I am an Assistant Professor at the University of Wisconsin-Madison and an American Institute of Mathematics Five-Year Fellow. I am also supported by a Sloan Fellowship and National Science Foundation grant DMS-1301690.
The main focus of my research is in number theory and algebraic geometry, but it also involves work in probability, additive combinatorics, and algebraic topology. My PhD work found new explicit descriptions of moduli spaces for algebras and modules for those algebras. In number theory, these descriptions are useful for parametrizing orders in number fields and ideal classes of those orders. In algebraic geometry, the work can be viewed as understanding moduli of abstract points, or alternatively as parametrizing finite covers and line bundles on those covers. I am also interested in the applications to questions of counting number theoretical objects such as number fields and class groups. Motivated by this interest, I have been developing tools in probability theory to study randomly arising finite groups, such as the Jacobians of random graphs. In recent years, I have been very interested in the question of the distribution of the number of points on curves over a fixed finite field, both in special families, where we are able to prove results, and in general, where developing heuristics and conjectures is the cutting edge. This has also motivated my interest in building stronger general tools to determine limiting distributions of points on varieties over finite fields. I am also interested in algebro-geometric analogs of these counting questions, where the "counting" happens in the Grothendieck ring of varieties and there are strong connections to homological stability in algebraic topology.
I completed my PhD at Princeton University in 2009 under the supervision of Manjul Bhargava, and was a Szego Assistant Professor at Stanford University from 2009-2011.
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In Fall 2014, I am teaching Math 748, Algebraic Number Theory.
In Spring 2014, I am teaching Math 222 (two classes), Calculus and Analytic Geometry.
In Spring 2012, I taught Math 490: Collaborative Undergraduate Research Lab.
In Fall 2011, I taught Math 847: Algebraic Curves and Varieties over Finite Fields.
In Summer 2013 and Fall 2013 I supervised three undergraduates and a graduate student on a research project funded by NSF grants DMS-1147782 and DMS-1301690.
I have supervised undergradutes in the past working on research projects with me through the Undergraduate Research Scholars Program.
I also supervised a group of undergraduates in Summer 2012 on research projects continuing the work from Math 490.
The UW Algebraic Geometry Seminar