Many problems in pure and applied mathematics lead to the basic problem of solving systems of polynomial equations with real or complex coefficients in several variables. The situation where the set of complex solutions is finite is fundamental and we will focus on that. While iterative numerical methods (for instance, the multivariable Newton's Method) are usually effective for approximating individual solutions, they give no direct information about how many solutions a system actually has. Using them, it is sometimes difficult to know when to stop looking for new solutions!
Other, more algebraic, techniques for determining (or bounding) the total number of solutions have been much studied. In this talk, we will discuss some of these ideas, starting with the most basic result of this type -- the Fundamental Theorem of Algebra, and its generalization to the classical Bezout theorem. We will then turn to some quite recent work and show how a more refined estimate, called the BKK bound, comes from the combinatorics of the particular "shape" of the terms in the polynomials (encoded via the so-called Newton polytopes).
Problems in biological fluid dynamics typically involve the interaction of an elastic structure with the surrounding fluid. Examples of these coupled fluid-structure systems include blood flow in the heart, air flow through the lungs, and sperm motility in the reproductive tract. These biological processes can each be described by a system of time-dependent, coupled, nonlinear partial differential equations. While the explicit solution of these complex equations is impossible, scientists are making progress in understanding these systems using computational methods.
In this talk, we will present an overview of biological fluid dynamics, and discuss the interdisciplinary nature of the research in this field. We will focus on the examples of ciliary and flagellar beating in microorganisms, as well as the swimming of nematodes and leeches.
An important paradigm of science at the beginning of the twenty-first century is the view of our social, physical, and technological world as an interconnected collection of networks. It is important to understand these networks as a whole rather than as a collection of their parts. For instance, the World Wide Web is more than just a collection of linked web sites; it takes on a life of its own that cannot be explained through the functioning of its parts. Modern genetics has taught us that the processes in our body are controlled by a network of genes acting together. Computers perform vast calculations by distributing the task over a network of processors that work together as a team.
This talk will describe several such networks, analysis methods, and scientific challenges we are facing in understanding the networked world around (and inside) us.
The Hecke algebras of type A arise naturally in the study of knot theory, quantum groups, and Von Neumann algebras. Their relation to the symmetric and braid groups allows for their study using combinatorics and low dimensional topology.
In this talk I will give an introduction to Hecke algebras of type A and B and show their relation to the symmetric group and the braid group (this groups will be defined in this talk).
I will also construct a beautiful homomorphism from a specialization of the Hecke algebra of type B onto a reduced Hecke algebra of type A. This homomorphism has proven to be an useful tool to reduce questions about the Hecke algebra of type B To the Hecke algebra of type A. If time permits, I will give applications of this homomorphism.
Grete Hermann proved in 1926 that for any ideal I in an n-dimensional polynomial ring over the field of rational numbers, if I is generated by k polynomials each of which has degree at most d, then it is possible to write each element f of I as a linear combination of the given generators such that the coefficients of the generators have degree at most (kd)2^n more than the degree of f. In other words, the ideal membership problem is doubly exponential in the number of variables. There are ideals for which singly exponential degree can be easily verified. For a long time there was hope that singly exponential bound was indeed the upper bound. However, in 1982, Mayr and Meyer found (generators of) ideals for which a doubly exponential bound in n is indeed achieved. This talk will be about these Mayr-Meyer ideals, the properties they do and do not satisfy, and how one can approach the computational complexity problem from the point of view of algebra.
We consider a continuous space that models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature (i.e., it is CAT(0) in the sense of Gromov), giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or otherwise doing statistical analyses on sets of trees on a common set of species. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.
Implementing this model requires computational techniques that make use of the dual combinatorial and continuous nature of this space. Such techniques are currently being developed.
This is joint work with Susan Holmes and Karen Vogtmann: http:www.math.cornell.edu/~vogtmann/papers/Trees/lap.pdf
The distinguished place that the squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ... hold within the set of all natural numbers is discussed in connection with Gauss' law of quadratic reciprocity. This will serve as an introduction to exponential sums, also known as trigonometric sums, and their applications in number theory (Gauss sums). The application of exponential sums in computational mathematics ("the fast Fourier transform"-FFT) and the application of Galois' theory of finite fields in cryptography (the new "Advanced Encryption Standard"-Rijndael) will also be discussed. Furthermore, the elementary level of the talk will discuss the historical relation between these developments and the rest of mathematics