Project 1: Navier-Stokes equations. This project involves mathematically and physically understanding the motion of incompressible fluids (e.g. water) in certain domains, such as a ball in three dimensional space, through basic differential equations such as diffusion and Laplace equations. In this project we'll do this analytically and/or probabilistically.

Project 2: Stability Problems. This project deals with the following interesting question: “Under what conditions must a mathematical object that satisfies a certain property approximately, e.g. a differential equation, be close to satisfying the property exactly?"

Project: Analyzing the generalized notions of skewness and standardized central moments. We will use Frechet means and the stochastic dominance method in the analysis of generalized skewness. Additionally, skweness, excess kurtosis, and other standardized central moments will be considered using the Stein operator. Students are expected to be familiar with advanced calculus, upper division probability and/or statistics, and know either MATLAB or R.

Project: Completely integrable systems. Completely integrable partial differential equations appear in classical mechanics, quantum mechanics, fluid mechanics and oceanography. The goal of this project is to do computational studies on simple models in soliton physics and quantum mechanics using methods that require only a knowledge of rational functions. Instead of being hard to implement, these methods are hard to derive, and the reason it is possible to use them comes from the theory of completely integrable systems. The mathematics and physics of completely integrable systems relate to some important recent discoveries in combinatorics. Thus, this project would be beneficial to most students hoping to pursue modern mathematics and physics research.

This project will require proficiency with rational functions, series, complex numbers and matrix operations typically found in courses on calculus, series, differential equations and linear algebra. A conceptual understanding of Fourier series as well as familiarity with MATLAB or Mathematica and/or a course that involves coding in a scientific context (such as numerical analysis) would also be helpful.

Project: Secure Computation. Suppose n users each hold private data, and they would like to compute some agreed-upon function of that data. Secure multi-party computation (MPC) is a cryptographic tool that allows them to do so, in a way that releases *only* the output of the computation, and leaks nothing else about the private inputs. In this project we will explore characterizations of which functions can be securely computed in various settings (security definition, number of users, communication channels between users).

Project: Inferring a metric species tree from topological gene trees. There are many ways to infer a species tree relating populations from gene trees relating biological sequences. But most of these methods do not give a metric species tree, meaning that we just get a topology of a tree, not its branch lengths. Also, the conflict between gene trees and species trees makes the inference problem challenging as standard phylogenetics methods applied to different gene trees for the same collection of species often give different trees. In this project, we will investigate the relationship between gene trees and species trees, and develop a statistically consistent method to infer a metric species tree from topological gene trees under a specific model. We will work both theoretically and practically. Students with a background in either mathematics and/or computer science are encouraged to apply.

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