The Undergraduate Summer School provides opportunities for talented undergraduate students to enhance their interest in mathematics. This program is open to undergraduates at all levels, from first-year students to those who have just completed their undergraduate education. Based on the backgrounds of the accepted students, they will be divided into two groups - introductory and advanced. There will be several activities organized for these groups, with some specifically intended for either the introductory or advanced group. There will be time for study groups and individual projects guided by advisors, as well as other activities.

2008 Course Descriptions:

1. (First week) Algorithmic Fewnomial Theory; J. Maurice Rojas, Texas A & M University

Systems of polynomial equations are ubiquitous in applications ranging from computational biology to mathematical physics. Every polynomial system has its own personality, so structured polynomial system solving means using (and designing) algorithms for solving that take advantage of the underlying structure.

One of the simplest structures one can study is sparsity, i.e., having few monomial terms. Fewnomial Theory, as pioneered by Descartes in 1637 and Khovanski in the 1980s, is a framework that tells us how systems of equations with "few" monomial terms have "few" real roots. However, can we actually design algorithms that take advantage of sparsity and enable significant speed-ups? Also, can we make a similar use of sparsity over different settings, such as finite fields or the rational numbers?

In these lectures, we will see positive answers to these questions. We begin our introduction from scratch, so no background in algebraic geometry, complexity theory, or number theory is assumed. Furthermore, we will see many cute pictures and animations revealing the personality of fewnomials.

2. (Second week) Introductory Course: Toric Surfaces Herb Clemens, Ohio State University

In this course we will give an introduction to toric surfaces. We will take a very concrete point of view and discuss several classical examples including the Veronese surface and rational normal scrolls. We will emphasize the idea that many abstract notions in algebraic geometry specialized to the setting of toric varieties can be understood using linear algebra and combinatorics. We will illustrate this principle with a discussion of singularities on toric surfaces and their resolutions.

3. (Third week) Introductory Course: Toric Surfaces; Jessica Sidman, Mt. Holyoke College

In this course we will give an introduction to toric surfaces. We will take a very concrete point of view and discuss several classical examples including the Veronese surface and rational normal scrolls. We will emphasize the idea that many abstract notions in algebraic geometry specialized to the setting of toric varieties can be understood using linear algebra and combinatorics. We will illustrate this principle with a discussion of singularities on toric surfaces and their resolutions.

4. (All three weeks) Introduction to Algebraic Geometry; David Perkinson, Reed College

Algebraic geometry is the study of solutions to systems of polynomial equations. It is a central and very active area of modern mathematics with deep connections to commutative algebra, complex analysis, number theory, combinatorics, and topology. It has applications in physics, robotics, coding theory, optimization theory, and more recently in biology and statistics. We will begin by developing the standard "dictionary" for translating between algebraic properties of polynomials and geometric properties of their solution sets. Special topics may include Hilbert functions and resolutions of ideals, enumerative geometry, Grassmannians and the Schubert calculus, and computational aspects involving Groebner bases. No background beyond linear algebra will be assumed.

More information coming soon!

PCMI's Undergraduate Summer School is supported in part by the National Security Agency and in part by the National Science Foundation grant # DMS- 0437137.