Summer Research in Mathematics

Math Research Experience 2015

The mathematics department will be hosting an internal research experience for undergraduates and early graduate students in Summer 2015. If your idea of fun is thinking about mathematics, then this may be your chance for getting paid for having fun.

Projects

• The Schmidt arrangement of the Eisenstein integers. The first picture here shows the Schmidt arrangement of the Eisenstein integers. It's an intricate arrangement of circles that reflects the structure of the Eisenstein integers, a subring of the complex numbers that has properties similar to the integers but a much more geometric flavour. This project will combine geometry with number theory. The required background is Math 3140. You will learn about Mobius transformations and number theory in quadratic fields.
  Mentor: Kate Stange.

• Random walks on unipotent groups. This project seeks to use algebraic techniques to help understand random walks arising from groups. In particular, we will study a family of unipotent groups called Heisenberg groups and the combinatorial walks they generate. Last summer studied a complementary family and made significant progress; we hope to build on this success. Math 3140 is a required background, but all other mathematics can be learned in the program.
  Mentor: Nat Thiem.

• Geometry of composition laws. In 1801 Gauss showed that equivalence classes of 2x2 integer matrices under row and column operations can be given a group structure. Today we can give an elegant geometric description of Gauss's composition law arising from the geometry of the projective line. Recently Bhargava gave a similar composition law on equivalence classes 2x2x2 cubes of integers (this time, equivalence is from row, column, and plane operations). We will study one approach to understanding this composition law geometrically. Comfort with commutative algebras and their modules will be necessary in this project. Math 4140 is a prerequisite.
  Mentor: Jonathan Wise.

• 2-cocycles and higher-rank graphs. Higher-rank graphs are a generalization of directed graphs that can be used to construct examples of C*-algebras (the main focus of my research). 2-cocycles are a particular kind of function on a higher-rank graph; different 2-cocycles give you, in principle, different C*-algebras. The goal of htis project is to improve our understanding of the phrase "in principle" in the previous sentence. That is, we will investigate several different equivalence relations on the set of all 2-cocycles, and try to understand the relationships between them, as well as what they tell us at the level of the C*-algebras. MATH 2001, plus at least one additional proof-based math class, is a prerequisite for this project, but all other necessary background will be covered during the project (including what a higher-rank graph is, and a C*-algebra if you're curious).
  Mentor: Elizabeth Gillaspy.

• Liber Mathematicae and Mathematics presentation on the web. Properly presenting Mathematics on the web is still a challenging task. The proposed REU addresses this problem by extending the Liber Mathematicae project and writing appropriate code in ruby on rails.
  Mentor: Markus Pflaum.

• Understanding the Recent Proof of the Poincare Conjecture in Simple Examples. The recent proof of Perelman of the long-standing Poincare' Conjecture amazed the mathematical world; indeed (after a lot of deep mathematics) what 'looks like a 3-sphere' was finally actually proven to be a 3-sphere!! We will be looking at several summaries of this astonishing proof, especially in the case of two-dimensional manifolds, to get familiar with the main ideas involved in this astonishing proof. Some knowledge of topology, ODE, and differential geometry is required.
  Mentor: Carla Farsi.

• Numerical Solutions of Reaction-Diffusion Partial Differential Equations. This project aims to understand the physics behind reaction-diffusion equations and then to numerically solve them. Convergence and stability properties of algorithms will also be studied. Math 4470 is a prerequisite, and some knowledge of programming in MATLAB, Mathematica, Maple or the like is also required.
  Mentor: Divya Vernerey.

Dates