## Mixing Times for Glauber Dynamics of the Ising Model on Random Regular

Hypergraphs

More details to come!

## Robust Graph Ideals

Our paper in the

Fix a monomial term order. A

Let

A

*Annals of Combinatorics*is available through Springer here.

Fix a monomial term order. A

*Gröbner basis* *G*of an ideal*I*is a generating set of*I*such that the ideal given by the leading terms of polynomials in*I*is itself generated by the leading terms of*G*. Bernd Sturmfels has an excellent short introduction to this type of basis. A*universal Gröbner basis*is a Gröbner basis with respect to all monomial term orders; equivalently, it is the union of all reduced Gröbner bases.Let

*k*be a field. Consider the graph pictured. Define a homomorphism between polynomial rings k[A, B, C, D] and k[a, b, c, d] by sending each edge to the product of its vertices. That is,*A*is sent to*ab*, B is sent to*bc*, etc. We call the kernel*I*of this homomorphism the*toric ideal*of the graph. It is known that toric ideals are prime ideals that are generated by binomials.A

*Graver basis*of*I*is the set of all binomials in*I*that are primitive. Now, we say that*I*is*robust*if its universal Gröbner basis is a minimal generating set. In our research, we showed that any robust toric ideal arising from a graph is also minimally generated by its Graver basis. Furthermore, we completely characterized all graphs which give rise to robust ideals using graph-theoretic conditions; specifically, these conditions are on the*circuits*of the graph, where a*circuit*is a closed walk allowing repetitions of vertices and not edges.## Reed College Senior Thesis

The senior thesis is a year-long, rigorous project required for graduation at Reed College. My thesis was advised by Jerry Shurman and heavily indebted to the work of Paul Garrett at the University of Minnesota. The actual document can be found here.

The thesis analyzed irreducible representations of GL(2,

The thesis analyzed irreducible representations of GL(2,

*k*) for finite fields*k**—*a subject which was generally hashed out by the late 1960s. However, the algebraic techniques employed are far from classical. Instead, these methods*scale-up*in the sense that they play roles in the study of representations of infinite-dimensional matrix groups over some infinite fields (the real, complex, and*p*-adic numbers). Specifically, I examined*principal series representations*of GL(2,*k*).

## Irreducible Restricted Disjoint Covering Systems (Cornell Summer Math Institute)

While the coursework portion of this summer program involved a fast-paced traversal through Rudin's "Principles of Mathematical Analysis", the research segment focused on elementary number theory. Our group paper can be found here. The research was then presented at the Nebraska Conference for Undergraduate Women in Mathematics in a poster session. An electronic copy of the poster is available here.