## Current Research Projects

Students will be divided into three groups, and each group will work closely with two faculty mentors on an open research problem. Faculty mentors will introduce students to one of the following research projects.

**Number Theory***Title*: Polynomial Divisor Graphs.*Expected Background*: Completion of either abstract algebra or a course in number theory.*Project Description*: The divisor graph on the integers up to*n*is the graph with integers labelled 1,2,...,*n*in which two vertices are connected by an edge if one divides another. This graph plays a role in a variety of number theoretic questions. One well known problem is to find the longest path in this graph that doesn't repeat a vertex. For example, the longest such path in the divisor graph up to 10 is 9-3-6-2-4-8-1-5-10. These graphs are also a useful tool for enumerating sets of integers with interesting properties. We will investigate versions of these questions for divisor graphs of polynomials rather than integers. Polynomials (with coefficients from a finite field) behave much like the integers in a variety of ways, but are sometimes easier to work with. The methods needed to investigate these problems draw from analytic number theory.**Group theory***Title*: Vertex minimal graphs with automorphism group of prescribed order.*Expected Background*: Elementary knowledge of groups, such as in an introductory abstract algebra course; exposure to graph theory would be helpful but not necessary.*Project Description*: Given a simple graph — consisting of a finite set of points called*vertices*and a collection of unordered pairs of those vertices which make*edges*— an automorphism of the graph is a permutation of the vertices which takes any edge of the graph to another edge. The set of all automorphisms forms a finite group under composition of permutations.

Assume that G is a finite group. A graph whose full automorphism group is isomorphic to G is called a G-graph, and α(G) is the minimal number of vertices among all G-graphs. We consider a related function*mv*, the*minimum vertex function*. Let*n*be a positive integer greater than 2. Define*mv(n)*to be the smallest value of α(G) from among all groups G of order*n*. The value of α(G) has been established for numerous infinite families of groups. This gives an upper bound for*mv(n)*for many*n*. There are also a lot of papers on the graphical regular representations (GRR) of a group that collectively show that in "most" cases*mv(n)*≤*n*.

This research project will compute the minimal vertex function for "small" values of*n*and the value for an infinite number of integers satisfying certain conditions.**Probability***Title*: Functional Dimension of Neural Networks.*Expected Background*: Completion of a first course in linear algebra. While a previous course in probability is helpful, it is not required.*Project Description*: Neural networks (particularly deep networks) are currently the most successful and prevalent technique for many machine-learning tasks. This project will study neural networks, particularly those with a ReLU activation function, and the expressiveness of the underlying function that is realized by a choice of parameters for the network.

We will have interest in understanding the functional dimension of the network parameters (introduced in work of Grigsby, Lindsey, Meyerhoff, and Wu) from a probabilistic point of view. Large functional dimension is desirable, since networks with a large functional dimension have the greater degree of responsiveness during training. In recent work of Grigsby, Lindsey, and Rolnick, it was shown that with a widening network architecture there is a positive probability of achieving the theoretical maximum functional dimension. Working with standard initialization schemes on the parameters, one direction for the project is to study properties of the probability distribution of the functional dimension for certain network architectures.