Department of Mathematics & Statistics

Abstract:
We use the method of Maxwell to set up balance equations for the transport of mass, momentum and energy in a simple gas with hard core. The particles follow free motion between collisions, which are random and are determined by the local free path.

We assume that after a collision, the particle joins the cohort of thermalised particles. By expanding the transport up to first power of the mesn free time, we obtain the compressible Navier-Stokes equations with temperature,except that one extra term appears in the heat transport equation, a Dufour effect. We do not assume the Boltzmann equation.

Friday November 29, 2002

Robert Myers
Perimeter Institute

Title: "Two Faces of Anti-deSitter Space: The Maldacena Conjecture and Positive Energy Theorems"

Abstract:
Anti-de Sitter (AdS) space is the simplest solution of Einstein's equations with a negative cosmological constant or alternatively the maximally symmetric geometry with constant negative curvature. This spacetime has long been of interest in both physics and mathematics. In particular, gravitational theories in AdS space have recently been under intense study because of the Maldacena conjecture which relates such theories to conformal field theories in one dimension less. I provide a description of this conjectured equivalence. Further, I will discuss implications for positive energy theorems. The latter play a central role in gravitational theories as they determine the classical stability of physical spacetimes, however, we will see that asymptotically AdS spaces seem to present interesting new challenges.

Friday November 15, 2002

Duncan Murdoch
University of Western Ontario

Title: "Perfect Sampling Algorithms: Connections"

Abstract:
The arrival of Propp and Wilson's (1996, Random Structures and Algorithms) coupling from the past (CFTP) algorithm caused a big stir in the Markov chain Monte Carlo community, because it did the seemingly impossible task of using a finite run of a Markov chain to obtain samples from the steady-state distribution of the chain. At around the same time Fill (1998, Annals of Applied Probability) developed quite a different algorithm that accomplished the same thing using rejection sampling.
In this talk (which is loosely based on Fill, Machida, Murdoch and Rosenthal, 2000, Random Structures and Algorithms) I will summarize both algorithms, and show how Wilson's (1999, Random Structures and Algorithms) ``read-once'' variation on CFTP is in some sense also a variation on Fill's algorithm.

Friday November 8, 2002

Bruce Williams
Notre Dame

Title: "Localization of Geometric Invariants"

Abstract:
A central theme in geometry is that global invariants can often be computed in terms of local data. Two classical examples are the Euler characteristic of a closed manifold and the Lefschetz invariants of an endomorphims of such a manifold. Later examples are the Atiyah-Singer Index theorem and Lefschetz-type theorems of Atiyah-Bott-Segal. More recent examples are given by algebraic K-theory index theorems, and topological Hochschild homology Lefschetz type theorems. These last result are inspired by the pioneering work of Geoghegan and Nicas.

Abstract:
We discuss old and new results and open problems on various types of colourings of finite projective planes and Steiner systems, ranging from "classical" colouring (no monochromatic lines) to its more modern variations and generalizations. This talk should be easily understandable to a non-specialist. Needless to say, this also will be a 'very colourful' talk.

Friday October 25, 2002

Alexander Kechris
Professor of Mathematics
California Institute of Technology
EVELYN NELSON LECTURES

Title: "Applications of ergodic theory to set theory"

Abstract:
In recent years there has been increasing interaction between ergodic theory and descriptive set theory, the area of set theory
concerned with the study of definable sets and functions (like Borel, analytic, etc.) on complete separable metric spaces. In particular, ideas and methods of ergodic theory have been used to solve a number of basic set theoretic problems in this area. In this talk, I will present a number of examples of such applications.

Friday October 18, 2002

Andrew Comech
University of North Carolina

Title: "Purely Nonlinear Instability of Minimal Energy Standing Waves"

Abstract:
For a variety of nonlinearities, the nonlinear Schroedinger equation is known to possess localized quasistationary solutions ("standing waves"). We prove that in the generic situation the standing wave of minimal energy among all other standing waves is unstable. This case was falling out of the scope of the classical paper by Grillakis, Shatah, and Strauss on orbital stability of standing waves. An interesting feature of the problem is the absence of (exponential) instability in the linearized system; in this sense, the resulting instability is ``purely nonlinear''. Essentially, the instability is caused by higher algebraic degeneracy of zero eigenvalue in the spectrum of the linearized system.The result can be generalized to abstract Hamiltonian systems with U(1) symmetry.

