Fuchsian lattices with small trace sets are arithmetic: the noncompact case
Considering a subgroup A of SL(2, R), the trace set of A, Tr(A) is the set of all traces of elements in A. It is known that the trace set of arithmetic lattices in SL(2, R) has linear growth. Shmutz conjectured that the inverse also holds. In this talk, we will give a positive answer to the conjecture in the case of non-uniform lattices.

Escape of Mass of Sequences
One way to study the distribution of nested quadratic number fields satisfying fixed arithmetic relationships 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 construct counterexamples to their conjecture in every characteristic. 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 works with Erez Nesharim and Uri Shapira and with Steven Robertson

Limit Theorems for Siegel Theta Series
Given a quadratic form, Q, in k variables, and fixed weight function, f, we define its associated Siegel theta series as S^Q_N(t) := \sum_{n \in \Z^k} f(N^{-1} n) e^{\pi i Q(n) t}, where t is a real number in the interval [0,1]. Such sums have found application in various problems from number theory to mathematical physics. We discuss how dynamical methods can be used to prove the existence of the limit distribution for appropriately normalized theta series in the case when Q is a generic quadratic form, and f is a Schwartz function. This is part of work in progress with J. Griffin and J. Marklof.

A truncated inner product formula in the geometry of numbers
We provide an asymptotic inner product formula for truncations of the pseudo-Eisenstein series on $\SL(n,\Z)\backslash\SL(n,\R)$ of the form
$$ E_{f}(g) = \sum_{L \subseteq \Z^n} f(\det Lg) $$
for $f \in C^\infty_0(\R_{\geq 0})$, where the sum ranges over all primitive rank $k < n$ sublattices of $\Z^n$. We consider both the Arthur and the ``harsh'' truncations, with uses in the geometry of numbers in mind. As an application, we present an improvement over an estimate concerning the number of primitive sublattices of a fixed lattice of bounded determinant. This is a joint work with Seokho Jin (Chungang U.).

Dimension of Lyapunov spectrum for nonuniformly hyperbolic subsets of flows
We investigate the Hausdorff dimension of the set of points with both forward and backward Lyapunov exponent equal to a given value \alpha, within the class of recurrently hyperbolic points in a flow with upper semicontinuity of entropy on the space of invariant measures. These points are modeled using the countable-state symbolic dynamics developed by Lima, Sarig, and collaborators. Our results extend classical dimension formulas from uniformly hyperbolic systems to this broader, non-uniformly hyperbolic setting.