No spoilers!

Let $\Lambda<SL(2,\mathbb{Z})$ be a finitely generated, non-elementary Fuchsian group of the second kind, and $\bf{v},\bf{w}$ be two primitive vectors in $\mathbb{Z}^2-\bf{0}$. We consider the set $\mathcal{S}=\{\langle \bf{v}\gamma,\bf{w}\rangle_{\mathbb{R}^2}:\gamma\in\Lambda\}$, where $\langle\cdot,\cdot\rangle_{\mathbb{R}^2}$ is the standard inner product in $\mathbb{R}^2$. Using Hardy-Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich and Sarnak, together with Gamburd's $5/6$ spectral gap, we show that if $\Lambda$ has parabolic elements, and the critical exponent $\delta$ of $\Lambda$ exceeds $0.995371$, then a density-one subset of all admissible integers (i.e. integers passing all local obstructions) are actually in $\mathcal{S}$, with a power savings on the size of the exceptional set (i.e. the set of admissible integers failing to appear in $\mathcal{S}$). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when $\Lambda$ is free, finitely generated, has no parabolics and has critical exponent $\delta>0.999950$.

Several long-standing open problems in the theory of C*-algebras reduce to whether for a given class of C*-algebras there is a locally universal one among them with certain nice properties. I will discuss how techniques from model theory, in particular model-theoretical forcing, can be used to shed light on these problems. This is joint work with Isaac Goldbring.

We introduce a general framework for studying a class of randomized load balancing models in a system with a large number of servers that have generally distributed service times and use a first-come-first serve policy within each queue. Under fairly general conditions, we use an interacting measure-valued process representation to obtain hydrodynamics limits for these models, and establish a propagation of chaos result. Furthermore, we present a set of partial differential equations (PDEs) whose solution can be used to approximate the transient behavior of such systems. We prove that these PDEs have a unique solution, use a numerical scheme to solve them, and demonstrate the efficacy of these approximations using Monte Carlo simulations. We also illustrate how the PDE can be used to gain insight into network performance.

I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the min-max method. I will explain the basic ideas behind the min-max theory as well as our new contributions.

We prove that any quasi-isometry between hyperbolic manifolds is homotopic to a harmonic quasi-isometry.

I will discuss joint work with Gabriele Di Cerbo on boundedness
of Calabi-Yau pairs. Recent works in the minimal model program suggest
that pairs with trivial log canonical class should satisfy some
boundedness properties. I will show that Calabi-Yau pairs which are not
birational to a product are indeed log birationally bounded, if the
dimension is less than 4. In higher dimensions, the same statement can be
deduced assuming the BAB conjecture. If time permits, I will discuss
applications of this result to elliptically fibered Calabi-Yau manifolds.

The Lerch zeta function is a three variable zeta function,
with variables $(s, a, c)$,
which generalizes the Riemann zeta function and has
a functional equation, but no Euler product. We discuss its properties.
It is an eigenfunction of a linear partial
differential equation in the variables $(a, c)$
with eigenvalue $-s$, and it is also preserved under a
a commuting family of two-variable Hecke-operators $T_m$
with eigenvalue $m^{-s}$. We give a characterization
of it in terms of being a simultaneous eigenfunction of
these Hecke operators.
We then give an automorphic interpretation of the Lerch zeta
function in terms of Eisenstein series taking
values on the Heisenberg nilmanifold, a
quotient of the real Heisenberg group modulo
its integer subgroup. Part of this work is joint with
W.-C. Winnie Li.