Geometry & Dynamics Seminar 2023-24


Some seminar talks will take place in Schreiber Building, room 309, and in addition 

will be broadcasted via the zoom app, while other talks will run entirely via the zoom app.

The seminar will take place on Wednesdays at 14:10.

Please check each announcement since this is sometimes changed. 

The zoom link is available upon request by email.

 

 

Upcoming Talks        Previous Talks        Previous Years






08.11.2023, 14:10 (Wednesday)
Pierre-Alexandre Arlove (Ruhr University Bochum)



Title: Contact orderability and spectral selectors
Location: Zoom session



Abstract: Some contactomorphisms groups and isotopy classes of
Legendrians admit a natural partial order first studied by Eliashberg
and Polterovich. In this talk I will use this partial order to define
functions on the latter spaces analogous to the spectral invariants in
Symplectic Geometry coming from Lagrangian Floer homology. For
Legendrians, I will show that these functions are spectral selectors
and, from there, derive dynamical applications such as the existence of
translated points and interlinked Legendrians. After discussing the
non-degeneracy of the selectors, I will present applications to the
geometric study of these spaces, namely : construction of time-functions
and the study of various metrics. This is a joint work with Simon
Allais.








15.11.2023, 14:10 (Wednesday) Dan Mangoubi (Hebrew University)



Title: On the inner radius of nodal domains
Location: Zoom session



Abstract: By the well known Faber-Krahn inequality the first Dirichlet Laplace
eigenvalue of a bounded domain \Omega in R^d is bounded from below
by Vol (\Omega)^{-2/d}, for some positive constant C_d depending on
dimension.

Consider a closed Riemannian manifold of dimension d. Let u be an
eigenfunction of the Laplace-Beltrami operator with eigenvalue \lambda.
Every connected component \Omega of $u\neq 0$ is called a nodal domain
of $u$. It follows from the Faber-Krahn inequality that Vol(\Omega)>= C
\lambda^{-d/2}.
A refined question due to Leonid Polterovich is whether one can inscribe
in \Omega a ball of radius C\lambda^{-1/2}.
The answer is positive in dimension two (M., 2006). In higher dimensions we
show that this is almost true: One can inscribe in \Omega a ball of radius
C\lambda^{-1/2}(\log\lambda)^{-(d-2)/2}.
I will explain several ideas which go into the proof.
The talk is based on joint work with Philippe Charron.








22.11.2023, 14:10 (Wednesday) Vincent Humiliere (Sorbonne University, Paris)



Title: Morse/Floer theory with DG-coefficients and periodic orbits in magnetic cotangent bundles.
Location: Zoom session



Abstract: In joint work (in progress) with Jean-François Barraud, Mihai Damian and Alexandru Oancea, and building on the work of Barraud and Cornea (2004), we develop a Morse/Floer theory with coefficients in a DG-local system. This allows for instance to recover the homology of a fibration (whose fiber does not necessarily have finite dimension) from Morse/Floer data on the base of the fibration. I will present an application to Hamiltonian dynamics, namely existence results for periodic orbits on certain energy hypersurfaces in magnetic cotangent bundles.







29.11.2023, 14:10 (Wednesday) Igor Uljarevic (University of Belgrade)



Title: Spectral invariants for a contact Hamiltonian
Location: Zoom session



Abstract: In this talk, I will introduce a persistence module
associated with a contact Hamiltonian on a fillable contact manifold
and discuss numerical invariants that can be extracted from it.
In particular, I will talk about spectral invariants and their
properties. This talk is based on a joint work with Danijel Djordjevic
and Jun Zhang.








