Mathematics and Statistics Departmental Colloquium Room 1634, Lederle Graduate Research Tower University of Massachusetts, Amherst
Refreshments at 3:45pm.     Talks begin at 4:00pm. Driving Directions   and   Campus Map

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Autumn 2002 Schedule:

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		12 September	Jonathon Weitsman, University of California at Santa Cruz
		4:00-5:00	The topology of Hamiltonian Loop Group spaces.
		19 September	Markus Schmies, Technische Universität Berlin
		4:00-5:00	Java Oorange: a laboratory for experimental programming in science
		26 September	Paola Sebastini, U Mass
		4:00-5:00	Statistical challenges in the post-genome era
		3 October	Allen Knutson, University of California at Berkeley and Gregory Warrington, University of Massachusetts
		4:00-5:00	Mathematics of Juggling
		10 October	Amherst College CVC Refreshments at 4:00 PM in 206 Seeley-Mudd
		4:30-5:30	Frank Morgan, Williams College (208 Seeley-Mudd)
			Double Bubbles Past, Present, and Future.
		17 October	Randy McCarthy, University of Illinois at Urbana-Champaign
		4:00-5:00	Classical algebra via Calculus
		24 October	William Graham, University of Georgia
		2:30-3:30	Degeneracy loci: linear algebra, geometry, and combinatorics
			Time Change Due to Talk of John Nash
		31 October	No colloquium
		6 November	Alexandar Bobenko,
		4:30-5:30	Cancelled: Delay in Visa
		7 November	Wendell Fleming, Brown University
		4:00-5:00	Max-Plus Stochastic Control
		14 November	Augustin Banyaga, Penn State University
		4:00-5:00	Locally conformal symplectic structures
		21 November	Joel Lebowitz, Rutgers University     Earlier than usual!
		3:30-4:30	The Meaning and Uses of Entropy
		28 November	No Coloquium: Thanksgiving
		5 December	Nantel Bergeron, York University, Toronto, Canada
		4:00-5:00	Quasi-symmetric polynomials and Temperley-Lieb invariants and covariants
		12 December	Constantine Dafermos, Brown University
		4:00-5:00	A new approach to the Riemann problem for hyperbolic conservation laws

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Abstracts

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Abstract
Let M be a Hamiltonian G-space; that is, a symplectic manifold equipped with a Hamiltonian action of a compact Lie group G. A theorem of Kirwan, inspired by the ideas of Atiyah and Bott, shows that the square of the moment map is an equivariantly perfect Morse function on M. As a consequence, the equivariant cohomology of M provides a set of generators for the cohomology of the reduced space M//G (Kirwan's surjectivity theorem).

We study an analog of this theorem in the case of infinite dimensional symplectic manifolds, equipped with a Hamiltonian action of a loop group LG, where G is a compact Lie group. We show that, in an appropriate sense, the square of the moment map is an equivariantly perfect Morse function on such a space. Similarly, there is an analog of Kirwan's surjectivity theorem.

Examples of such spaces are coadjoint orbits of the loop group (where the Morse function is the classical energy functional of Morse and Bott) and spaces arising from Yang-Mills theory in two dimensions.

(joint work with R. Bott and S. Tolman)
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Abstract
Experiments play a key role in science. This is reflected in the way that most scientific software is developed. On the one hand there is the scientific question that has to be solved, on the other the need to verify the suitability and correctness of the methods used. Together these aspects combine to a workflow we call experimental programming. The experience of more than ten years research at the TU Berlin has yielded a tool offering an environment which hopefully matches the requirements resulting from the workflow described above: Oorange.

This talk will briefly discuss the concepts of Oorange and the technical background, including reference to Java. The workflow of experimental programming with Oorange will be demonstrated by examples. These include one illustration developed completely from scratch, and real-life examples from current research projects.
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Abstract
On February 12, 2001 leaders of the Human Genome Project announced the completion of a draft of the human genome. The result of these efforts is a map of the human genes, and because about 50% of discovered genes have known functions, the challenge now is to annotate this map, by discovering the functions of genes, and their interplay with proteins and the environment to create complex, dynamic living systems. This is the goal of functional genomics.

