Non-linear Dynamics and Applied Mathematics

• Complex Analysis Professor Igor Krichever. Graduate student leader: Anton Dzhamay: ad@math.columbia.edu This project centers around the Riemann mapping theorem, which says that any simply connected domain U in the complex plane, which is not the entire complex plane, is analytically isomorphic to the unit disk. Recent work of Wiegmann and Zabrodin shows that if U is bounded by an analytic curve, then it satisfies an interesting differential equation. This seminar will discuss generalizations of this result. Preferred background: undergraduate complex analysis.
• Dimensional Topology Professor Walter Neumann. Graduate student leader: Abhijit Champanerkar The project will address computation of so-called A-polynomials of 3-manifolds. The main prerequisites and techniques involve abstract algebra. The necessary knowledge of three-manifold topology can be picked up quickly. The A-polynomial is a very important recent invariant for 3-manifolds, but it has been hard to compute, and one of the issues would be to experiment with numerical and computer algebra approaches to computing it. It also appears to have deep connections with other invariants. Experimental testing of such connections, as well as the theoretical investigation of them, will be part of the project. The project will thus involve experimentation using such programs as snap and snappea for 3-manifold computation, and pari/gp, mathematica, macaulay, and the like for algebraic computation. Some programming will be involved. Even though the emphasis is computational geometry, the students will learn about the beautiful interplay between algebraic geometry and topology from the examples at hand.
• Grobner bases and algebraic geometry (not offered) Professor Henry Pinkham; Graduate student leader: Ye Tian The problem we will work on here is elementary, and can be stated completely: let H be a subgroup of Zn (=the product of n copies of the integers). Let M be the monoid generated by the intersection of H with any orthant (= generalization to n dimensions of a quadrant in the plane). The goal is to find a good algorithm for computing a basis for M as a monoid, in other words, find a collection of vectors v_1 ... v_m in M so that any element in M can be written as a_1 v_1 + a_2 v_2 + ... a_m v_m, where the a_i are *positive* integers. This is called a Hilbert basis. The current algorithms blow up very quickly. The techniques used to study this problem involve an interesting called a Hilbert basis. The current algorithms blow up very quickly. The techniques used to study this problem involve an interesting combination of linear algebra, ring theory, geometry and combinatorics, especially that of polyhedra. A good algorithm would have many applications in pure and applied mathematics. We will start out by working through the relevant material in the text: "Ideals, Varieties and Algorithms" by Cox-Little-O'Shea (2nd edition, Springer-Verlag). The state of the art is given in either of two books: "Using algebraic geometry" again by Cox-Little-O'Shea (GTM 185, 1998, Springer) or "Grobner Bases and Convex Polytopes" by B. Sturmfels (AMS 1995). We will also look at the algorithm proposed by Loic Pottier in a 3 page paper, "Euclid's algorithm in dimension n" (ISSAC 96, ACM Press, July 1996). The computer component of the project will be to use the computer program Macaulay to do computations in rings. Prerequisite: undergraduate algebra.

For more information contact Henry C. Pinkham .