06.12.2023, 14:10 (Wednesday) Alejandro Vicente (Hebrew University)



Title: Integrable systems, Lagrangian fibrations and symplectic embedding problems
Location: Zoom session



Abstract: Toric domains are a special class of symplectic manifolds
in C^n, invariant by the standard circle action in each of the
one-dimensional complex planes. This abundance of symmetries makes
the study of embedding problems into toric domains and computations of
symplectic capacities of toric domains, often possible. So when
considering these kinds of problems in general manifolds (for example:
disk cotangent bundles of surfaces), a strategy could be to produce
a toric domain out of your given manifold and "reduce" the original
problem to a similar one in toric domains.

In this talk, I will explain how to carry out this idea for computing
the biggest ball that can be symplectically embedded into the disk
cotangent bundle of an ellipsoid of revolution. The idea to obtain such
a related toric domain comes by studying an integrable system for the
disk cotangent bundle of an ellipsoid of revolution and using
Arnol'd-Liouville Theorem to obtain action-angle coordinates. We then
use the obtained toric domain to suggest a candidate to the best
symplectically embedded ball. Finally, to show that this is, as a
matter of fact, the best possible ball, we use some obstructional tools,
more specifically, ECH capacities. This part of the talk is joint work
with Brayan Ferreira and Vinicius Ramos.

We will also see how to obtain these results from another point of view,
namely, by using the rotational symmetry to realize these disk cotangent
bundles as almost toric fibrations. With this last technique, we will
recover some old results from Vinicius Ramos on the Lagrangian bidisk 
B^2x B^2. Furthermore, we construct some interesting Lagrangian fibrations
in the Lagragian bidisk B^3x B^3 and the disk cotangent bundle of S^3.
This is work in progress with Santiago Achig-Andrango and Renato Vianna.








13.12.2023, 17:10 (Wednesday) Dylan Cant (University of Montreal)




Title: A symplectic cohomology persistence module for contact isotopies of the ideal boundary of a Liouville manifold
Location: Zoom session



Abstract: Let W be a Liouville manifold with ideal contact boundary Y.
It is well-known that the contact geometry of the Y is influenced by
the symplectic geometry of W. Notably, every contact isotopy of Y is
the "ideal restriction" of some Hamiltonian isotopy of W. The associated
Floer cohomology group depends only on the contact isotopy. Any contact
isotopy can be "wrapped" by the Reeb flow associated to a choice of
contact form (the wrapping depends on some parameter), and the associated
Floer cohomology groups can be organized into a persistence module.
As one varies the contact isotopy, the resulting barcode varies
Lipshitz-continuously with respect to Shelukhin's Hofer norm (with
Lipshitz constant 1). This structure is used to prove various
existence results for "translated points," and to construct
spectral invariants for contactomorphisms.








20.12.2023, 14:10 (Wednesday) Matthias Meiwes (Tel Aviv University)



Title: Orbit growth in link complements and 3D Reeb flows
Location: Zoom session



Abstract: Recently, Alves and Pirnapasov studied the growth of contact homology in the complement of a link of closed Reeb orbits on contact 3-manifolds and discovered orbit forcing phenomena that are special to Reeb flows. Motivated by those findings, they asked a question which can be formulated roughly as follows: Can we, by taking a suitable sequence of links of closed orbits, recover the topological entropy of any 3D Reeb flow by the growth of contact homology in the complement of those links? In this talk, I will explain a result that gives a positive answer to that question for generic Reeb flows. I will also discuss some applications of (variants of) that result. One is in a joint work with M. Alves, L. Dahinden, and A. Pirnapasov where we studied robustness features of the topological entropy of Reeb flows.







27.12.2023, 14:10 (Wednesday) Sara Tukachinsky (Tel Aviv University)



Title: Open-closed maps as classifiers in open Gromov-Witten theory
Location: Zoom session



Abstract: Open-closed maps relate cohomology theories of a symplectic manifold and a Lagrangian submanifold that make use of pseudo-holomorphic curves. Bounding chains are differential forms on a Lagrangian submanifold that solve a particular (Maurer-Cartan) equation, also involving pseudo-holomorphic curves. Such forms are used in defining Lagrangian Floer cohomology and open Gromov-Witten invariants. Under non-trivial topological assumptions, a version of the open-closed map turns out to give a full classification of the space of all bounding chains, up to a natural equivalence relation. Partly based on joint work with Pavel Giterman.