The modern approach to functional genomics takes advantage of the new technology of microarrays to observe the genome of an entire organism in action by simultaneously measuring the level of expression of thousands of genes under the same experimental condition. Microarray technology is used in simple comparative experiments, when the goal is to identify the genes that are differentially expressed in cells of the same tissue in two different conditions. More ambitious experiments try to discover gene functions or gene interactions from temporal or multifactor experiments. The design and analysis of these experiments require the development of new modeling and computational techniques, and this talk will describe some of the recent methodology we developed for the analysis of microarray data.
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Abstract
Around 1985, several jugglers independently invented the same mathematical theory of juggling patterns. In addition to aiding in the categorization, recording and enumeration of patterns, this description of patterns has facilitated the discovery of worthwhile new ones. The theory will be presented with ample demonstrations. As time permits, we will introduce the combinatorial and probabilistic questions suggested by this viewpoint. No juggling knowledge will be assumed.
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Abstract
Goodwillie's calculus of functors has a similar goal as the ordinary calculus of functions: use linear (additive) methods to study non-linear phenonema. Its context though is in the realm of modern algebra, and a basic object is the Taylor tower of a functor. New criteria are developed jointly with Kristen Baxter for Taylor towers to split into their layers.

Our primary objective is to introduce the audience to this calculus of functors via a simple construction due to work with Brenda Johnson and to show via examples what kind of information the associated Taylor tower produces. We will also discuss how this splitting unifies several classical results of homological algebra.
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Abstract
This talk concerns a broad class of algebraic varieties defined by conditions involving linear algebra -- more precisely, the linear algebra of vector bundles. For example, let X denote the set of all n by n complex matrices, and let X_r denote the set of matrices whose rank is at most r. Then the set of matrices with rank n is a dense open set in X -- a "generic" matrix has rank n. Each of the X_r is a set of degenerate matrices, the most degenerate being X₀, which is simply the zero matrix. Algebraic varieties defined by conditions such as these are called degeneracy loci. They lie at the intersection of linear algebra, geometry, and combinatorics.

In this talk I will discuss three types of questions about degeneracy loci. (1) When are sets defined by degeneracy conditions nonempty? (2) If a degeneracy locus is a finite set of points, how can one calculate the number of points? More generally, how can one make cohomology calculations? (3) Some topological questions about degeneracy loci have been reduced to combinatorics. For example, in some cases one can use combinatorics to tell if a degeneracy locus is smooth. Are there ways to make the combinatorial calculations more efficient?
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We discuss symplectic, contact and locally conformal symplectic structures. We show how they are connected and how they organize themselves inside the category of Jacobi structures.

An emphasis is put on the role of their automorphism groups since they encode the corresponding geometries in the spirit of the Erlanger Programme.
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Abstract
According to the second law of thermodynamics (Clausius 1857) the entropy S(M) of an isolated macroscopic system passing from an equilibrium state specified by parameters M₁ (energy, volume, etc.) to another equilibrium state with parameters M₂ must satisfy the inequality S(M₂) \geq S(M₁): the entropy S(M) of an equilibrium system is defined operationally. A microscopic expression for S(M), which also provides an understanding of the origin of the second law, was given by Boltzmann: S(M) = {log of the phase space volume corresponding to M}. These equilibrium expression for S(M) naturally generalize to systems in local equilibrium with parameters M(x,t) varying slowly (on a microscopic scale) in space and time: this includes fluids in turbulent motion. Note that S(M) is defined for individual macroscopic systems and does not require probability distributions or ensembles: the generalization of the microscopic definition to quantum systems requires some care.

Boltzmann also defined an entropy function S_gas({f}) for a dilute gas far from local equilibrium, where f(x, v,t) is the density of gas particles in the six-dimensional m-space, and showed that this increases monotonically with time. S_gas({f}) = {log of phase space volume corresponding to f}: S_gas({f_eq}) = S(M). A similar useful definition of entropy for general systems far from local equilibrium, i.e. for systems in which correlations are important, is still very much an open problem. I will describe recent work on large deviations in stationary nonequilibrium states which has some bearing on this problem.
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Abstract
Quasi-symmetric polynomials were introduced in enumerative combinatorics by Gessel and Stanley for the enumeration of P-partitions. Recent developments show that these polynomials can be view as Temperley-Lieb polynomials invariants. We will recall the basic facts on quasi-symmetric polynomials and survey some of the striking recent developments concerning them. We will also look at the Temperley-Lieb analogue of the diagonally symmetric polynomials and the diagonally symmetric harmonics.
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Abstract
This lecture will survey the classical approach for solving the Riemann problem for hyperbolic systems of conservation laws, and will discuss a new method of solution, based on a variational principle. TOP