Applicable Algebraic Geometry

This program is for graduate students of IMA Participating Institutions only. In order to participate, students need to fill out the application form and need to be nominated by their department chair.

From Monday, July 23 through Friday, August 10 in 2007, Texas A&M University in College Station, Texas will be the host of the Institute for Mathematics and its Applications (IMA) Summer Graduate Program in Mathematics. The course will concentrate on Applicable Algebraic Geometry.

Program Description:

Recent years have seen applications of many ideas and techniques from algebraic geometry to problems in applied mathematics and engineering. Part of this is a recognition of essential algebraic structures in applied problems and part is a need in applications for exact/certifiable results. It is also due in no small measure to modern, simplified presentations of algebraic geometry, interest in particular examples, and the growing use of computers in algebraic geometry. The 2006-2007 IMA Thematic Year "Applications of Algebraic Geometry" is showcasing these trends and will lead to further, deeper applications of algebraic geometry. This IMA PI Summer Graduate Program will help prepare the ground for the future by introducing graduate students from IMA participating institutions to some of these exciting developments and new perspectives.

The program will be structured around a course in applicable algebraic geometry, treating foundational material as well as current applications. The foundations will include Gröbner bases, toric varieties, and real algebraic geometry, while the applications will be drawn from optimization, non-linear computational geometry, algebraic statistics, and mathematical biology. We will emphasize computational aspects by including computer tutorials and laboratories on relevant software. We will also have guest lectures explaining current research topics. This multi-tiered menu, ranging form introductory material through current research, will ensure that every student gains something from their experience.

The core of this summer Graduate Program will be two series of lectures. One by Sottile and Theobald will consist of 25 lectures, provide foundational background, and cover current applications in optimization and in non-linear computational geometry. Serkan Hosten and Seth Sullivant will give a joint series of 10 lectures on algebraic statistics and applications to biology. Both lecture series will include regular assigned problems and a joint daily discussion section, possibly run by postdoc mentors. These mentors will include Texas A&M Postdoc Luis Garcia (algebraic statistics, mathematical biology, and geometric modeling).

Many applications of algebraic geometry are facilitated by computer experimentation, calculation, and user-friendly software. Because of this, some of the problems from the course will include computer work, in levels varying from the simple computation of examples, to full-blown computer laboratory projects. To familiarize the students with this aspect of the course, the program will feature regular tutorials on installing and using relevant mathematical software, (such as Maple, Singular, Macaulay 2, SosTools, and GloptyPol).

Details of course: The lecture series of Sottile and Theobald will consist of five parts, each taking roughly 5 lectures. The first three parts are foundational and the last two are advanced topics. These are based on parts of a graduate textbook on Applicable Algebraic Geometry that Sottile and Theobald are developing. Participants will be provided with copies of the text.

1. Introduction to basic concepts from algebraic geometry. Projective and affine varieties, ideals, Gröbner bases, and standard examples.
2. Basic ideas and algorithms from real algebraic geometry. This prominence of real algebraic geometry is because in applications, real solutions are often much more important than complex ones.
3. Deformation and numerical techniques in algebraic geometry. Applications to solving systems of equations and connections to tropical geometry.
4. Connections between real algebraic geometry (positivity of polynomials) and semidefinite programming in optimization.
5. Algebraic geometry in nonlinear computational geometry and geometric modeling.

The lecture series of Hosten and Sullivant will consist of 10 lectures, delivered in the last two weeks of the Graduate Program. This series will have two parts.

1. Algebraic statistics, which uses algebraic geometry for making statistical inferences, as many statistical models for discrete random variables are classical algebraic varieties. The course will explain this connection and discuss some of the interesting geometry, as well as the statistical consequences of the algebraic analysis, such as in maximum likelihood estimation.
2. Computational biology, particularly on the relevance of algebraic statistical models to genome sequence analysis.

Common themes in this course will be the role of symbolic computation and of classical and concrete algebraic varieties, such as toric varieties. Also, while not apparent from the course descriptions, geometric combinatorics, particularly polytopes and vector configurations, will play a fundamental role.

Program:  Schedule

In addition to the course, we will have a series of guest lectures on further applications of algebraic geometry. This will result in four hours each day of classroom instruction and lectures. We will also have 75 minutes each day alternating between a discussion session and a computer lab. This program will be organized around coffee breaks, and the lecturers will be available outside of lectures.

Housing:

The participants will stay in on-campus conference housing which is also used for summer REU programs. Among the amenities offered are community kitchens, pool and fitness center, aerobics room, multimedia room, big-screen TV lounges and surround sound movie theater. There are also individual and group study rooms with ethernet, and ten computer study rooms. More information is available at www.livethetradition.com/.

Contact Information
Organizers:
Professor Frank Sottile
sottile@math.tamu.edu 
Professor Laura Felicia Matusevich
laura@math.tamu.edu 
Professor Thorsten Theobald
theobald@math.uni-frankfurt.de 

TAMU contact staff:
Ms. Rhonda Faust
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
rhonda@math.tamu.edu