Titles and Abstracts

Kathryn Lindsey, Boston College

Title: Rational maps built by mixing two polynomials

Abstract:  A mating of two quadratic polynomials (with connected Julia sets) is, roughly speaking, a rational map that is built by “gluing” the two filled Julia sets together along their boundaries to form a topological sphere, and then defining the map on this sphere to be whichever polynomial is defined. 

Together with Sarah Koch and Tom Sharland, I introduced a new and related way to view some rational maps as the result of “gluing” together two filled Julia sets: a mixing.  In contrast to a traditional mating of Julia sets (in which the two filled Julia sets are invariant), a mixing intertwines the dynamics on the two Julia sets, in such a way that the second iterate of the mixing coincides with the second iterate of the mating.  In order to form this definition, we prove theorems describing pairs of bicritical rational maps that have shared iterates. 

Chenxi Wu, University of Wisconsin

Title: Zeroes of zeta functions and polynomial core entropies 

Abstract: I will discuss some prior work with Kathryn Lindsey, Giulio Tiozzo, Ethan Farber, Harrison Bray and Diana Davis on core entropies of certain families of complex polynomial maps, and the generalization of these results to more general setting of families of sub shifts of finite types arising from certain linear orders.

Sean Li, UConn

Title: Quantitative rectifiability in the Heisenberg group

Abstract: The Heisenberg group can be thought of as a non-Abelian generalization of Euclidean space and has been the target of intense research in the past three decades. We will discuss the Heisenberg group and recent progress in the quantitative study of rectifiability for dimension-1 and codimension-1 sets in this setting.  Time permitting, we will also discuss applications to singular integrals on subsets of the Heisenberg group.

Lam Pham, Brandeis

Title: Short closed geodesics in higher rank arithmetic orbifolds

Abstract: A well-known conjecture of Margulis predicts the existence of a uniform lower bound on the systole of any irreducible arithmetic locally symmetric space. In joint work with F. Thilmany, we proved that this conjecture is equivalent to a weak version of the Lehmer conjecture, a well-known problem from Diophantine geometry.
Taking the analysis of the length spectrum further, in joint work with M. Fraczyk, we established a uniform lower bound for the lengths of closed geodesics in the case of simple Lie groups of any rank conditional on a uniform lower bound on Salem numbers, a much weaker — but still open — problem. In particular, our work establishes an unconditional uniform lower bound for the lengths of many closed geodesics. I will discuss these results and highlight the structure of the bottom of the length spectrum of arithmetic orbifolds.

Noelle Sawyer, Southwestern

Title: Unique Equilibrium States for Geodesic Flows on Translation Surfaces

In this talk I will discuss some known results about the geodesics and equilibrium states of the geodesic flow in negative curvature. After I will introduce some of the tools and techniques needed to show the uniqueness of equilibrium states in the setting of translation surfaces. This is joint work with Benjamin Call, Dave Constantine, Alena Erchenko, and Grace Work. Grace will continue the description of our work in her talk that directly follows. 

Grace Work, University of Wisconsin

Title: Properties of Equilibrium States for Geodesic flow on Translation Surfaces

Abstract: Using the previous work, as discussed in Noelle’s talk, we will show that some equilibrium states have mixing properties – the $K$-property and the Bernoulli property. This is recent work with Benjamin Call, Dave Constantine, Alena Erchenko, and Noelle Sawyer. We will also mention some open questions and possible directions to extend this work to more general $\text{CAT}(0)$ spaces.

Diane Davis, Exeter

Title: Periodic billiard paths on regular polygons

Abstract: With my colleague Samuel Lelièvre, I’ve been studying periodic billiard paths on regular polygons for several years. Recently, we’ve expanded our Sage program so that it can draw every periodic billiard path on every regular polygon. This has opened up an enormous (indeed, infinite) world of beautiful pictures. We are working to explore the most interesting corners of this new world, which raise questions about the structure and symmetries of periodic billiard paths. I’ll show lots of pictures and explain our recent observations and results.

Alena Erchenko, Dartmouth

Title: Flexibility and rigidity for Cantor repellers

Abstract: We will consider dynamical systems that we call Cantor repellers which are expanding maps on invariant Cantor sets coming from iterated function systems. Cantor repellers have two natural invariant measures: the measure of full dimension and the measure of maximal entropy. We show that dimensions and Lyapunov exponents of those measures are flexible up to well understood restrictions. We will also discuss the boundary case for the range of values of the considered dynamical data. This is joint work with Jacob Mazor.

Patrick Hooper, CCNY

Title: Stellar Foliation Structures on Surfaces 

Abstract: A translation surface is a singular geometric structure on a surface modeled on the plane where transition maps are translations. Some recent research has focused on extending results known for translation surfaces to dilation surfaces, where we broaden allowable transition maps to include dilations of the plane. While there is no natural metric on a dilation surface, we will show that there are natural analogs of geodesics in this context. For example, we’d like to understand when a homotopy class of closed curves on dilation surface has a canonical representation. We will see that the quest to answer questions of this nature naturally leads to an even more general geometric structure on surfaces. This new structure still has directional foliations and an SL(2, R)-action on the “space of structures.” This is joint work with Ferrán Valdez and Barak Weiss. 

Dubi Kelmer, Boston College

Title: Intrinsic Diophantine approximations on the sphere with and without dynamics.

Abstract: Intrinsic Diophantine approximations is concerned with the question of how well can one approximate a real point on an algebraic variety by rational points lying on the same variety. In this talk I will focus on the case of the sphere and describe several problems in intrinsic Diophantine approximations and some results on these problems using methods from dynamics and number theory. I will then describe some refinements of these results, based on recent joint work with Shucheng Yu, relying on a second moment formula of light cone Siegel transform.

Dmitry Kleinbock, Brandeis

Title: Dimension drop conjecture in homogeneous dynamics

Abstract: Let (X,\mu,T).png be an ergodic probability measure preserving system on a metric space X.png, and let U.png be a non-empty open subset of X.png. Consider the (\mu.png-null) set of points in X.png whose trajectory misses U.png . When can one prove that this exceptional set has Hausdorff dimension less than the dimension of X.png? This dimension drop phenomenon has been conjectured for actions on homogeneous spaces and proved in several special cases, for example when X.png is compact or has rank one. I will sketch a proof of a fairly general case of the conjecture – for arbitrary \textrm{Ad}.png-diagonalizable flows on irreducible quotients of semisimple Lie groups. Two main ingredients of the proof are effective mixing and the method of integral inequalities for height functions on X.png. Joint work with Shahriar Mirzadeh.