03.01.2024, 14:10 (Wednesday) Iosif Polterovich (University of Montreal)



Title: Stability of isoperimetric eigenvalue inequalities on surfaces
Location: Schreiber 309 and zoom



Abstract: Optimisation of Laplace eigenvalues on Riemannian manifolds is a fascinating topic in spectral geometry. In the past decade, significant progress has been achieved on maximisation of eigenvalues on surfaces under the area constraint. I will discuss some recent advances on this subject, with an emphasis on the stability estimates for sharp isoperimetric inequalities. The talk is based on a joint work with M. Karpukhin, M. Nahon and D. Stern.







10.01.2024, 14:00 (Wednesday)
Xujia Chen (Harvard University)



Title: Why can Kontsevich's invariants detect exotic phenomena?
Location: Zoom session



Abstract: In topology, the difference between the category of smooth manifolds and the category of topological manifolds has always been a delicate and intriguing problem, called the "exotic phenomena". The recent work of Watanabe (2018) uses the tool "Kontsevich's invariants" to show that the group of diffeomorphisms of the 4-dimensional ball, as a topological group, has non-trivial homotopy type. In contrast, the group of homeomorphisms of the 4-dimensional ball is contractible. Kontsevich's invariants, defined by Kontsevich in the early 1990s from perturbative Chern-Simons theory, are invariants of (certain) 3-manifolds / fiber bundles / knots and links (it is the same argument in different settings). Watanabe's work implies that these invariants detect exotic phenomena, and, since then, they have become an important tool in studying the topology of diffeomorphism groups. It is thus natural to ask: how to understand the role smooth structure plays in Kontsevich's invariants? My recent work provides a perspective on this question: the real blow-up operations on a smooth manifold depends on the smooth structure in an essential way; thus, the topology of the spaces obtained by doing some blow-ups on a smooth manifold/fiber bundle X encodes information of the smooth structure on X.







17.01.2024, 14:00 (Wednesday)
Robert Cardona (University of Barcelona)



Title: Contact topology and time-dependent hydrodynamics: non-mixing and spectral invariants
Location: Zoom session



Abstract: The well-known connection between contact topology and the Euler equations for ideal fluids, introduced in the seminal work of Etnyre and Ghrist, is restricted to the study of stationary solutions, and usually for adapted instead of fixed ambient metrics. In this talk, we broaden the scope of contact hydrodynamics by presenting a new framework that allows assigning contact/symplectic invariants to large sets of time-dependent solutions to the Euler equations on any three-manifold with an arbitrary fixed Riemannian metric. Applications include a general non-mixing result for the infinite-dimensional dynamical system defined by the equations and the existence of new non-trivial first integrals of the PDE obtained from spectral invariants in embedded contact homology. This is based on joint work with Francisco Torres de Lizaur.







24.01.2024, 17:10 (Wednesday) Filip Brocic (University of Montreal)



Title: Arnold’s chord conjecture for conormal Legendrian lifts
Location: Zoom session



Abstract: The chord conjecture, due initially to Arnold in the case of the standard contact three-sphere, asserts the existence of a Reeb chord with boundary on every closed Legendrian submanifold of a closed contact manifold for every contact form. This conjecture was established in various settings by Cieliebak, Mohnke, Hutchings and Taubes, and others. In this talk, I will sketch a proof of the chord conjecture for conormal bundles of closed submanifolds of any closed manifold seen as Legendrians in the co-sphere bundle. This generalizes a result of Grove in Riemannian geometry regarding the existence of geodesics normal to the submanifold. The method of proof involves wrapped Floer cohomology with local coefficients. This talk is based on a joint work with Dylan Cant and Egor Shelukhin.







