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.

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