Area of Research:Algebraic Number Theory
I have been an Acting Assitant Professor and VIGRE fellow at the University of Washington since September 2006. I received a BS in Mathematics in the Spring of 2000 from the University of Arizona. My PhD work started the following Fall with a VIGRE fellowship at the University of Michigan and was completed in the Summer of 2006 under the direction of Chris Skinner.
I am interested in algebraic number theory, specifically Iwasawa theory and related topics. Much of the motivation for my research stems from the conjecture of Birch and Swinnerton-Dyer and its generalizations. For an arithmetic object (e.g., a number field or an elliptic curve defined over Q), these conjectures predict a precise relationship between algebraic invariants (pieces of Galois cohomology groups) and analytic invariants (special values of complex L-functions) attached to the object.
One approach to problems of this nature is, roughly speaking, to study how these invariants behave in (p-adic) families. For example, given an elliptic curve E defined over Q, one can study the behavior of the group E(K) (of points on E with coordinates in a number field K) as the field K varies over certain p-power order extensions of Q. This behavior can in many cases be related to a p-adic object which encodes special values of the complex L-function of E. My research applies this circle of ideas in an attempt to make progress towards Birch and Swinnerton-Dyer type conjectures for modular forms.
Area of Research: Algebraic Geometry, Representation Theory, Mathematical Physics
I am a VIGRE acting assistant professor in the Department of Mathematics at UW. I received B.S. degrees in both Mathematics and Physics (1999) at Penn State before attending Princeton University for graduate school; I received my Ph.D in 2004 from Princeton University. I spent 2004-2005 as an EPSRC postdoctoral research assistant at the University of Oxford in England.
Broadly speaking, my interests pertain to the geometry of spaces with group actions. Many interesting spaces arising in mathematical physics and representation theory are constructed as quotients of spaces with group actions (including many moduli spaces). Usually the presence of a group action provides a deep link between concepts in representation theory, and the geometric structure of the space in question.
My thesis work studied the problem of classification of equivariant vector bundles on spherical varieties and more general ``prehomogeneous varieties." More recently, I have been interested in computation of motivic cohomology (a very general cohomology theory for algebraic varieties) for varieties with group action. In particular, together with Brent Doran and Frances Kirwan, I have been interested in studying moduli spaces of bundles on algebraic varieties via these methods.
Department: Applied Mathematics
Area of Research: Geophysical Fluid Dynamics
Since the fall 2005, I have been working as a VIGRE postdoctoral fellow in the Department of Applied Mathematics at the University of Washington. After completing my B.S. with a major in mathematics at the University of Notre Dame in 1999 (and spending a year as a volunteer with a development organization in Zambia), I joined the Ph.D. program at the Courant Institute of New York University with the intention of studying something in topology or differential geometry. An introductory class to physical oceanography I audited my first year there, however, persuaded me to switch gears completely and to research sea ice dynamics instead. I received my Ph.D. in 2005.
My research interests generally fall into the area of geophysical fluid dynamics. In particular, I am interested in gaining a better understanding of the processes controlling the physics of the oceans and the atmosphere through modeling. In my work I employ a range of models, from simplified ones that permit analytic solutions to highly complex (and more realistic) numerical models. Occasionally, I dabble with data. (After all, applied mathematics is supposed to have some relation to the "real" world...) Currently, I am continuing my work on sea ice, extending the new framework for calculating the internal pressure term I developed in my thesis to a full two-dimensional model. I have also become involved in a state estimation project, whose goal is to recreate past climate states based on limited and averaged available observations and numerical model output. Along the way, we hope to gain insight into the relative importance of various locations and types of observations to determine global climate, which may in turn tell us something about the processes at work.
In my free time, I help to run a non-profit called Deep Roots, which gives scholarships to underprivileged kids in developing countries. In my really free time, I enjoy traveling, reading, music and dancing.