31.01.2024, 14:10 (Wednesday) Sara Tukachinsky (Tel Aviv University)



Title: Relative quantum cohomology of complete intersections
Location: Schreiber 309 and zoom session



Abstract: Quantum cohomology of a symplectic manifold X is the usual cohomology, but with wedge product deformed by adding contributions coming from pseudo-holomorphic spheres. Given a Lagrangian submanifold L, there is a relative version of quantum cohomology. It can be thought of as dual to the homology of the complement of L in X. Here, the product is deformed using a combination of pseudo-holomorphic spheres and disks. In this talk, I will recall the above structures, then give explicit computations for the ring structure of relative quantum cohomology of some particularly convenient complete intersections. Joint work with Kai Hugtenburg.







14.02.2024, 14:10 (Wednesday) Yoav Zimhony (Tel Aviv University)



Title: Commutative control data for smoothly locally trivial stratified spaces
Location: Schreiber 309 and zoom session



Abstract: For a compact Lie group $G$ and a Hamiltonian $G$-space $M$ with momentum map $\mu:M\to \g^*$, we prove that the zero level set $\mu^{-1}(0)$ and the critical set $\text{Crit}\norm{\mu}^2$ of the norm squared momentum map are neighbourhood smooth weak deformation retracts. To this end we show that these subsets, stratified by orbit types, satisfy a condition stronger than Whitney (B) regularity --- \textit{smooth local triviality with conical fibers}. Using this condition we construct control data in the sense of Mather with the additional properties that the fiber-wise multiplications by scalars, coming from the tubular neighbourhood structures, preserve strata and commute with each other. We use this control data to obtain the neighbourhood smooth weak deformation retraction. Finally, such structures for the zero level set $\mu^{-1}(0)$ reduce to similar structures for the reduced space $\mu^{-1}(0)/G$, yielding a similar result for the reduced space and its stratified subspaces.







27.03.2024, 14:10 (Wednesday) Susan Tolman (University of Illinois Urbana-Champaign)




Title: Non-Hamiltonian circle actions with minimal fixed points
Location: Schreiber 309 and zoom session



Abstract: Let the circle act on a closed manifold $M$, preserving a symplectic form $\omega$. We say that the action is Hamiltonian if there exists a moment map, that is, a map $\Psi: M \to R$ such that $i_\xi \omega = - d \Psi$, where $\xi$ is the vector field that generates the action. In this case, a great deal of information about the manifold is determined by the fixed set. Therefore, it is very important to determine when symplectic actions are Hamiltonian. There has been a great deal of research on this question. It's easy to see that every Hamiltonian action has fixed points. McDuff proved that the converse wasn't true by constructing a non-Hamiltonian action with fixed tori. She then raised the following question, usually called the ``McDuff conjecture": Does there exist a non-Hamiltonian symplectic circle action with isolated fixed points on a closed, connected symplectic manifold? I was able to construct such an example with 32 fixed points, but this raised another question. What is the minimal number of possible fixed points? I will discuss my work with D. Jang on reducing the number of fixed points. We have already constructed an example with as few as 10 fixed points, and are now working on constructing an example with only two fixed points, which is the smallest possible number.







15.05.2024, 14:10 (Wednesday) Victor Ivrii (University of Toronto), joint session with Analysis seminar



Title: Complete Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Coefficients Operators and Bethe-Sommerfeld Conjecture in Semiclassical Settings
Location: Schreiber 209, no zoom



Abstract: 1. Under certain assumption complete semiclassical asymptotic holds for integrated density of states for periodic and almost periodic perturbations constant coefficient operators.
2. In dimension 2 or higher almost all spectral gaps are missing (in contrast to 1-dimension).









