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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)
|
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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| 15.05.2024, 14:10 (Wednesday) |
Victor Ivrii (University of Toronto),
joint session with Analysis seminar |
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| Title: |
Complete
Spectral Asymptotics for Periodic and Almost Periodic Perturbations of
Constant Coefficients Operators and Bethe-Sommerfeld Conjecture in
Semiclassical Settings |
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| Location: |
Schreiber 209, no zoom |
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| 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).
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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: |
On typical rotation numbers for families of circle maps with singularities |
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| Location: |
Schreiber 309 and zoom session |
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| 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. |
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| 29.05.2024, 17:10 (Wednesday) |
Andre Neves (University of Chicago) - Blumenthal
Lecture in Geometry |
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| Title: |
Abundance of minimal hypersurfaces. |
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| Location: |
Zoom session |
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| 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. |
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| 05.06.2024, 14:10 (Wednesday) |
Jean Gutt (University of Toulouse and INU Champollion) |
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| Title: |
Coarse distance from dynamically convex to convex |
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| Location: |
Zoom session |
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| 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. |
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| 19.06.2024, 14:10 (Wednesday) |
Baptiste Serraille (Orsay) |
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| Title: |
The sharp C^0-fragmentation property for Hamiltonian
diffeomorphisms and homeomorphisms on surfaces
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| Location: |
Room 309 and zoom session
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| 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.
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| 26.06.2024, 14:10 (Wednesday) |
Lenya Ryzhik (Stanford) - MINT Distinguished Lecture |
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| Title: |
Diffusion of knowledge and the state lottery society
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| Location: |
Room 309 and zoom session
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| 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.
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| 26.06.2024, 17:10 (Wednesday) |
Egor Shelukhin (Universite de Montreal) |
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| Title: |
A symplectic Hilbert-Smith conjecture
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| Location: |
Zoom session
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| 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.
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| 2.07.2024, 14:10 (Tuesday) |
Lenya Ryzhik (Stanford) - MINT Distinguished Lecture, joint with Analysis seminar |
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| Title: |
Shape defect function and convergence to traveling waves
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| Location: |
Room 209
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| 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.
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| 3.07.2024, 14:10 (Wednesday) |
Lev Buhovsky (TAU) |
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| Title: |
Asymptotic Hofer geometry on surfaces
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|
| Location: |
Room 309 and zoom session
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| 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.
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| 10.07.2024, 14:10 (Wednesday) |
Marco Mazzucchelli (ENS Lyon) |
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| Title: |
Locally maximal closed orbits of Reeb flows
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| Location: |
Zoom session
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| 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.
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| 24.07.2024, 14:10 (Wednesday) |
Pazit Haim-Kislev (TAU) |
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| Title: |
A counterexample to Viterbo's conjecture
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| Location: |
Room 309 and zoom session
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| Abstract: |
Viterbo's volume-capacity conjecture asserts that among all convex bodies
of the same volume, the ball has the largest capacity. This conjecture has been highly influential in the study
of symplectic capacities since its introduction in 2000, sparking extensive research.
In this talk, I will present a counterexample to Viterbo's conjecture in every dimension, demonstrating that not all
capacities coincide on the class of convex domains. This is a joint work with Yaron Ostrover.
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| 31.07.2024, 14:10-15:00 (Wednesday) |
Adi Dickstein (TAU) |
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| Title: |
Constraints on symplectic quasi-states
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| Location: |
Room 309 and zoom session
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| Abstract: |
Symplectic
quasi-states, introduced by Entov and Polterovich, are non-necessarily
linear functionals on the space of real-valued continuous functions on
closed symplectic manifolds. Currently, in dimensions greater than two,
the only known constructions of non-linear symplectic quasi-states rely
on Floer theory.
In this talk, I will explore the question: "Is there a simpler way to
construct non-linear symplectic quasi-states?" I will present a
construction of a more general object known as a topological
quasi-state, introduced by Aarnes, and show that such topological
quasi-states are symplectic only if they are already linear. The proof
uses new result on symplectic embeddings. This talk is based on joint
work with Frol Zapolsky.
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| 31.07.2024, 15:10-16:00 (Wednesday) |
Oleg Sheinman (Steklov Mathematical Institute, Moscow) |
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| Title: |
On reversion of the Abel--Prym map and its applications to integrable
systems
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| Location: |
Room 309 and zoom session
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| 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.
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| 7.08.2024, 14:10 (Wednesday) |
Matthias Meiwes (TAU) |
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| Title: |
The fine curve graph and Hamiltonian diffeomorphisms
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| Location: |
Room 309 and zoom session
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| Abstract: |
The
fine curve graph of a closed oriented surface S of positive genus g was
introduced by Bowden, Hensel, and Webb, as a variant of the curve
graph. Its vertex set is the set of non-contractible simple closed
curves and its edges are disjoint curves (when g>=2) or curves
intersecting at most once (g=1). This graph is Gromov hyperbolic and it
is a useful object for studying the identity component of the group of
diffeomorphisms on S, which acts on the graph by isometries. The above
authors constructed unbounded quasimorphisms on that group, answering a
question by Burago-Ivanov-Polterovich. Subsequently, the distinct types
of group elements (hyperbolic, parabolic, elliptic), defined according
to their action on the fine curve graph, have been investigated by
several researchers and are now quite well understood. In my talk, I
will first review some of those recent results, and then ask about
specific features of that action by elements in the subgroup of
Hamiltonian diffeomorphisms. In particular, I will discuss the
behaviour of hyperbolic action with respect to the Hofer metric, and a
certain variant of asymptotic translation length of a Hamiltonian
element. Based on joint work in progress with Arnon Chor and Marcelo
Alves.
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