University of Wisconsin-Madison

I am a Professor at the University of Wisconsin-Madison. I am also supported by a Packard Fellowship for Science and Engineering, a Sloan Research Fellowship, National Science Foundation CAREER grant DMS-1652116, National Science Foundation grant DMS-1301690, and a Vilas Early Career Investigator Award.

The main focus of my research is in number theory and algebraic geometry, but it also involves work in probability, additive combinatorics, random groups, 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 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. Trying to understand distributions of class groups and Galois groups in number theory has also led to my interest in different models of random groups and their properties.

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. I was an American Institute of Mathematics Five-Year Fellow from 2009-1017.

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My CV.

In Fall 2016, I am teaching Math 748, Algebraic Number Theory.

In Fall 2014, I taught Math 748, Algebraic Number Theory.

In Spring 2014, I taught 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.

The UW Algebraic Geometry Seminar

Notes for those who are asking me to write a letter of recommendation

A short reminder to myself about How to determine the splitting type of a prime (from the permutation representation of the decomposition and inertia groups).