Title: Teichmüller curves as the vanishing locus of modular forms Abstract: Teichmüller curves are totally geodesic curves $C \subset \mathcal{M}_g$ inside the moduli space of curves with respect to the Teichmüller metric. We will focus on a special class of Teichmüller curves arising from Hecke triangle groups and embed them into the locus of real multiplication inside $\mathcal{A}_g$, the…

Title: p-Adic asymptotic subalgebra enumeration Abstract: Given a finite-dimensional p-adic algebra, we are interested in the number of subalgebras of index p^i as a function of i. To study this sequence of numbers we use a generating series, called the local subalgebra zeta function. Its poles describe the growth of the number of subalgebras as i goes to infinity with the largest pole, in particular, dominating the asymptotics. From the…

Title: Cospectrality of random graphs and arithmetic statistics Abstract: A classical question in spectral graph theory asks which graphs are determined up to isomorphism by the spectrum of their adjacency matrix. Both positive and negative examples are known. But, what happens in the "typical case" of a random graph? Haemers' conjecture posits that the probability that the graph is determined by its spectrum should tend to one as the number…