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08.11.2023, 14:10 (Wednesday)
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Pierre-Alexandre Arlove (Ruhr University Bochum) |
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| Title: |
Contact orderability and spectral selectors |
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| Location: |
Zoom session |
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| 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. |
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| 15.11.2023, 14:10 (Wednesday) |
Dan Mangoubi (Hebrew
University) |
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| Title: |
On the inner radius of nodal domains |
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| Location: |
Zoom session |
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| 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.
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| 22.11.2023, 14:10 (Wednesday) |
Vincent Humiliere (Sorbonne University, Paris) |
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| Title: |
Morse/Floer theory with DG-coefficients and periodic orbits in magnetic
cotangent bundles. |
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| Location: |
Zoom session |
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| 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. |
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| 29.11.2023, 14:10 (Wednesday) |
Igor Uljarevic (University of Belgrade) |
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| Title: |
Spectral invariants for a contact Hamiltonian |
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| Location: |
Zoom session |
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| 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. |
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| 06.12.2023, 14:10 (Wednesday) |
Alejandro Vicente (Hebrew University) |
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| Title: |
Integrable systems, Lagrangian fibrations and
symplectic embedding problems |
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| Location: |
Zoom session
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| 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.
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| 13.12.2023, 17:10 (Wednesday) |
Dylan Cant (University of Montreal)
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| Title: |
A symplectic cohomology persistence module for
contact isotopies of the ideal boundary of a Liouville
manifold |
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| Location: |
Zoom session |
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| 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. |
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| 20.12.2023, 14:10 (Wednesday) |
Matthias Meiwes
(Tel Aviv University) |
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| Title: |
Orbit growth in link complements and 3D Reeb flows |
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| Location: |
Zoom session |
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| 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.
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| 27.12.2023, 14:10 (Wednesday) |
Sara Tukachinsky (Tel Aviv University) |
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| Title: |
Open-closed maps as classifiers in open Gromov-Witten theory |
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| Location: |
Zoom session |
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| 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. |
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| 03.01.2024, 14:10 (Wednesday) |
Iosif Polterovich (University of Montreal) |
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| Title: |
Stability of isoperimetric eigenvalue inequalities on surfaces |
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| Location: |
Schreiber 309 and zoom |
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| 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. |
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10.01.2024, 14:00 (Wednesday)
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Xujia Chen (Harvard University) |
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| Title: |
Why can Kontsevich's invariants detect exotic phenomena? |
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| Location: |
Zoom session |
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| 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. |
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17.01.2024,
14:00 (Wednesday)
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Robert Cardona (University of Barcelona) |
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| Title: |
Contact topology and time-dependent hydrodynamics: non-mixing and spectral invariants |
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| Location: |
Zoom session |
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| 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. |
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| 24.01.2024, 17:10 (Wednesday) |
Filip Brocic (University of Montreal) |
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| Title: |
Arnold’s chord conjecture for conormal Legendrian lifts |
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| Location: |
Zoom session |
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| 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.
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| 31.01.2024, 14:10 (Wednesday) |
Sara Tukachinsky (Tel Aviv University) |
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| Title: |
Relative quantum cohomology of complete intersections |
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| Location: |
Schreiber 309 and zoom session |
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| 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.
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| 14.02.2024, 14:10 (Wednesday) |
Yoav Zimhony (Tel Aviv University) |
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| Title: |
Commutative control data for smoothly locally trivial stratified spaces |
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| Location: |
Schreiber 309 and zoom session |
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| 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.
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| 27.03.2024, 14:10 (Wednesday) |
Susan Tolman (University of Illinois Urbana-Champaign)
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| Title: |
Non-Hamiltonian circle actions with minimal fixed points |
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| Location: |
Schreiber 309 and zoom session |
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| 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. |
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| 03.04.2024, 14:10 (Wednesday) |
TBA |
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TBA |
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TBA |
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TBA |
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| 10.04.2024, 14:10 (Wednesday) |
TBA |
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| Title: |
TBA |
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| Location: |
TBA |
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TBA |
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| 08.05.2024, 14:10 (Wednesday) |
TBA |
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| Title: |
TBA |
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| Location: |
TBA |
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TBA |
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| 15.05.2024, 14:10 (Wednesday) |
TBA |
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| Title: |
TBA |
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| Location: |
TBA |
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TBA |
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| 22.05.2024, 14:10 (Wednesday) |
Konstantin Khanin (University of Toronto),
Distinguished Lecture in Pure Mathematics |
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| Title: |
TBA |
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| Location: |
Schreiber 309 and zoom session |
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| Abstract: |
TBA |
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| 29.05.2024, 14:10 (Wednesday) |
Andre Neves (University of Chicago) - Blumenthal
Lectures in Geometry |
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| Title: |
I: Abundance of Minimal hypersurfaces. II: Minimal surfaces in negatively curved manifolds |
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| Location: |
Schreiber 309 and zoom session |
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| Abstract: |
I: 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.
II: For negatively curved manifolds the study of closed geodesics is fairly advanced and one can answer such questions as what is the growth rate of closed geodesics and what does a random closed geodesic look like. In comparison almost nothing is known about the behavior of minimal surfaces in negatively curved manifolds but in the last years some progress has been made. I will survey the recent results. |
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| 05.06.2024, 14:10 (Wednesday) |
TBA |
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TBA |
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TBA |
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TBA |
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| 19.06.2024, 14:10 (Wednesday) |
TBA |
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| Title: |
TBA
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TBA
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TBA
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