22.05.2024, 14:10 (Wednesday) Konstantin Khanin (University of Toronto)- Distinguished Lecture in Pure Mathematics



Title: On typical rotation numbers for families of circle maps with singularities
Location: Schreiber 309 and zoom session



Abstract: I shall discuss how one can define in a natural way the notion of typical rotation numbers for families of circle maps with singularities. This problem is related to a well known fact that in the case of maps with singularities the set of parameters, corresponding to irrational rotation numbers, has zero Lebesgue measure. Our approach is based on the hyperbolicity of renormalizations. I shall also discuss a natural setting for the Kesten theorem in the case of maps with singularities.







29.05.2024, 17:10 (Wednesday) Andre Neves (University of Chicago) - Blumenthal Lecture in Geometry



Title: Abundance of minimal hypersurfaces.
Location: Zoom session



Abstract: Minimal surfaces are physical objects which appear naturally in math and applied science. In the 80’s Yau conjectured that any closed Riemannian manifold should have an infinite number of closed minimal hypersurfaces. For 30 years little progress was made but over the last 10 years a renewed interest on the problem led to its complete solution. I will survey the results, the new ingredients, and the current state of the art.







05.06.2024, 14:10 (Wednesday) Jean Gutt (University of Toulouse and INU Champollion)



Title: Coarse distance from dynamically convex to convex
Location: Zoom session



Abstract: Chaidez and Edtmair have recently found the first examples of dynamically convex domains in R^4 that are not symplectomorphic to convex domains (called symplectically convex domains), answering a long-standing open question. In this talk we shall present new examples of such domains without referring to Chaidez-Edtmair’s criterion. We shall show that these domains are arbitrarily far from the set of symplectically convex domains in R^4 with respect to the coarse symplectic Banach-Mazur distance by using an explicit numerical criterion for symplectic non-convexity. This is joint work with J.Dardennes, V.Ramos and J.Zhang.










19.06.2024, 14:10 (Wednesday) Baptiste Serraille (Orsay)



Title: The sharp C^0-fragmentation property for Hamiltonian diffeomorphisms and homeomorphisms on surfaces

Location: Room 309 and zoom session




Abstract: Fragmentation properties have been used by Banyaga, Fathi and Thurston in order to study the algebraic structure of groups of diffeomorphisms, volume preserving diffeomorphisms/homeomorphisms and Hamiltonian diffeomorphisms. This property has been improved on surfaces by Entov, Polterovich and Py and later by Seyfaddini in order to give better control on the "fragments". Still on surfaces, I will present a sharper version of those results.








26.06.2024, 14:10 (Wednesday) Lenya Ryzhik (Stanford) - MINT Distinguished Lecture



Title: Diffusion of knowledge and the state lottery society

Location: Room 309 and zoom session




Abstract: Diffusion of knowledge models in macroeconomics describe the evolution of an interacting system of agents who perform individual Brownian motions (this is internal innovation) but also can jump on top of each other (this is an agent or a company acquiring knowledge from another agent or company). The learning strategy of the individual agents (jump probabilities) are obtained from an additional optimization problem that involves the current configuration of particles and is a solution to a forward-backwards in time mean-field game. We will discuss some preliminary results on the basic properties of this system.








26.06.2024, 17:10 (Wednesday) Egor Shelukhin (Universite de Montreal)



Title: A symplectic Hilbert-Smith conjecture

Location: Zoom session




Abstract: I will discuss a recent proof of new cases of the Hilbert-Smith conjecture for actions by homeomorphisms of symplectic nature. It rules out faithful actions of the additive p-adic group and provides further obstructions to group actions in symplectic topology. The proof relies on a new approach to this circle of questions combined with power operations in Floer cohomology and quantitative symplectic topology.








