The Department of Mathematics and the Department of Statistics at Texas A&M will host an eight-week Research Experience for Undergraduates (REU) during the summer of 2014. We anticipate running the following programs:
- Number Theory Mentored by Riad Masri and Matt Young.
- Mathematical Biology mentored by Jay Walton, and Jean Marie Linhart at Texas A&M, June and July
- Algorithmic Algebraic Geometry mentored by Maurice Rojas, June and July
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 Jay Walton and Jean Marie Linhart
In this interdisciplinary program, we shall explore mathematical aspects of a wide variety of models arising in mathematical biology. In the modeling of complex ecosystems, the main emphasis will lie on the impact of changes in the landscape topology on populations, including its distribution and dynamics. Two subareas will be emphasized - deterministic modeling using differential equations and stochastic modeling using techniques from statistics.
In the beginning of the program, various modeling problems and useful mathematical and statistical techniques will be presented, and participants will learn how to apply the mathematical techniques and theories to the biological problems. Participants will then choose research problems and form small research teams to work on the problems.
One topic we shall explore is the habitat fragmentation and re-connection question, which is concerned with the following issue. Decades of world wide wilderness land use policy decisions have resulted in many wilderness areas being broken into chains of isolated patches of national parks, forests, refuges or other protected zones. While the total area of these patches might seem considerable, their fragmented structure into relatively small patches might not provide viable habitats for many species, especially large predators. Many ecologists argue that the health of the predatory species at the top of the food chain provide a reliable barometer of the health of the entire ecological system. Consequently, there are large scale programs being proposed and in some cases already under way to connect fragmented habitats via corridors which will permit certain species, especially large predators, to travel between formerly isolated patches. The mathematical problem is to model this process in order to try to gauge the likelihood that it will be effective in making for more robust wildlife habitats, or whether it could cause a negative disruption to a habitat system and make matters worse.
Another important topic to be explored concerns the biodiversity/eco-stability debate which addresses the assertion that greater biodiversity leads to increased stability of an ecosystem, i.e. the more diverse the gene pool, the more robust the ecosystem is to perturbations. This has been a long running politically and ethically charged debate in need of rigorous quantitative scientific input.
Among the mathematical tools to be used are notions from finite dimensional dynamical systems, partial differential equations and differential geometry. Although sophisticated models could involve difficult concepts from these subjects, simplified models can be investigated by bright undergraduates with minimal background (but with a healthy enthusiasm to learn). The participants will be introduced to a variety of deterministic approaches to modeling ecosystems beginning with the classical Lotka/Volterra system of interacting species or predator/prey models. Very elementary topological ideas are used to quantify the fragmented structure of habitats from the point of view of individual species. More specifically, different species might see a given habitat system as having different connected components or patches. For example, certain bird species might see a habitat as connected whereas some plant or small, crawling animal species might see it as highly fragmented. Thus, the first task in studying an ecosystem is to determine its patches, or connected components, from the point of view of each of the species selected for study. Then one introduces interacting species models on the landscape.
The simplest level of modeling of species interaction dynamics on a fragmented habitat with corridors involves using ordinary differential equations for the densities of the species on the system of patches permitting migration of certain species between certain patches. These problems are ideal for undergraduate students to tackle, and preliminary results from the program suggest that selective migration can indeed qualitatively alter the dynamics over isolated patches without migration. Some groups of REU participants might want to choose projects extending these ODE models with migration to more than two patches and to more than two interacting species. Among the many possible questions to pursue is whether or not it matters which of the species has the ability to migrate? For example, in predator/prey type interactions, does one see similar dynamics if just the predator can migrate or if just the prey can migrate?
Other groups of participants might choose projects modeling the spatial distribution and movement of species within patches as well as between patches. These models will be of two types, continuous time partial differential equation models (the simplest being of diffusion type) and discrete time, convolution integral models (as proposed recently by Alan Hastings). Some groups will use simple diffusion operators to model the movement of species around a flat landscape, while other groups will study generalizations of these operators to non-flat surfaces using ideas from differential geometry.
As an alternative, some groups of the REU participants might wish to pursue statistical approaches to ecosystem modeling. Again, many avenues are open for these projects including stochastic population models. These are particularly useful for modelling the transient behavior of populations and also the behavior of small populations.
The basic ideas of stochastic population models will be introduced. The methodology is based on Markov models. Basic assumptions about the population lead to a system of Kolmogorov differential equations for the probability functions. Generating functions are useful in obtaining the moments or cumulants of the population size. The Kolmogorov differential equations lead naturally using the generating function approach to systems of differential equations for the cumulants. These can be solved analytically or numerically to describe the population. This methodology can be applied to models for a single population including both linear birth-immigration-death models and nonlinear birth-immigration-death models.
Our goal is to extend the basic ideas illustrated in single-population models to model multiple populations such as predators and prey. We also want to bring spatial aspects into the modelling by allowing for several locations. This leads us to developing models for multiple populations. Joint moments and cumulants for the sizes of multiple populations are introduced. In a manner analogous to that for a single population, bivariate Kolmogorov differential equations lead naturally using the generating function approach to systems of differential equations for the bivariate cumulants. This approach is illustrated by applying the methodology to multiple population models that can include births, deaths, immigration, and migration.
Much of our interest will be on the bivariate distribution of the number of predator and prey animals over the course of time. The above methodology enables us to obtain the cumulants of the bivariate distribution for any time. To approximate the corresponding bivariate density, such methods as saddlepoint approximations and series expansions will be used. Such research is helpful in the formulation of management strategies. Researchers can apply the stochastic models to determine the effects of various control procedures such as reducing birth rate, increasing a death rate, and restricting immigration on the bivariate distribution of the numbers of predators and prey.
The emphasis in mathematical physiology will be to the cardiovascular system. Among the topics discussed will be modeling the mechanical behavior of arteries and the heart muscle, and the role that mechanical behavior plays in the development and progression of cardiovascular disease. Another topic will be modeling the biology and chemistry involved with vascular diseases such as atherosclerosis and aneurisms. Recently, mathematical models have played a key role in understanding and predicting the onset, progression and treatment of these and many other diseases. The participants selecting to work in this area will be introduced to the relevant biological and chemical concepts as well as pertinent mathematical techniques and theories. As with the ecosystems modeling, they will then form small research groups to work on specific topics and problems.
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.
- [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.
- [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).