My mathematical research lies at the intersection of the the areas known as algebraic number theory and algebraic geometry.
Generally speaking, algebraic number theory studies the relation of integers and polynomials. For instance, we may ask is are there integers x,y, and z such that 
(the answer is yes! these are known as pythagorean triples). In general it is very difficult (even undecidable!) to know whether there are integers so that a given polynomial equation is satisfied. Algebraic geometry, on the other hand, attaches geometric spaces to systems of polynomials as means of understanding and studying them. When we combine algebraic geometry and algebraic number theory, purely geometric considerations from algebraic geometry can give a lot of insight about the integer solutions of polynomial equations.
The descriptions are dripping in jargon: proceed with caution!
A Topology on Points on Stacks
I define a topology on isomorphism class of p-adic points on algebraic stacks, generalizing the usual definitions for varieties. I generalize this definition to include adelic points of stacks over the adeles. Finally, I use the topology to formulate and prove a strong approximation result for certain stacky curves over number fields.
This paper has been submitted. Preprint
Specialization of Néron-Severi Groups in Characteristic p>0
I prove that for any smooth, proper family of algebraic varieties over a field of characteristic p>0, if the Newton polygon of the second crystalline cohomology is constant, then there exists a bound on the exponent of p in the torsion of the cokernel of any specialization of Néron-Severi groups coming from that family. I use this to prove that in any smooth, proper family of algebraic varies over a field of characteristic p, there is a closed fiber which has the same Néron-Severi rank as the generic fiber.
This paper has been submitted. Preprint