My research interest lies in homogeneous dynamics and their interactions with number theory and geometry. You can find my research papers/preprints below. (Click the title for PDF)

We prove the first case of polynomially effective equidistribution of closed orbits of semisimple groups woth non trivial centralizer. The proof relies on uniform spectral gap, builds on and extends work of Einsiedler, Margulis, and Venkatesh.

We settle the following conjecture of David Masser using homogeneous dynamics: if two given integral quadratic forms are equivalent and both have three or more variables, then one form can be translated into the other by a unimodular integral matrix whose norm is bounded above by a polynomial of the height of the two given forms.

In the study of Fuchsian groups, it is a nontrivial problem to determine a set of generators. Using a dynamical approach we construct for any cocompact arithmetic Fuchsian group a fundamental region in SL(2, R) from which we determine a set of small generators.

We prove some effective equidistribution results for translates of maximal horospherical measures in the space of unimodular lattices, which, for this special case, establishes a quantitative version of the recent work of Mohammadi and Salehi-Golsefidy [Amer. J. Math. 2014].

Let x be a primitive integral vector whose coordinates are at least 2. The Frobenius number of x is the largest integer which cannot be expressed as a non-negative integer linear combination of the coordinates of x. Marklof [Invent. 2010] proved the existence of the limit distribution of the Frobenius numbers. In this paper we established an effective version of his result.

We define the notion of spectral gap for the actions of discrete groups on von Neumann algebras and study their relations with the states of the von Neumann algebras which are invariant under the group action. Using this notion we show that a finitely generated ICC group is inner amenable if and only if there exist more than one inner invariant states on its group von Neumann algebra.