Math Research Experience 2017
The mathematics department will be hosting an internal research experience for undergraduates and early graduate students in Summer 2016. If your idea of fun is thinking about mathematics, then this may be your chance for getting paid for having fun.
Projects

An action of the binary tetrahedral group and an application to algebraic topology. A certain group, called the Binary tetrahedral group, is of particular interest to some algebraic topologists. It appears as the group of automorphisms of an elliptic curve defined over the "field with four elements." There is a way to extract an action of this group on a certain power series ring. This action is part of the action of a much larger group, and is one of the key players in beautiful story called chromatic homotopy theory These come into play when computing certain topological invariants. We will study these group actions and use them to make some algebraic computations that have topological meaning. In the process, we will be learning about the padic integers, algebra in power series rings, elliptic curves and formal group laws.
Mentor: Agnes Beaudry. Prerequisite. Math 3140. 
Local and global wellposedness of wave equations. Partial Differential Equations are used to model the world around us. In this project we investigate mathematical properties of wellposedness of the solutions related to systems of equations appearing in the electromagnetism and/or the Standard Model of particle physics. Wellposedness questions ask for example if solutions exist, do they exist globally in time or if they end with a blowup in some specific mathematical sense.
Mentor: Magda Czubak. Prerequisite. Math 3001. 
Study of invariants of singularities. The Milnor number is an invariant of a function germ and provides information of the singularity type the germ represents. In this proposed REU project certain singular spaces will be studied, their Milnor numbers computed and it will be attempted to construct further invariants of singularities. Depending on the students interests and knowledge, the project can go more into a computational or more into a theoretical direction.
Mentor: Markus Pflaum. Prerequisite. Math 3001, Math 3450. 
The Geometry of Number Fields. Motivated by cryptographic applications, we investigate some statistics of number fields. In particular, we consider the ring of integers of a number field embedded into R^n using the Minkowski embedding. We will ask some questions about the geometry of the lattice and the relationship to its ideal sublattices. The project will have a strong experimental/computational component. We will use Sage mathematics software to generate data and conjectures. No prerequisites besides the algebra pillar sequence; expect to learn some algebraic number theory, some lattice geometry, and how to use Sage and program in Python.
Mentor: Kate Stange. Prerequisite. graduate student only. 
Unipotent Rook Polytopes. Unipotent polytopes are a family of geometric objects (such as the hypercube) that arise naturally out of algebra and combinatorics. Being at the intersection of so many areas gives many possible tools to study the structure of these polytopes. Past REU projects have explored various families of unipotent polytopes from numerous points of view: random walks, face lattice, representation theory, etc. One of the more tractable families seems to be the family of rook polytopes, whose vertices are given by placements of nonattacking rooks on an mxn chessboard. This summer we will use all the hard work of past summers to finally understand the algebraic and combinatorial foundations of rook polytopes.
Mentor: Nat Thiem. Prerequisite. Math 3140. 
Numerical Solutions of Partial Differential Equations. This project aims to understand the physics behind elliptic, parabolic or hyperbolic PDEs and then to numerically solve them. Convergence and stability properties of algorithms will also be studied. Applications to PDEs are in steady state flow of inviscid fluids, heat conduction, and waves on a drumhead.
Mentor: Divya Vernerey. Prerequisite. Math 3430, and knowledge of programming in MATLAB, Mathematica, Maple or the like. 
Visualizing the arithmetic of the rational numbers. In this project, we will develop tools for visualizing Conway's topograph as a visual representation of the rational numbers (actually the rational projective line). We will use these tools to illustrate the Euclidean algorithm (for calculating the greatest common divisor of two integers), continued fractions, the arithmetic of polynomial functions, the structure of irrational numbers (especially quadratic irrationals), and some exceptional group isomorphisms. This will involve learning some hyperbolic geometry and will require comfort with linear algebra.
Mentor: Jonathan Wise. Prerequisite. Math 3130, Math 2001 + one higher level, and some programming.