Friday October 11, 2002.
Please note this colloquium has been rescheduled to Term 2 -- dates TBA

P.M. Fitzpatrick
University of Maryland

Title: "Spectral Flow Detects Bifurcation of Critical Points"

Abstract:
Spectral flow is a well-known integer invariant associated to a path of self-adjoint Fredholm operators that has invertible end-points. For a one parameter family of Fredholm functionals on a Hilbert space, the spectral flow of the path of Hessians along a trivial branch of critical points determines bifurcation of ctitical points from the trivial branch. A novel construction of spectral flow will be described that leads to the proof of this bifurcation theorem for critical points. Some applications of the theorem to the bifurcation of solutions of paths of Hamiltonian systems will be mentioned. The results described are joint work with Jacobo Pejsachowicz and Lazero Recht.

Friday October 4, 2002

A Weiss
University of Alberta

Time:   2:30-3:30 pm

Title:   "Galois structure and Iwasawa theory"

Abstract:
Some of the Galois structure of algebraic number fields can be described in terms of special values of zeta functions,at least
conjecturally.Such conjectures can sometimes be proved by using a new 'equivariant'refinement of the Main Conjecture of Iwasawa theory.This refinement seems to work because it replaces the standard Iwasawa modules by 'approximations'of finite projective dimension.The focus of the talk will be the comparison of 'equivariant' and 'classical'Iwasawa theory.

The following talk is part of the Ontario Topology Conference
Jacques Hurtubise
McGill University and CRM

Time: 4:00-5:00 pm

Title: "Algebraically integrable systems"

Abstract:
The local geometry of a real integrable Hamiltonian system is quite simple: a fibration by tori. Things become much more interesting when one goes to complex variables, and asks that the tori be algebraic. In particular, when the algebraic tori are associated to Riemann surfaces, there is a beautiful link to the geometry of surfaces.

Friday September 27, 2002

Huai-Dong Cao
IPAM at UCLA

Title: "Translating Kaehler-Ricci solutions and dimension reduction"

Abstract:
Translating Kaehler-Ricci solitons arise as a blow-up limit of singularities of Hamilton's Ricci flow on Kaehler manifolds and can be regarded as a natural extension of Calabi-Yau metrics on noncompact complex manifolds. In this talk we will discuss geometric and complex analytic properties of translating Kaehler-Ricci solitons of nonnegative curvature, and the method of dimension reduction for the Kaehler-Ricci flow.

David Brydges, UBC

Date:     Friday April 5, 2002
Room:   GS/209
Time:    3:30 pm

Title: "Branched Polymers in D+2 dimensions equals the hard core gas in D dimensions"

Abstract:
In 1981 Parisi and Sourlas conjectured that the typical size and other features of large self-avoiding branched polymers in $D+2$ dimensions is determined by another statistical mechanical problem in D dimensions and on the basis of this conjecture they were able to calculate critical exponents for branched polymers in two and three dimensions. I will explain what these terms mean in terms accessible to a general audience and give some details on recent progress on this problem. The colloquium may be of interest to people who like combinatorics and probability. There is also a small connection with topology, in particular special integrals involving differential forms.

The colloquium is based on a paper by John Imbrie and myself posted at http://xxx.lanl.gov/abs/math-ph/0107005

Date:    Thursday March 28, 2002     (Please note new day)
Time:    3:30 pm
Room:   GS/209

Title: "Blow-ups of the Toda lattices and their intersections with the Bruhat cells"

Abstract:
I will discuss the topology of the set of singular points (blow-ups) in the solution of the nonperiodic Toda lattice defined on real split semisimple Lie algebra $\mathfrak g$. The set of blow-ups is called the Painlev\'e divisor. The isospectral manifold of the Toda lattice is compactified through the companion embedding which maps the manifold to the flag manifold associated with the underlying Lie algebra $\mathfrak g$. The Painlev\'e divisor is then given by the intersections of the compactified manifold with the Bruhat cells in the flag manifold. I will give explicit description of the topology of the Painlev\'e divisor for the cases of all the rank two Lie algebra, $A_2,B_2, C_2, G_2$, and $A_3$ type. The results are obtained by using the Mumford system and the limit matrices introduced originally for the periodic Toda lattice.

This is a joint work with L. Casian. The talk contains some physics, geometry, algebra and pictures.

SPECIAL COLLOQUIUM
Professor Paul Rabinowitz
University of Wisconsin-Madison
Title:         "Spatial heteroclinics for a class of semilinear elliptic PDE's"
Date:        Monday March 25, 2000   (Please note new day)
Time:        3:00 pm
Room:      GS/209

Adrian Nachman
Mathematics Department
University of Toronto
Title:    "The inverse boundary value problem of Calderon and a nonlinear Fourier transform"
Date:    Friday March 15, 2002
Time:    3:30 pm
Room:   GS/209

Richard Wentworth
Johns Hopkins University
Title:    "Some recent results concerning mapping class groups"
Date:    Friday, March 8, 2002
Time:    3:30 pm
Room:   GS/209

****Please note, that for today only, refreshments will be served at 3:30 pm instead of 3:00 pm****

Professor Jianhong Wu
Senior Canada Research Chair
Department of Mathematics and Statistics
York University

Date:     Friday March 1, 2002
Time:     4:00 pm (new time)
Room:   GS/209

Title: "Neural Networks with Delay: Pattern Recognition, Associative Memory and Attractors"

Abstract: We start with a short discussion of the connections between pattern recognition, associative memory, and the global dynamics of a network of neurons. We then present some recent work on the structure of the global attractor for a simple excitatory network of neurons with delayed recurrent loops. We address the issue of the coexistence of multiple limit cycle attractors or periodic solutions which are unstable but with large domains of attraction, and we discuss some potential applications to dynamic memory storage and encoding.

