Research

My area of research is number theory, specifically arithmetic geometry. My main interest is in the geometry of Shimura varieties of in characteristic p, and the related subjects of Rapoport-Zink spaces and affine Deligne-Lusztig varieties. I'm also interested in the local Langlands program and in mod-p and p-adic automorphic forms. 

Preprints and Publications:

This paper concerns the characteristic-p fibers of 𝖦𝖴(q−2,2) Shimura varieties, which classify abelian varieties with additional structure. These Shimura varieties admit two stratifications of interest: the Ekedahl-Oort stratification, based on the isomorphism class of the p-torsion subgroup scheme, and the Newton stratification, based on the isogeny class of the p-divisible group. It is natural to ask which Ekedahl-Oort strata intersect the unique closed Newton stratum, called the \emph{supersingular locus}. In this paper, we present several novel techniques that give information about the interaction between the two stratifications for general signature (q−2,2), and as an application, we completely answer this question for the signature (3,2)..

We describe the structure of the supersingular Rapoport-Zink space associated to the group of unitary similitudes of signature (2,n-2) for an unramified quadratic extension of p-adic fields. In earlier work, two of the authors described the irreducible components in the category of schemes-up-to-perfection. The goal of this work is to remove the qualifier "up-to-perfection".

We study the supersingular locus of a reduction at an inert prime of the Shimura variety attached to GU(2,n-2). More concretely, we realize irreducible components of the supersingular locus as closed subschemes of flag schemes over Deligne-Lusztig varieties defined by explicit conditions. Moreover we study the intersections of the irreducible components. A stratification og the Deligne-Lusztig varieties defined using a power of Frobenius action appears in the description of the intersections.

In this note, we study Shimura varieties for the groups GU(V), where V is a Hermitian space relative to a CM extension E/F. We give a description of the supersingular locus of the fiber at a prime v over p of such a Shimura variety, under the assumptions that the dimension of V over E is less than or equal to 4 and that the prime p splits completely in F and is unramified in E.

We give a description of the GL4 Rapoport-Zink space, including the connected components, irreducible components, intersection behavior of the irreducible components, and Ekedahl-Oort stratification.  As an application of this, we also give a description of the supersingular locus of the Shimura variety for the group GU(2,2) over a prime split in the relevant imaginary quadratic field.

My thesis (2018) consists primarily of the above result, with a more general introduction.

Here are some previous talks I've given at conferences and seminars:

On Compatibility with Cuspidal Support in Local Langlands Correspondences 

Expository, Women in Numbers 5 Power Day

On Structure of Supersingular Loci:

Recording and slides can be found here.

On Supersingular Loci of Some Unitary Shimura Varieties:

Slides: Supersinular Loci in Moduli Spaces of Abelian Varieties

On The GL(4) Rapoport-Zink Space

Slides: Supersingular Loci and the GL(4) Rapoport-Zink Space

Some General Audience Talks:

Portland State University Mathematics Colloquium

University of Oregon Women in Graduate Studies

Reed Mathematics Teacher-Scholar Symposium

     Science Slam,  University of Oregon (general science audience)

Women in Mathematics in New England, Smith College (undergraduate math audience)

 Women in Mathematics in New England, Smith College (undergraduate math audience)

My research is in number theory, but I'm also happy to supervise reading/research with undergraduates in number theory or in other topics. My student Morgan Presson recently made this excellent poster about the p-adic integers.