# REU Programs

The Department of Mathematics at Texas A&M will host an eight-week Research Experience for Undergraduates (REU) during the summer of 2015. We will run the following programs:

- Number Theory Mentored by Riad Masri and Matt Young.
- Mathematical Modeling in Biology Mentored by Anne Shiu.
- Algorithmic Algebraic Geometry Mentored by J. Maurice Rojas

## 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: Mathematical Modeling in Biology

**Mentored by Anne Shiu**

The participants in this program will investigate mathematical models from three areas of mathematical biology: biochemical reaction networks, immunology, and neuroscience. The emphasis will be on systems amenable to analysis by algebraic and combinatorial methods. No background in biology is required, and all projects are accessible to undergraduate participants.

* Biochemical reaction networks. A reaction network is a directed graph in which the nodes are labeled so that each directed edge represents a chemical reaction such as, for instance, A+B ---> C, in which one unit of A and one of B react to form one unit of C. The focus of this component is on dynamical systems that arise from reaction networks taken with mass-action kinetics. The ODEs that govern these systems are multivariate polynomials and therefore are amenable to algebraic techniques. An example project question is the following: Can we develop a criterion for when the quasi-steady state assumption (QSSA) technique works or for when it fails? QSSA is a popular elimination method for model reduction in chemical systems that has been used for nearly a century. This simple technique first solves for certain "fast'' (quickly reacting) species and then substitutes those quantities into the remaining "slow'' species. Although QSSA is a widely used technique, Pantea et al. proved recently that sometimes the first step of QSSA is impossible, that is, it is possible that the "fast'' species can not always be solved explicitly in terms of the "slow'' species.

* Multistationarity in models of immune response. Many ODE models of immune response take the form of reaction networks with mass-action kinetics. Therefore, theorems from chemical reaction network theory directly apply in this setting. However, this approach has received little attention, and moreover such results have been limited to cases in which steady states are unique. Participants in this component will learn about existing techniques for analyzing multiple steady states, and then apply them to analyze models arising in immunology.

* Convex neural codes. One goal of President Obama's BRAIN (Brain Research through Advancing Innovative Neurotechnologies) Initiative is to determine the wiring diagram of the brain. To this end, a starting principle is that "neurons that fire together, wire together''. The aim of this REU component is to clarify which sets of neurons can fire together under the assumption that each neuron of interests codes for some convex region of some Euclidean space. This assumption is valid in certain biological settings, for instance, when each neuron has a corresponding place field, a convex region in space (this space might be a tabletop on which a laboratory rat is walking) and that neuron fires precisely when the subject (rat) is in that region. This research is motivated by place cells in neuroscience, which won its discoverers the 2014 Nobel Prize in Medicine.

## 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 orbital design and computational biology, investigating some recent techniques from flower constellations and algebraic statistics. 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, resultants, homotopy methods, connections to robotics, contingency tables, and maximum likelihood estimation.

**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.
- [PS04] ``The Mathematics of Phylogenomics,'' by Lior Pachter and Bernd Sturmfels, Math ArXiV paper, downloadable from xxx.arxiv.org/math.ST/0409132.
- [PRW00] Algebraic Statistics: Computational Commutative Algebra in Statistics, by G. Pistone, E. Riccomagno, and H.P. Wynn, CRC Press (2000).