The lengths of closed geodesics on a hyperbolic manifold are determined by the traces of its fundamental group. When the latter is a Zariski dense subgroup of an arithmetic group, the trace set is contained in the ring of integers of a number field, and may have some local obstructions. We say that the surface's length set ``saturates" (resp. ``asymptotically saturates") if every (resp. almost every) sufficiently large admissible trace appears. In joint work with Xin Zhang, we prove the first instance of asymptotic length saturation for punctured covers of the modular surface, in the full range of critical exponent exceeding one-half (below which saturation is impossible).

Let $A$ be a (unital) algebra over the complex numbers and $a \in A$. At a very high level, the term \textit{functional calculus} refers to constructions of the form, ``Take some collection $\mathcal{F}$ of scalar functions, and, for all $f \in \mathcal{F}$, define $f(a) \in A$ in a sensible way." One can always take $\mathcal{F} = \mathbb{C}[t]$ with the obvious definition of $p(a) \in A$ for $p \in \mathbb{C}[t]$, but this is pretty much the end of the construction when $A$ has no additional structure. When $A$ has some analytic structure -- as is frequently the case in functional analysis and operator algebras -- one can construct functional calculi for much larger classes of functions. In this slightly experimental talk, it is possible that I will discuss functional calculus in Banach algebras, $C^*$-algebras, and/or von Neumann algebras. The talk will be in a ``choose your own adventure" style, so the audience will decide the exact trajectory of the talk democratically. (I offer my thanks to Max Johnson for the idea to give this kind of talk.) Prerequisites will be minimal: passing familiarity with norms, inner products, bounded/continuous linear maps, completeness, etc. should suffice.

The theory of canonical divisors on curves has witnessed an explosion of interest in recent years, motivated by the recent developments in the study of limits of canonical divisors on nodal curves. Imposing conditions on canonical divisors allows one to construct natural geometric subvarieties of moduli spaces of pointed curves, called strata of canonical divisors. The strata are in fact the projection on moduli spaces of curves of incidence varieties in the projectivized Hodge bundle. I will present a graph formula for the class of the restriction of such incident varieties over the locus of pointed curves with rational tails. The formula is expressed as a linear combination of tautological classes indexed by decorated stable graphs, with coefficients enumerating appropriate weightings of decorated stable graphs. I will conclude with some applications. Joint work with Iulia Gheorghita.

We introduce a new class of permutations, called web permutations. Using these permutations, we provide a combinatorial interpretation for entries of the transition matrix between the Specht and web bases, which answers Rhoades's question. Furthermore, we study enumerative properties of these permutations. This is based on the work with Byung-Hak Hwang and Jihyeug Jang.