Abstracts

Howard Masur: Dynamical systems on translation surfaces and on moduli spaces

Abstract: A polygon in the plane with pairs of sides identified by translations give rise to a surface with a Euclidean structure except at a finite number of cone points. For each slope there is a flow on the surface by straight lines of that slope. Translation surface of a fixed genus and cone angles fit together to form a moduli space. There is an action of the group SL(2,R) on each such moduli space. There is a deep connection between these two dynamical systems. Knowing the SL(2,R) orbit of a surface in its moduli space often gives information about the dynamics of the straight line flows on the surface itself. In this introductory talk I will introduce translation surfaces, these two dynamical systems, and explain a few of the connections.

Jasmine Bhullar: Equilibrium states and d-bar continuity beyond subshifts of finite type

Abstract: The concept of d-bar distance on the space of invariant measures on a shift space was introduced by Ornstein to study the isomorphism problem for Bernoulli shifts. For mixing subshifts of finite type, Coelho and Quas showed that the map that sends a Hölder continuous potential to its equilibrium state is d-bar continuous. In this talk, we will discuss an extension of this result for countable-state Markov shifts.

Karen Butt: Quantitative marked length spectrum rigidity

Abstract: The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. Conjecturally, the marked length spectrum determines the metric up to isometry (Burns--Katok). This is known to be true in some special cases, namely in dimension 2 (Otal, Croke), in dimension at least 3 if one of the metrics is locally symmetric (Hamenstadt, Besson--Courtois--Gallot), and in any dimension if the metrics are assumed to be sufficiently close in a suitable C^k topology (Guillarmou--Knieper--Lefeuvre). Even in these cases, there is more to be understood about to what extent the marked length spectrum determines the metric. Namely, if two manifolds have marked length spectra which are not equal but are close, is there some sense in which the metrics are close to being isometric? In this talk, we will provide some (quantitative) answers to this question, refining the known rigidity results for surfaces and for locally symmetric spaces of dimension at least 3.

Akshat Das: A three gap theorem for the adeles

Abstract: In order to understand problems in dynamics which are sensitive to arithmetic properties of return times to regions, it is desirable to generalize classical results about rotations on the unit circle to the setting of rotations on adelic tori. The classical three gap theorem (or Steinhaus conjecture) states that, there are at most three distinct gap lengths in the fractional parts of the sequence a, 2a, ..., Na for any real number a and positive integer N. One of the more recent proofs of this has been given by Marklof and Strömbergsson, in which they use a lattice based approach to gaps problems in Diophantine approximation. In this talk, we use an adaptation of this approach to prove a natural generalization of the classical three gap theorem, for rotations on adelic tori. This is joint work with Alan Haynes.

Alex Kapiamba: The Geometry of the Mandelbrot Set

Abstract: The Mandelbrot set is a complicated fractal object, and understanding its geometry has been an active area of research over the past forty years. In this talk, we will explore this set and survey some of what is known and unknown about its structure.

Kitty Yang: Symmetry groups of subshifts

Abstract: Let A be a finite alphabet. The set of bi-infinite sequences of over A, together with the left-shift map, is the full shift over A. A subshift is a closed, shift-invariant subset of the full-shift. We define two symmetry groups of a subshift - the automorphism group and the mapping class group - and discuss the relationship between them, as well as how dynamical complexity affects these symmetry groups.