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MSRI-UP 2008 Colloquia

MSRI-UP 2008: Experimental Mathematics

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2008 Colloquium

6/20/08 - Luis Melara, University of Colorado
A homotopy technique for TV-minimization methods for image denoising problems

Homotopy methods can often be used to make mathematical programming problems easier to solve. One source of notoriously difficult problems is total variation image denoising. In this talk, a direct relationship is established between the radius of the Kantorovich ball guaranteeing the convergence of Newton's method and the regularization parameter. The regularization parameter is used as a homotopy parameter. We will conclude by presenting numerical results and discussing a different optimization procedure.

6/27/08 - Rosa Orellana, Darmouth College
From braids to knots: An introduction to representation theory

In this talk I will give an introduction to representation theory. I will show how we can use representations of the braid group to answer questions about knots. Representation theory has many very important applications in physics, chemistry and engineering. The talk will be self-contained, no knowledge of knots, braids or representation theory will be assumed.

7/03/08 - Javier Arsuaga, San Francisco State University
Some applications of knot theory to molecular biology

Knot theory studies simple closed curves in space. Despite its traditional classification as a "pure math" subject, knot theory has been extensively applied in physics, chemistry and molecular biology. In this talk I will review some of the current applications of knot theory to molecular biology, in particular to the structure and maintenance of chromosomes. A chromosome consists of proteins and a single DNA molecule, which is usually very long and that is folded into a very small volume (e.g. the nucleus of the cell). This spatial confinement is related to the formation of knots (or links if several molecules are involved) which can be informative of the chromosome organization. Noteworthy is the example of DNA knots found in certain viruses that are unveiling new properties of the DNA organization inside the virus. During this presentation I will also introduce some of the experimental, mathematical and computational tools commonly used to study DNA knots.

7/11/08 - Matthias Beck, San Francisco State University
Discrete Volume Computations for Polytopes: An Invitation to Ehrhart Theory

Our goal is to compute the volume of certain easy (and fun!) geometric objects, called polytopes, which are fundamental in many areas of mathematics. Although polytopes have an easy description, e.g., using a linear system of equalities and inequalities, volume computation is hard even for these basic objects. Our approach is to compute the \emph{discrete volume} of a polytope P, namely, the number of grid points that lie inside P, given a fixed grid in Euclidean space such as the set of all integer points. A theory initiated by Ehrhart implies that the discrete volume of a polytope has some remarkable properties. We will exemplify Ehrhart theory with the help of several families of polytopes whose discrete volumes are connected with some of our friends in various mathematical areas, such as binomial coefficients, Eulerian, Stirling, and Bernoulli numbers.

7/11/08 - Robin Wilson, California State Polytechnic University
Knots, Surfaces, and 3-Manifolds

The areas of knot theory and 3-manifold topology are closely related. In fact, the exterior of a knot is itself an example of a 3-manifold. One approach to studying 3-manifolds is to understand the surfaces that are contained in them. In this talk I will give an introduction to knot theory, its connections with 3-manifold topology, and the study of surfaces in 3-manifolds and knot exteriors. I will also discuss some recent research about bridge surfaces in knot complements.

7/18/08 - Pamela Williams, Sandia National Laboratories
Energy-related Applications at Sandia

Recently, the surging cost of gasoline and its effect on Americans' driving habits have dominated the headlines. Related applications of interest at Sandia National Laboratories range from transportation to alternative energy sources such as hydrogen and biofuels. During this talk, we will present problems and proposed solutions related to mass transit and hydrogen energy. The common theme underlying these applications is the need for data analysis. Although transit districts such as San Francisco's Bay Area Rapid Transit (BART) have controlled their trains automatically for decades, the control systems have provided limited capability in terms of train position location and speed control. Increases in capacity now require trains to run closer together than these systems can accommodate. Therefore new systems, such as the Advanced Automatic Train Control (AATC) system, are under development. 

Our goal is to use optimization within the AATC system to smooth out train operations and to reduce energy consumption and power infrastructure requirements. We are initially focusing on a schedule-constrained problem with the primary objective of improving passenger comfort. The Hydrogen Macrosystem Model (MSM) is a decision support tool for the Department of Energy. The MSM contains several models including hydrogen production and distribution as well as greenhouse gas emissions. We use statistical methods to determine the effect of selected input parameters on the system's output. We will present results of our analysis.