# REU Programs

The Department of Mathematics at Texas A&M will host an eight-week Research Experience for Undergraduates (REU) during May 30 - July 21, 2017. We will run the following programs:

- Mathematics of Topological Quantum Computation Mentored by Eric Rowell
- Number Theory Mentored by Riad Masri and Matt Young
- Algorithmic Algebraic Geometry Mentored by J. Maurice Rojas

## Summer Program: Mathematics of Topological Quantum Computation

**Mentored by Eric Rowell**

Quantum computation refers to computational models using quantum states. This topic sits in the intersection of physics, mathematics and computer science. The Nobel Prize in Physics 2016 was awarded for theoretical discoveries of topological phase transitions and topological phases of matter, which are closely related to topological quantum computation. There are many interesting mathematical questions related to this subject with connections to algebra, topology, complexity theory, among other areas. Knot theory, braid group representations and Galois theory are some of the main topics involved. Students with interest in any of these areas are encouraged to apply. The emphasis will be given to the algebraic aspects of these problems. No background in quantum computation is required.

## Summer Program: Number Theory

**Mentored by Riad Masri and Matt Young**

Much of modern number theory revolves around two different types of functions: L-functions and modular forms. The simplest example of an L-function is the Riemann zeta function which, despite over 150 years of research, still has many unproved conjectures such as the famous $1,000,000 Riemann Hypothesis. Other types of L-functions encode properties of algebraic equations like y^2 = x^3 + ax + b. Modular forms are amazingly symmetric functions that are closely related to L-functions. They also have applications to solving certain algebraic equations and also have more exotic connections to physics.

The participants of this REU will have a variety of options to explore this beautiful area of number theory at an accessible level. Based on the interests of the participants, possible projects could include:

- Studying zeros of modular forms
- Developing numerical tools to study modular forms
- Studying central values of L-functions of modular forms
- Studying rational points on elliptic curves

## Summer Program: Algorithmic Algebraic Geometry

**Mentored by J. Maurice Rojas**

Born over two millenia ago, algebraic geometry sought to understand the solution of polynomial equations. Now, numerous applications (including computational biology, complexity theory, signal processing, satellite orbit design, robotics, coding theory, optimization, game theory, and statistics, just to name a few) call for the solution of massive systems of equations. Modern algorithmic algebraic geometry gives us the tools to solve such systems. Algorithmic algebraic geometry is also a vibrant field where students can profitably pursue any number of rich directions.

Assuming only a linear algebra background, we begin with a brief introduction to some of the computational tools from algebra, combinatorics, and geometry that we'll need. In parallel, we also give an introduction to applications coming from algorithmic complexity, orbital design, and optimization. Students will be expected to embark on computational experiments almost immediately.

The core technical topics we will cover include the following: Basic convex and tropical geometry, fewnomial theory over the real and p-adic numbers, homotopy methods for solving polynomial systems, and A-discriminants.

**MAIN REFERENCES:**

- [Stu02] Solving Systems of Polynomial Equations, by Bernd Sturmfels, CBMS Lecture Series, AMS Press, 2002.
- [SW05] The numerical solution of systems of polynomials arising in engineering and science, by A. Sommese and C. Wampler, World Scientific, 2005.

** SUPPLEMENTAL REFERENCES: **

- [CLO97] Ideals, Varieties, and Algorithms, by David A. Cox, John B. Little, and Donal O'Shea, Springer-Verlag, 1997.