Spring 2023

Speakers, topics and mentors:

Vaughn Climenhaga on Specification and Measures of Maximal Entropy
Mentors: JinCheng Wang & Leyla Yardimci

Transitive uniformly hyperbolic systems satisfy the specification and expansivity properties, which Rufus Bowen used to prove uniqueness of the measure of maximal entropy, and of equilibrium states for a broad class of potential functions. These measures have strong stochastic behavior and play an important role in the Margulis asymptotics for periodic orbits. I will survey the ingredients of Bowen's argument, and explain how Dan Thompson and I extended his approach to some non-uniformly hyperbolic systems, including geodesic flows in nonpositive curvature (with Keith Burns and Todd Fisher).

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Osama Khalil on Mixing, Counting and Equidistribution in Lie Groups after Eskin and McMullen
Mentor: Anthony Sanchez

I will present ideas introduced by Duke-Rudnick-Sarnak and Eskin-McMullen addressing many natural counting problems using fundamental tools from homogeneous dynamics. These ideas have had major impact beyond the setting of their inception. 

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Bryna Kra on Infinite Patterns in Sets of Positive Density: Translating Combinatorics to Dynamics
Mentors: Ethan Ackelsberg & Anh Le

Furstenberg used ergodic theory to prove Szemeredi’s Theorem that any subset of integers with positive upper density contains arbitrarily long arithmetic progressions.  Among other tools, his proof introduced the Correspondence Principle, a general technique for translating a combinatorial problem into a dynamical one.  While the original formulation suffices for certain patterns, refinements of the translation are needed for finding infinite patterns in sets of positive upper density. This is one of the ingredients used in recent work joint with Joel Moreira, Florian Richter, and Donald Robertson resolving conjecture of Erdos. (Focusing on the paper Infinite Sumsets in Sets with Positive Density to be followed by A proof of Erdos’s B + B + t conjecture)

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Amie Wilkinson on Pathological Foliations
Mentor: Davi Obata

Groups will read Absolutely Singular Dynamical Foliations and/or Pathological Foliations and Removable Zero Exponents. Amie Wilkinson will give a survey talk on pathological foliations and introduce some helpful ideas. 

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