Tuesdays, 4:30 - 5:30pm in JWB 335
The goal of this colloquium is to encourage interaction among graduate students, specifically between graduate students who are actively researching a problem and those who have not yet started their research. Speakers will discuss their research or a related introductory topic on a level which should be accessible to nonspecialists. The discussions will be geared toward graduate students in the beginning of their program, but all are invited to attend. This invitation explicitly includes undergraduate students.
Talks will be held on Tuesdays at 4:30pm in JWB 335, unless otherwise
Speaker: Graeme Milton
Title: A variety of problems in composite materials
Abstract: In this lecture I will discuss a variety of problems which currently interest me. One is creep in composites; a second is extrapolating measurements of the real and imaginary parts of the dielectric constant beyond the measured frequency interval; a third is exploring the possible stress-strain pairs in linear composites (this work was initiated with Sergey Serkov and Sasha Movchan); and a fourth is exploring the properties of a whole new class of composite called partial differential microstructures. These problems involve a variety of mathematical tools and many open questions remain.
Title: A (brief) survey of constant mean curvature surfaces in Euclidean space
Abstract: Constant mean curvature surfaces model a film dividing two regions of space with a pressure difference accross the interface. (No pressure difference corresponds to zero mean curvature.) People have studied these surfaces for over 200 years (going back to Gauss and Lagrange), yet they remain somewhat mysterious. I will begin this talk by explaining several notions of mean curvature and some of the basic tools one uses to study constant mean curvature surfaces (particularly Alexandrov reflection). During the remainder of the talk I will survey some of the existence and classification results of the last 10-15 years. Hopefully, this lecture will be accessible to anyone familiar with vector calculus and some basic geometry.
Speaker: Mladen Bestvina
Title: How to average trees?
Abstract: I will describe the recent work of Billera (combinatorialist), Holmes (biologist) and Vogtmann (topologist) on the following problem from biology. Genetic relationships between species can be described in terms of phylogenetic* trees -- these are (usually) binary trees with leaves representing species. Various ambiguities lead to several candidate trees and the problem is to "average" them to a single tree.
The set of possible trees has the structure of a metric space which happens to be nonpositively curved. In spaces of nonpositive curvature there is a good notion of a "centroid" of a set of points, and in the space of trees it can be found by a fast algorithm that provides a solution (according to the biologists) to the above problem.
This space of trees is also closely related to certain polyhedra called associahedra studied 40 years ago by Stasheff**. The vertices correspond to different ways of parenthesizing the expression 1+2+...+n and their cardinality is a Catalan number.
*All Greek and Latin words will be defined upon request.
Title: A Window into Mirror Symmetry
Abstract: Enumerative geometry is a classical subject in algebraic geometry
that, roughly speaking, deals with counting the number of geometric
objects satisfying some conditions. For example, we may want to find
the number of lines (or quadrics or cubics or ...) through a given
number of points.
As can be expected, some problems have been solved in ancient times,
In this talk I will try to sketch some of the ideas that underlie
Title: An incursion into Commutative Algebra on the footsteps of a simple example
Abstract: Consider f,g,h three polynomials in two variables, with complex coefficients. It is true that there exist a,b,c polynomials in two variables with complex coefficients such that (fgh)^2 =a*f^3+b*g^3+c*h^3. The only proof I know uses some nontrivial commutative algebra techniques. I will use this elegant example to talk about some standard objects in commutative algebra such as local rings, localization, systems of parameters, dimension, integral closure etc.. The talk will illustrate how commutative algebra can be some times used in establishing elementary statements as the one above.
Title: What is Tropical Algebraic Geometry?
Abstract: Consider the set of real numbers with two operations, taking the maximum for addition and taking the sum for multiplication. This is an example of a Tropical Semiring. (The name "Tropical" was given by French computer scientists in honor of the Brazilian mathematician Imre Simon who pioneered the subject.) Note that 1+1=1 tropically.
The talk is devoted to the geometric objects described by tropical
equations. It turns out that the tropical objects are handier than their
classical algebro-geometric counterparts. For instance, tropical algebraic
curves are graphs. A curious fact is that, even though this geometry is
based only on a semiring, it resembles in many ways complex algebraic geometry
(which is based on an algebraically closed field).
Title: The American Option Valuation Problem
Abstract: Suppose you own one share of Microsoft stock and speculate that the price will go down. One way of safeguarding your investment is to purchase an
(American) put option, which gives you the right to sell your stock for
some prescribed price at any time before a certain date. How much would
you be willing to pay for the option, and when is it optimal to exercise
First, we will derive the famous Black-Scholes formula for pricing
European put options. Second, we will show how the American option
problem can be formulated as a free boundary problem. Lastly, we will
reformulate the problem as a linear complementarity problem and solve it
numerically using the projected SOR algorithm.
Department of Mathematics
University of Utah
155 South 1400 East, JWB 233
Salt Lake City, Utah 84112-0090
Tel: 801 581 6851, Fax: 801 581 4148