
focus on what happens when we have lower regularity. The motivation for studying low
regularity is that when a Tonelli Hamiltonian has no conjugate points, only the existence of
continuous integrals can be proved. More precisely, we will raise the question of  which  continuous
Lagrangian foliations are symplectically homeomorphic to the standard one and prove that when
the integrals are just C1 and when the Hamiltonian is Tonelli,  we indeed obtain a continuous
Lagrangian foliation that is symplectically homeomorphic to the standard one and that
Arnol’d-Liouville theorem remains true with a symplectic homeomorphism instead of a
C1 change of coordinates.







30.05.2018, 15:10-16:00 (Wednesday)
Vinicius Gripp B. Ramos (IMPA)



Title: Symplectic embeddings, lagrangian products and integrable systems
Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: Symplectic embedding problems are at the core of the study of symplectic topology.
There are many well-known results for so-called toric domains,
but very little is known about other kinds of domains.
In this talk, I will mostly speak about a different kind of domain, namely a
lagrangian product. These domains are of a very different nature and are
related to billiards, as discovered by Artstein-Avidan and Ostrover. I will explain how
to use integrable systems to see that some of these products are secretely toric domains
and how to use symplectic capacities to obtain sharp obstructions to many symplectic
embedding problems.










06.06.2018, 14:10 (Wednesday) Shira Tanny (TAU)



Title: The Poisson bracket conjecture in dimension 2
Location: Schreiber bldg., room 309, Tel-Aviv University




Abstract: I will discuss the Poisson bracket invariant of covers and explain how elementary
geometric arguments can be used to prove Polterovich's pb-conjecture in dimension 2.
This is a joint work with L. Buhovsky and S. Logunov. We express our gratitude to
F. Nazarov for his contribution to this work.











13.06.2018, 14:10 (Wednesday) Misha Bialy (TAU) 



Title: Around Birkhoff conjecture for convex billiards


Location: Schreiber bldg., room 309, Tel-Aviv University



Abstract: In this talk I will discuss a very old conjecture attributed to G.Birkhoff
on convex billiards. It states that the only integrable convex billiards in the plane are ellipses.
Here it is very important to specify what is understood by integrability.
This talk is about recent important developments toward a positive solution of this conjecture,
and on the geometric problems naturally arising on the way.
The talk is supposed to be elementary and does not require any prior knowledge.

















Organized by Misha Bialy, Lev Buhovsky, Yaron Ostrover, and Leonid Polterovich