Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Dynamics, Geometry, & Groups Seminar

Dynamics, Geometry, & Groups - Alena Erchenko (Stony Brook)

Thursday, April 1st, 2021

Time: 1:30 p.m Place:

Speaker: Alena Erchenko (Stony Brook University)

Title: Riemannian Anosov extension.

Abstract: Consider a smooth Riemannian manifold Σ with strictly convex spherical boundary, hyperbolic trapped set (possibly empty) and no conjugate points. We show that Σ can be isometrically embedded into a closed Riemannian manifold with Anosov geodesic flow. We will explain one of the main ingredients which is the analysis of the behavior of Jacobi fields. We also discuss some applications of the main result. This is a joint work with Dong Chen and Andrey Gogolev.

Dynamics, Geometry, & Groups - Maxime Fortier-Bourque (Glasgow)

Thursday, March 25th, 2021

Time: 1:30 p.m Place:

Speaker: Maxime Fortier-Bourque (University of Glasgow)

Title: Geometric inequalities via trace formulas.

Abstract: The systole of a metric space is the length of its shortest closed geodesic and its kissing number is the number of distinct homotopy classes of closed geodesics realizing the systole. Given a moduli space of metric spaces, like the space of flat tori of a given dimension or the space of closed hyperbolic surfaces of a given genus, the goal is to find how large these quantities can get. After stating what is known so far, I will describe ongoing joint work with Bram Petri in which we obtain new upper bounds on the systole and kissing number of hyperbolic surfaces. Our approach is based on the Selberg trace formula and is inspired by work of Cohn and Elkies on the maximal density of sphere packings in Euclidean spaces.

Dynamics, Geometry, & Groups - Ioannis Iakovoglou (IMB)

Thursday, March 18th, 2021

Time: 1:30 p.m Place:

Speaker: Ioannis Iakovoglou (Institut de Mathématiques de Bourgogne)

Title: Anosov flows in dimension 3: a dynamical game describing the actions of surgeries on the foliations.

Abstract: From every Anosov flow in dimension 3 it is possible to construct infinitely many others via Dehn-Goodman-Fried surgery. Similarly, to the theorem of Lickorish-Wallace stating that any (closed orientable and connected) 3-manifold is obtained by Dehn surgeries on the 3-sphere, conjecturally the same thing happens for (transitive with orientable foliations) Anosov flows in dimension 3. Motivated by this question, in a recent work with C.Bonatti we propose a dynamical game on the plane as a means to understand the foliations of an Anosov flow after surgery. In this talk, I will introduce this dynamical game on the plane together with some interesting questions around it and their relation with Anosov flows in dimension 3.

Dynamics, Geometry, & Groups - Anthony Sanchez (U Washington)

Thursday, March 4th, 2021

Time: 1:30 p.m Place:

Speaker: Anthony Sanchez (University of Washington, Seattle)

Title: Gaps of saddle connection directions for some branched covers of tori.

Abstract: Holonomy vectors of translation surfaces provide a geometric generalization for higher genus surfaces of (primitive) integer lattice points. The counting and distribution properties of holonomy vectors on translation surfaces have been studied extensively. In this talk, we consider the following question: How random are the holonomy vectors of a translation surface? We motivate the gap distribution of slopes of holonomy vectors as a measure of randomness and compute the gap distribution for the class of translation surfaces given by gluing two identical tori along a slit. No prior background on translation surfaces or gap distributions will be assumed.

Dynamics, Geometry, & Groups - Idris Assani (UNC)

Wednesday, February 24th, 2021

Time: 11:30 a.m Place:

Speaker: Idris Assani (University of North Carolina at Chapel Hill)

Title: Wiener Wintner dynamical systems.

Abstract: WW dynamical systems form a class of ergodic dynamical systems that we introduced and talked about in our 2003 book "Wiener Wintner ergodic theorems". This class offers simple proof of difficult pointwise results such as the return times and the a.e. double recurrence . We will present these simple proofs in this talk.

Dynamics, Geometry, & Groups - Anthony Genevois (CNRS)

Thursday, February 11th, 2021

Time: 1:30 p.m Place:

Speaker: Anthony Genevois (Centre national de la recherche scientifique)

Title: Asymptotic geometry of lamplighters over one-ended groups.

Abstract: This talk will be dedicated to the asymptotic geometry of wreath products of the form (finite group)wr(finitely presented one-ended group). After a general introduction, I will describe and explain a complete classification of these groups up to quasi-isometry. (Joint work with R. Tessera.)

Dynamics, Geometry, & Groups - Nir Lazarovich (Technion)

Thursday, February 4th, 2021

Time: 1:30 p.m Place:

Speaker: Nir Lazarovich (Technion Israel Institute of Technology)

Title: Flexible Stability of Surface Groups.

Abstract: Roughly speaking, a finitely presented group is said to be (flexibly) stable if any approximate action of the group on a finite set is an approximation of an action. Stability is closely related to local testability (in CS), soficity of groups, and residual finiteness. In this joint work with Arie Levit and Yair Minsky, we show that surface groups are flexibly stable using the geometry of CAT(-1) spaces and a new quantitative variant of LERF.

Dynamics, Geometry, & Groups - Marcin Sabok (McGill University)

Thursday, January 28th, 2021

Time: 1:30 p.m Place:

Speaker: Marcin Sabok (McGill University)

Title: Hyperfiniteness at Gromov boundaries.

Abstract: I will discuss recent results establishing hyperfiniteness of equivalence relations induced by actions on Gromov boundaries of various hyperbolic spaces. This includes boundary actions of hyperbolic groups (joint work with T. Marquis) and actions of the mapping class group on boundaries of the arc graph and the curve graph (joint work with P. Przytycki).

Dynamics, Geometry, & Groups - Macarena Arenas (Cambridge)

Thursday, January 21st, 2021

Time: 1:30 p.m Place:

Speaker: Macarena Arenas (University of Cambridge)

Title: Linear isoperimetric functions for surfaces in hyperbolic groups.

Abstract: One of the main characterisations of word-hyperbolic groups is that they are the groups with a linear isoperimetric function. That is, for a compact 2-complex X, the hyperbolicity of its fundamental group is equivalent to the existence of a linear isoperimetric function for disc diagrams D -->X.
It is likewise known that hyperbolic groups have a linear annular isoperimetric function and a linear homological isoperimetric function. I will talk about these isoperimetric functions, and about a (previously unexplored) generalisation to all homotopy types of surface diagrams. This is joint work with Dani Wise.

Dynamics, Geometry, & Groups - Hakan Doga (SUNY Buffalo)

Thursday, January 14th, 2021

Time: 1:30 p.m Place:

Speaker: Hakan Doga (SUNY Buffalo)

Title: A Combinatorial Description of the Knot Concordance Invariant Epsilon.

Abstract: Sitting at the intersection of 4-dimensional topology and knot theory, the knot concordance group is an important object in low-dimensional topology whose structure is not yet fully explored and understood. One approach to study knot concordance is to use knot Floer homology, introduced by Ozsvath-Szabo and Rasmussen independently in early 2000s, and the invariants obtained from this theory. In this talk, I will describe the knot concordance, introduce some basic definitions of the combinatorial knot Floer homology called the "grid homology", explain our method of computing the concordance invariant epsilon and talk about some results. This is a joint work with S. Dey.