One way to study the distribution of quadratic number fields is through the evolution of continued fraction expansions. In the function field setting, it was shown by de Mathan and Teullie that given a quadratic irrational $\Theta$, the degrees of the periodic part of the continued fraction of $t^n\Theta$ are unbounded. Paulin and Shapira improved this by proving that quadratic irrationals exhibit partial escape of mass. Moreover, they conjectured that they must exhibit full escape of mass. We show that the Thue Morse sequence is a counterexample to their conjecture. In this talk we shall discuss the technique of proof as well as the connection between escape of mass in continued fractions, Hecke trees, and number walls. This is part of ongoing work joint with Erez Nesharim.

In this talk, I will explore two methods for constructing Topological Quantum Field Theories (TQFTs). The first method is known as the universal construction, which traces its origins to the work of Blanchet, Habegger, Masbaum, and Vogel in 1995. The second method involves a generator and relation presentation of the category of cobordisms, as developed by Juhasz in 2018. I will introduce a concept called a chromatic morphism and demonstrate how it generates numerous examples for both construction methods. These examples yield interesting representations of mapping class groups and open new avenues for studying 4-manifolds. This talk is designed to be introductory and suitable for graduate students.

Given an irreducible element of Out(Fn), there is a graph and an irreducible "train track map" on this graph, which induces (a representative of) the outer automorphism on the fundamental group of the graph. The stretch factor of an outer automorphism measures the rate of growth of words in Fn under applications of the automorphism, and appears as the leading eigenvalue of the transition matrix of a train track representative. I'll present work showing a lower bound for the stretch factor in terms of the edges in the graph and the number of folds in the fold decomposition of the train track map. Moreover, in certain cases, a notion of the latent symmetry of the graph gives a lower bound on the number of folds required for any train track map on a given graph. We use this to classify all single fold train track maps.

A classical theorem of Furstenberg states that the only proper closed subsets of the circle which are invariant under multiplication by both two and three are finite. Several generalizations of this result have been proven since then. First I will give some background, and then discuss a class of systems motivated by this, called chaotic almost minimal systems. I’ll present some results joint with Kra and Cyr about such systems, including the existence of Z^d-actions which are chaotic almost minimal and possess multiple non-atomic ergodic measures for d>=1. Time permitting, I will list some more recent work, joint with Kra, about some related results.

A natural question asked in nearly all branches of mathematics is "When are two objects `equivalent'?" The notion of equivalence should reflect what type of structure within our objects we wish to compare. In dynamical systems the natural notion of equivalence comes from conjugacies: Two maps $f,g: X \rightarrow X$ are said to be conjugate if there exists an invertible mapping $h: X \rightarrow X$ such that $h\circ f=g\circ h$. The regularity of the conjugacy $h$ determines the sense in which the systems are equivalent, i.e. if $h$ and $h^{-1}$ are measurable, then $f$ and $g$ are measurably equivalent, and if $h$ is a homeomorphism then they are topologically equivalent, etc. In this talk, we will let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugate by a homeomorphism $h$. It was proved by de la Llave in 1992 that the conjugacy $h$ is automatically $C^{1+}$ if and only if the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all periodic orbits. We prove that if the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all points of some large period $N\in\mathbb{N}$, then $f$ and $g$ are ``approximately smoothly conjugate." That is, there exists a a $C^{1+\alpha}$ diffeomorphism $\overline{h}_N$ that is exponentially close to $h$ in the $C^0$ norm, and such that $f$ and $f_N:=\overline{h}_N^{-1}\circ g\circ \overline{h}_N$ is exponentially close to $f$ in the $C^1$ norm.

We consider the problem of classifying Kolmogorov automorphisms (or K-automorphisms for brevity) up to isomorphism or up to Kakutani equivalence. Within the collection of measure-preserving transformations, Bernoulli shifts have the ultimate mixing property, and K-automorphisms have the next-strongest mixing properties of any widely considered family of transformations. Therefore one might hope to extend Ornstein’s classification of Bernoulli shifts up to isomorphism by a numerical Borel invariant to a classification of K-automorphisms by some type of Borel invariant. We show that this is impossible, by proving that the isomorphism equivalence relation restricted to K-automorphisms, considered as a subset of the Cartesian product of the set of K-automorphisms with itself, is a complete analytic set, and hence not Borel. Moreover, we prove this remains true if we restrict consideration to K-automorphisms that are also C∞diffeomorphisms. In addition, all of our results still hold if “isomorphism” is replaced by “Kakutani equivalence”. This shows in a concrete way that the problem of classifying K-automorphisms up to isomorphism or up to Kakutani equivalence is intractable. These results are joint work with Philipp Kunde.

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