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2007 PI Summer Graduate Program

Applicable Algebraic Geometry

Applicable Algebraic Geometry

Texas A&M University in College Station, Texas

July 23-August 10, 2007 (Travel Days: July 22 and August 11)

July 23-August 10, 2007 (Travel Days: July 22 and August 11)

Frank Sottile | Mathematics, Texas A&M University |

Mathematics, Texas A&M University | |

Thorsten Theobald | FB Mathematik, JW-Goethe Universität Frankfurt |

Speakers: | |

Frank Sottile | Mathematics, Texas A&M University |

Thorsten Theobald | FB Mathematik, JW-Goethe Universität Frankfurt |

Serkan Hosten | Mathematics, San Fransisco State University |

Seth Sullivant | Mathematics, Harvard University |

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.

Schedule

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.

- Introduction to basic concepts from algebraic geometry.
Projective and affine varieties, ideals,
Gröbner bases, and
standard examples.
- 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.
- Deformation and numerical techniques in algebraic
geometry. Applications to solving systems of equations and
connections to tropical geometry.
- Connections between real algebraic geometry
(positivity of
polynomials) and semidefinite
programming in optimization.
- 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.

- 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.
- 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

Professor Laura Felicia Matusevich

Professor Thorsten Theobald

TAMU contact staff:

Ms. Rhonda Faust

Department of Mathematics

Texas A&M University

College Station, TX 77843-3368

Name | Department | Affiliation |
---|---|---|

Monique Azar | Department of Mathematics | Purdue University |

Florian Stefan Block | Department of Mathematics | University of Michigan |

Herivelto Martins Borges | Department of Mathematics | University of Texas |

Justin Bush | Department of Mathematics | Rutgers University |

Patrick Byrnes | School of Mathematics | University of Minnesota |

Daniel Cabarcas | Department of Mathematical Sciences | University of Cincinnati |

Dustin Alexander Cartwright | Department of Mathematics | University of California |

Xiaohui Chang | Department of Statistics | University of Chicago |

Philippa Charters | Department of Mathematics | University of Texas |

Teng Chen | Department of Mathematics | North Dakota State University |

Tianran Chen | Department of Mathematics | Michigan State University |

David Christopher Clark | Department of Mathematics | Michigan Technological University |

Crystal Clough | Department of Mathematical Sciences | University of Cincinnati |

Adam Crock | Department of Mathematics | City University of New York (CUNY) |

Rafael Del Valle-Vega | Department of Mathematical Sciences | University of Puerto Rico |

Daniel Erman | Department of Mathematics | University of California |

Luis Garcia-Puente | Department of Mathematics | Texas A & M University |

Jason E. Gower | Institute for Mathematics and its Applications | University of Minnesota |

Nickolas Jason Hein | Department of Mathematics | Texas A & M University |

Christopher Hillar | Department of Mathematics | Texas A & M University |

Serkan Hosten | Department of Mathematics | San Francisco State University |

Valerie Marie Hower | University of Georgia | |

Ashraf Ibrahim | Department of Mathematics | Texas A & M University |

Corey Foster Irving | Department of Mathematical Sciences | Texas A & M University |

Ning Kang | Department of Mathematics | University of Texas |

Edward D. Kim | Department of Mathematics | University of California |

Jimmy Kimball | Department of Mathematics | Texas A & M University |

Elizabeth Kleiman | Department of Mathematics | Iowa State University |

Ajit Kumar | Department of Mathematics | University of Houston |

Christopher Allen Kurth | Department of Mathematics | Iowa State University |

Juyoun Lee | Department of Statistics | Pennsylvania State University |

Myoungji Lee | Department of Statistics | University of Chicago |

Shaowei Lin | Department of Mathematics | University of California |

Abraham Martin del Campo | Department of Mathematical Sciences | Texas A & M University |

Federico Nicolás Martínez | Department of Mathematical Sciences | University of Buenos Aires |

Laura Felicia Matusevich | Department of Mathematics | Texas A & M University |

Gregory Steven McWhirter | Department of Mathematical Sciences | University of Pittsburgh |

Jason Morton | Department of Mathematics | University of California |

Ashish Myles | Department of Computer Science | University of Florida |

Christopher Neff | Department of Mathematics | University of California |

Linh Viet Nguyen | Department of Mathematics | Texas A & M University |

Ekin Ozman | Department of Mathematics | University of Wisconsin |

Casian Pantea | Department of Mathematics | University of Wisconsin |

Eric Patterson | Department of Mathematics | University of Chicago |

Sonja Petrovic | Department of Mathematics | University of Kentucky |

Fernando Piñero | Department of Mathematical Sciences | University of Puerto Rico |

Cordian Riener | Department of Computer Science and Mathematics | Johann Wolfgang Goethe-Universität Frankfurt |

Jim Ruffo | Department of Mathematical Sciences | Texas A & M University |

Korben Rusek | Department of Mathematics | Texas A & M University |

Alexandra Seceleanu | Department of Mathematics | University of Illinois at Urbana-Champaign |

Frank Sottile | Department of Mathematics | Texas A & M University |

Reinhard Steffens | Department of Computer Science and Mathematics | Johann Wolfgang Goethe-Universität Frankfurt |

Erik Stokes | Department of Mathematics | University of Kentucky |

Seth Sullivant | Department of Mathematics | Harvard University |

Chia-Liang Sun | Department of Mathematical Sciences | University of Texas |

Tsung-Hui Sun | ||

Theresa Ann Szczepanski | Department of Mathematical Sciences | University of Delaware |

Thorsten Theobald | Department of Computer Science and Mathematics | Johann Wolfgang Goethe-Universität Frankfurt |

Louis Theran | Department of Computer Science | University of Massachusetts |

Stefan Tohaneanu | Department of Mathematics | Texas A & M University |

Ya-lun Tsai | Department of Mathematics | University of Minnesota |

Shayne Michael Vargo | Department of Mathematics and Statistics | San Diego State University |

Alan A Veliz-Cuba | Department of Mathematics | Virginia Polytechnic Institute and State University |

Dominic Richard Walter | Department of Civil Engineering | Leopold-Franzens Universität Innsbruck |

Zeying Wang | Department of Mathematics | University of Delaware |

Erik Westlund | Department of Mathematical Sciences | Michigan Technological University |

Emily Elspeth Witt | Department of Mathematics | University of Michigan |

Han Xiao | Department of Statistics | University of Chicago |

Jiangwei Xue | Department of Mathematics | Pennsylvania State University |

Saeid Yasamin | Department of Mathematics | Indiana University |

Qian Yin | Department of Mathematics | University of Michigan |

Zhijun Yin | Department of Mathematical Sciences | University of Cincinnati |

Seonguk Yoo | Department of Mathematics | University of Iowa |

Josephine Yu | Department of Mathematics | University of California |

Li-Hua Yu | ||

Wenliang Zhang | Department of Mathematics | University of Minnesota |

Ruijun Zhao | Department of Mathematics | Purdue University |

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