The following projects will be part of the REU. The years in parentheses indicates which years the projects will run.

### Can data analytics help the poor? with Dr. Wayne Tarrant (2020, 2021, 2022)

With the awarding of the 2019 Nobel prize in Economics, it is hoped that there will be a greater focus on the economics of poverty. Although the recent laureates are known for their use of randomized controlled trials, there are many other ways to consider economic questions. During this summer we will focus on Kenya due to Dr. Tarrant's connections to the country. Kenya's technological innovations of M-PESA and M-Shwari will guide what we do, but we won't be afraid to delve into whatever subject arises. Students may use computational, mathematical, statistical, and programming tools, among others. We will value inquisitiveness, creativity, and tenacity above all else.

### Spatial graph embeddings with Dr. Kenji Kozai (2020, 2021)

The study of topological properties of graph embeddings and the study of random knots have each existed since 1980s and 1990s. Random knots have been of particular interest as ways to model long polymers such as DNA and their topological configurations when confined into a tight space like a cell. Until recently, the random graph embeddings and their topological properties had not been studied. This REU project will look at the topology of random graph embeddings and applications to biological molecules that are non-linear and better modeled through graphs than as mathematical knots.

### Discrete logarithms with Dr. Joshua Holden (2021)

Many of the cryptographic algorithms used today are built upon variations of a common transformation, namely:

g^{x}=y (mod n)

where g, x, y, and n are integers. This transformation is known as *discrete exponentiation modulo n*. For instance, Diffie-Hellman key exchange, ElGamal digital signatures, and the Blum-Micali pseudorandom bit generator all use discrete exponentiation. This project will use p-adic numbers to try to look for patterns in this transformation and similar transformations that can be exploited.

### Ehrhart Theory and Chip Firing with Dr. McCabe Olsen (2022)

Ehrhart theory, first developed in the 1960’s, is the study of enumeration of integer points (or lattice points) inside of polytopes with integer point vertices. The foundational result of Ehrhart theory, the Ehrhart polynomial, encodes a connection between this enumeration of the integer lattice and the Euclidean volume of the polytope itself. A combinatorial pursuit at its core, Ehrhart theory has been applied to solve problems in other areas of mathematics such as algebraic geometry, commutative algebra, and number theory. This project will investigate algebraic and geometric properties of families of polytopes related to the chip-firing game on finite graphs.