October 4: Amie Wilkinson
Title: An Introduction to Centralizer Rigidity
Abstract:The centralizer of a diffeomorphism (flow) is the set of all diffeomorphisms (flows) that commute with it under composition. It has been conjectured by Smale (and proven in some settings) that the generic diffeomorphism/flow has the smallest possible centralizer, containing only the original dynamics. I'll describe a project with Danijela Damjanović and Disheng Xu to understand the un-generic situation: what happens when the centralizer is larger than expected?
October 11: Danny Calegari
Title: Combinatorics of the Tautological Elamination
Abstract:The Shift Locus is the space of degree q normalized polynomials in one complex variable all of whose critical points are in the attracting basin of infinity. One (complex) dimensional slices of this space — “Butcher’s Slices” — are described combinatorially and holomorphically by certain objects called {\em Tautological Elaminations}. The combinatorics of these objects are rather intricate, and have two very different descriptions related by the so-called “Sausage map”. We will try to describe these objects and the correspondence, explain some of the implications for the combinatorics, and (if time permits) state a couple of conjectures.
October 18: Seung uk Jang
Title: Kummer Rigidity for Hyperbolic Hyperkaehler Automorphisms
Abstract:We show that a hyperbolic (i.e., has positive entropy) holomorphic automorphism on a projective hyperkaehler manifold, with volume being the measure of maximal entropy, is necessarily a Kummer example. This generalizes the analogous results for compact complex surfaces, e.g. (Cantat-Dupont 2020) or (Filip-Tosatti 2018). The structure of dynamical degrees of hyperkaehler manifolds, together with a trick with Jensen's inequality, discovers that the stable and unstable distributions are holomorphic and exhibit uniform rate of contraction and expansion. Together with some complex-geometric facts, we then can show that our hyperkaehler manifold and automorphism are the normalization of a torus quotient and its automorphism, i.e., a Kummer example.
October 25: Jon DeWitt
Title: Periodic Data Rigidity of an Anosov Automorphism with a Jordan Block
Abstract: We consider the regularity of a conjugacy between an Anosov automorphism of a torus and a C^1-small perturbation. We say that an Anosov automorphism is locally periodic data rigid if this conjugacy is C^{1+} whenever the perturbation is C^{1+} and the return maps of the perturbed differential are conjugate to those of the automorphism. We give an example of an Anosov automorphism that is periodic data rigid despite not being irreducible.
November 1: Alex Wright (University of Michigan)
Title: Billiards, renormalization, and Hurwitz spaces
Abstract: We produce a new subvariety of the Hodge bundle in genus 8 that is invariant under the GL(2,R) action. The translation surfaces in this orbit closure admit a somewhat miraculous map to the Riemann sphere with monodromy a certain solvable group of order 400, which is totally invisible from the point of view of flat geometry and dynamics. The general outline of our argument is soft and algebro-geometric, but it is an open question which of the group theoretic parameters can be modified to yield new invariant subvarieties. Joint work with Vincent Delecroix and Julian Rueth.
November 8: Frederik Benirschke
Title: Orbit closures, cylinder deformations and meromorphic differentials
Abstract: Strata of translation surfaces have a natural GL(2,R)-action and the study of its orbit closures has seen tremendous progress in recent years. One of the big open problems is the classification problem: Can one write down a list of all orbit closures? We motivate one possible approach to the classification problem using degeneration techniques and explain how this approach naturally leads to meromorphic differentials. Translation surfaces of meromorphic differentials are much less well behaved than their holomorphic counterparts but some of the structure is preserved. As an example we show that Wright’s cylinder deformation theorem can be generalized to meromorphic differentials. This is partly joint work with Benjamin Dozier and Samuel Grushevsky.
November 15: Kasra Rafi (University of Toronto)
Title:The sublinearly Morse boundary as a Model for the Poisson boundary.
Abstract: In algology with the Gromov boundary of a Gromov hyperbolic space, we define a notion of boundary that identifies the hyperbolic directions in a proper geodesic mantric space X. It turns out many arguments in the setting of Gromov hyperbolic spaces can be carried out with sub-linear error terms instead of uniform. For example, the notion of Morse geodesics can be replaces with \kappa-Morse geodesics for a given a sublinear function \kappa. We define the \kappa-boundary of X to be the space of all \kappa-Morse quasi-geodesics rays. We show that this boundary, equipped with the corse visual topology, is QI-invariant, metrizable and large. Namely, for a large class of groups, the generic direction is represented by a \kappa-Morse geodesic and the \kappa-boundary can be used as a topological model for the Poisson boundary of random walk in the group. The talk is based on several joint projects with Ilya Gekhtman, Yulan Qing and Giulio Tiozzo.
November 29: Michael Hutchings (UC Berkeley)
Title:Quantitative C^\infty closing lemmas for area-preserving surface diffeomorphisms
Abstract:We show that an area-preserving diffeomorphism of a closed surface satisfying a "rationality" property has the "C^\infty closing property". The latter property asserts that for any nonempty open set, one can make a C^\infty small Hamiltonian perturbation supported in the open set to obtain a periodic orbit intersecting the open set. Moreover we obtain quantitative results, asserting roughly speaking that during a given Hamiltonian isotopy, within time \delta a periodic orbit must appear of period at most O(\delta^{-1}). The proof uses spectral invariants in periodic Floer homology. This is a joint work with Oliver Edtmair.