2.07.2024, 14:10 (Tuesday) Lenya Ryzhik (Stanford) - MINT Distinguished Lecture, joint with Analysis seminar



Title: Shape defect function and convergence to traveling waves

Location: Room 209




Abstract: It is well known that solutions to many semilinear parabolic equations can be interpreted in terms of various functionals of branching Brownian motion, most famously so for the Fisher-KPP equation. The shape defect function is a recently reinvented convenient PDE tool that allows to analyze the convergence of the solutions to traveling waves in a very straightforward fashion. Surprisingly, its behavior looks anomalous in the BBM context.








3.07.2024, 14:10 (Wednesday) Lev Buhovsky (TAU)



Title: Asymptotic Hofer geometry on surfaces

Location: Room 309 and zoom session




Abstract: The growth of the Hofer norm of iterations of a Hamiltonian diffeomorphism is still not well understood in general. In all known examples where the growth is not asymptotically linear, it appears to be bounded. This dichotomy phenomenon was previously demonstrated by Polterovich and Siburg for autonomous diffeomorphisms on (open) connected surfaces of infinite area. In my talk, I will discuss a new approach that extends this result to the 2-sphere and potentially to other closed surfaces. Based on a joint work in progress with Ben Feuerstein, Leonid Polterovich and Egor Shelukhin.








10.07.2024, 14:10 (Wednesday) Marco Mazzucchelli (ENS Lyon)



Title: Locally maximal closed orbits of Reeb flows

Location: Zoom session




Abstract: A compact invariant set of a flow is called locally maximal when it is the largest invariant set in some neighborhood. In this talk, based on joint work with Erman Cineli, Viktor Ginzburg, and Basak Gurel, I will present a "forced existence" result for the closed orbits of certain Reeb flows on spheres of arbitrary odd dimension: - If the contact form is non-degenerate and dynamically convex, the presence of a locally maximal closed orbit implies the existence of infinitely many closed orbits. - If the locally maximal closed orbit is hyperbolic, the assertion of the previous point also holds without the non-degeneracy and with a milder dynamically convexity assumption. These statements extend to the Reeb setting earlier results of Le Calvez-Yoccoz for surface diffeomorphisms, and of Ginzburg-Gurel for Hamiltonian diffeomorphisms of certain closed symplectic manifolds.








24.07.2024, 14:10 (Wednesday) Pazit Haim-Kislev (TAU)



Title: TBA

Location: Room 309 and zoom session




Abstract: TBA








31.07.2024, 14:10-15:00 (Wednesday) Adi Dickstein (TAU)



Title: TBA

Location: Room 309 and zoom session




Abstract: TBA








31.07.2024, 15:10-16:00 (Wednesday) Oleg Sheinman (Steklov Mathematical Institute, Moscow)



Title: On reversion of the Abel--Prym map and its applications to integrable systems

Location: Room 309 and zoom session




Abstract: Abel map transfoms a certain symmetric power of a Riemann surface to an Abelian variety called Jacobian of the Riemann surface. In the theory of integrable systems Abel map appeared as early as in Jacobi's "Lectures on dynamics". In frame of the method of Separation of Variables, the phase space of the system exfoliates into symmetric products of curves. We will consider the case when those curves are algebraic. Then the Abel map transforms that foliation into the Lagrangian foliation of the system. It is wellknown that the trajectories of integrable systems are straight line windings of the Lagrangian tori (the fibers of the last foliation). To get trajectories explicitly, in the original separation coordinates, we need to reverse the Abel map. This problem is known as Jacobi inversion problem. Its solution is classical for Jacobians. However, for majority of classical and new integrable systems Lagrangian tori are not Jacobians but different Abelian varieties called Prym varieties, or Prymians. In general, no analog of Jacobi inversion can be formulated for Prymians. We hilight the case when the last nevertheless is possible. As application, we represent the corresponding Prymians as symmetric powers of certain curves, and resolve some Hitchin systems in theta functions. All algebraic geometry preliminaries will be explained.








7.08.2024, 14:10 (Wednesday) Matthias Meiwes (TAU)



Title: TBA

Location: Room 309 and zoom session




Abstract: TBA









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