National Science Foundation

Research Experiences for Undergraduates (REU)

2018 Projects

Yevgeniy Kovchegov: Quantum Walks Kovchegov applies probability and stochastic processes to various contemporary problems, This summer, Kovchegov plans to work with REU students on projects in combinatorial probability involving tree self-similarity under the operation of leaf pruning, and random self-similar trees. In particular, to explore Horton and Tokunaga self-similarity for the level set trees generated by space homogeneous random walks on integers.

Clayton Petsche: Dynamics of Polynomial Maps Petsche is a number theorist with a primary focus on the area of arithmetic dynamics. This summer, Petsche proposes an exploration of topics in algebraic dynamical systems in several variables over p-adic and other non-Archiedean fields. Possible projects include:
1. Find a new, dynamical proof of the non-Archimedean Perron-Frobenius theorem established in the 2015 OSU REU, based on fixed-point theory. Does such an approach have further non-Archimedean applications in parallel with the real setting?
2. Classify the dynamics of non-Archimedean Henon maps in residue characteristic 2, the excluded case in the 2016 OSU REU.
3. Perform new and interesting calculations of dynamical invariants, such as entropy and Hausdorff dimension, in the context of p-adic plane automorphisms.

Mike Rosulek: Secure Computation Mike Rosulek is a researcher in cryptography, with specific interest in secure computation protocols. A secure computation protocol allows n parties, each of which has a private input, to learn some agreed-upon function of their inputs, while learning nothing else. A fundamental question is to find simple characterizations of which functions have unconditionally secure protocols. Such characterizations are known for only for the case of n=2. Our team will focus on developing a characterization for some special cases having n greater or equal to 3.

Holly Swisher: Partitions and Modular Forms Swisher's research is inspired by the work of Ramanujan with a primary focus on questions related to integer partitions and modularity. Notably, the subjects of mock modularity and most recently quantum modularity have gained a lot of current interest. In 2015, her REU students studied quantum modularity properties of a class of mock modular forms related to certain theta functions. The proposed project for next summer is to investigate quantum modularity properties for certain classes of generalized partition rank generating functions which have a particularly nice hypergeometric shape.

In general, the proceedings from the past few years will give an idea of the variety and general levels of the projects.