Advisor: Greg Kuperberg

A remarkable theorem of the Kirchhoff states that the number of spanning trees of a finite graph is given by the determinant of a certain associated matrix. There is an analogous result Kasteleyn which expresses the number of perfect matchings of a bipartite, planar graph as a determinant of a certain matrix. Both of these results are widely useful for counting various types of combinatorial objects, including labelled trees and plane partitions. In both cases, the associated has integer entries, and as a result, yields an abelian group with the same number of elements as the set being counted. In the case of spanning trees, the associated abelian group is called the Jacobian group of the graph; in the case of perfect matchings, it is called a Kasteleyn cokernel.

In this project, we will explore the structure of Jacobian groups and Kasteleyn cokernels for various types of graphs. Even when the enumeration problem itself has been solved, the structure of these groups is often unknown or only conjectured. One of the few cases that is well-understood is the Kasteleyn cokernel for the domino tilings of an Aztec diamond. The number of domino tilings is well-known to be 2^n(n+1)/2. The cokernel in this case is a direct sum of a progression of cyclic groups who sizes are a progress of powers of 2:

ℤ/2 ⊕ ℤ/4 ⊕ ℤ/8 ⊕ ... ⊕ ℤ/2ⁿ

Beyond the combinatorial setting of this project, we can also explore connections to representation theory of Lie groups, and to tropical algebraic geometry. For a more hands-on project, it is also valuable to obtain answers experimentally with computer software, in order to formulate conjectures or obtain clues for solving them.

See an illustration of the Artic Circle Theorem

Project 2: Tracking the molecular evolution of SARS-CoV2 and other coronaviruses

Advisors: Mariel Vazquez and Javier Arsuaga

Coronaviruses cause a wide variety of human respiratory diseases including SARS, MERS, and COVID-19. The genome of coronaviruses consists of a long single stranded RNA molecule that encodes for several structural proteins including the Spike (S) protein that decorates the surface of the virus in a crown-like fashion. The S protein is known to be the main player in the recognition and infection of human cells. The evolution of these viruses is driven by a relatively high point-mutation and recombination rates.

In this interdisciplinary project we will combine stochastic methods, topologically data analysis (TDA), machine learning, and molecular modeling to study the evolution of SARS-COV2 and its variants. Specific details of the project will be developed accoring to students' interests. For more information students can read the abstract of our recent NSF RAPID award on this topic DMS-2030491 .

Project 3: Markoff triples and the Markoff tree

Advisor: Elena Fuchs and Daniel Martin

Consider the Markoff equation,

x₁²+x₂²+x₃²-3x₁x₂x₃=0

First studied by Markoff in 1879 in the context of rational approximations to irrational numbers, integer solutions (x₁,x₂,x₃) ∈ ℤ³ to this equation are called Markoff triples. Suppose we ask, are there infinitely many prime numbers that appear as a member of some Markoff triple? Lurking behind this seemingly simple and yet unsolved question is an object called the Markoff tree, as well as mod p versions of this tree for primes p. In this project, we will consider connectivity properties of these trees.

These mod p Markoff graphs are easy to describe: consider the graph X=(V,E) of all nonzero integer Markoff triples, where the set of vertices corresponds to the set of triples (x₁,x₂,x₃) in ℤ³ satisfying the Markoff equation, and the edges are formed as follows: the vertex corresponding to (x₁,x₂,x₃) is connected by an edge to the vertex corresponding to (y₁,y₂,y₃) if and only if (x₁,x₂,x₃) =R_i((y₁,y₂,y₃)) for some 1 ≤ i ≤ 3, where

R₁(a,b,c):=(3bc-a,b,c) R₂(a,b,c):=(a,3ac-b,c) R₃(a,b,c):=(a,b,3ab-c).

It can be shown that this graph will have four connected components: one containing the triple (1,1,1), one containing the triple (-1,-1,1), one containing (-1,1,-1), and finally one containing (1, -1, -1). We will focus on the graph X_p where p is prime, in which the vertices correspond to solutions to the Markoff equation mod p, and edges are defined exactly as in the graph X.

Bourgain, Gamburd, and Sarnak studied these graphs with the aim of answering questions such as the one about infinitude of primes among Markoff numbers. In a first paper on this subject, they showed that, outside a small but infinite subset of primes p for which p²-1 has many prime factors in a certain sense, the mod p Markoff graphs are connected. Later, a result of Will Chen made it possible to prove that this is true outside a finite set. However, the bounds we have on this finite set right now are much too large to make it possible to prove via computer computation that the graphs corresponding to the primes in this set are connected. We will try to either improve the bounds and complete this proof, or shed light on what needs to be done to do so.

Project 4: Mathematical modeling of swimming

Advisors: Bob Guy, Becca Thomases, and Katie Link

Swimming requires coordination, and swimming in different environments or in a group requires sensing the environment and adapting. Single-cell, microscopic organisms have no nervous system to coordinate their movements or process environmental information. The coordination of body movements is an emergent phenomenon that results from interactions between molecular motors, the swimmer body, and the surrounding fluid environment. In our past work we have developed computational methods that solve the equations that describe the coupled mechanics of active structures with the surrounding fluids.

In this project we will learn about: the equations of fluid mechanics and algorithms for generating numerical solutions; the biological structure of the flagellum and mathematical descriptions of both its movement and molecular motors. We will extend existing computer code and mathematical models to address biological questions such as: How is the beat of the flagellum shaped by the surrounding fluid? How does spatially localized energy production shape the beat of a sperm flagellum? How do nearby beating flagella coordinate their beats? Addressing these questions involves mathematical modeling, scientific computation, nonlinear dynamics, and optimization. Students who are interested in learning about computational fluid dynamics may focus their work on learning about the numerical methods used to solve these equations, and they will work to develop new methodology for efficiently solving these equations.