Professor N. Bergeron
York University, Toronto

Title:      "Hopf algebra of quasi-symmetric function"
Date:     Friday February 15, 2002
Time:     3:30 pm
Room: GS/209

Abstract:
The algebra of symmetric functions (or symmetric polynomials in several variables) is a familiar object and indeed we have, at one point or another in our life, encountered such functions. In this talk I will introduce the algebra of quasi-symmetric functions. It is a larger algebra that still satisfies some very enjoyable properties and there is a growing interest in their study.

Title:     "Colouring finite projective planes and Steiner systems"
Date:     Friday February 1, 2002
Time:    3:30 pm
Room: GS/209

Mary Pugh
University of Toronto

Date:      Friday January 25, 2002
Time:     3:30 pm
Room:    GS/209

Title: "Long-wave unstable thin film equations
--- the modelling and analysis of painting your ceiling and spilling your coffee"

Abstract:
We consider long-wave unstable interface models of the type $$ h_t = - (h^n h_{xxx})_x - B (h^m h_x)_x $$ where
$B > 0$, and $n$ and $m$ are constants. I will discuss how these equations arise from fluid dynamics as well as the mathematical difficulties involved in analysing the equations. I will discuss the possiblitlity of finite-time singularities (joint work with Andrea Bertozzi of Duke University) as well as possible long-time behaviors of solutions (joint work with Richard Laugesen of the University of Illinois of Champaign-Urbana).

Title:   "Stability and instability of spiral waves"
Date:   Friday, November 30
Time:   3:30 pm
Room: GS/209

Date:     Friday, November 23, 2001
Time:     3:30 pm
Room:   GS/209

Title: "Interior Point Algorithms for Linear Optimization"
(based on joint work with J. Peng and C. Roos)

Abstract: Linear Optimization (LO) is probably the most successful and most intensively studied model in applied Mathematics. Advances in algorithms for LO have a great impact on the theory and applications of all fields of optimization. The last 15 years is dominated by the developments about Interior Point Methods (IPMs). IPMs are not only efficient, i.e., polynomial, in theory but also highly efficient in practice, especially when solving very large scale sparse problems.

Professor P. Monk
University of Delaware

Title:     "Forward and Inverse Electromagnetic Scattering"
Date:     Friday, November 16, 2001
Time:    3:30 pm
Room:   GS/209

Abstract:
When electromagnetic waves impinge on an object, they interact with the object and give rise to a scattered field. In a forward scattering problem, we are interested to compute the effects of the object or scatterer on a known incident field. I shall first discuss numerical methods for approximating this problem in the frequency domain and mention some of the issues that influence the choice of algorithm. In an inverse scattering problem, electromagnetic waves are used to determine properties of an inaccessible scatterer (for example mine-detection). This problem, in which the incident and resulting scattered field are
known but the scatterer is unknown, has quite different mathematical properties to the forward problem. Indeed such inverse problems are usually ill-posed and non-linear. However, I shall present a simple algorithm for determining the shape of an unknown scatterer and show some computational results.

Professor G. Slade
University of British Columbia

Title:      "Scaling limits and super-Brownian motion"
Date:     Friday, November 9
Time:     3:30 pm
Room:   GS/209

Abstract:
This lecture will discuss self-avoiding walks, lattice trees, and percolation. These elementary models are interesting both mathematically and for their applications in physics and chemistry. Physicists and chemists have had much to say about
them, but, at the level of mathematical theorems, much of the most interesting behaviour is not understood. In high spatial dimensions, a technique known as the lace expansion has resolved many of the mathematical issues. Scaling limits of high-dimensional lattice trees and percolation turn out to involve super-Brownian motion, which is the scaling limit of branching random walk (as ordinary Brownian motion is the scaling limit of ordinary random walk).

Speaker:   Dr. J. Scherk, University of Toronto
Title:         "Compactifying locally symmetric spaces"
Date:         Friday, October 26, 2001
Time:        3:30 pm
Room:      GS/209

Speaker:    Dr. E. Sawyer, McMaster University
Title:          "Regularity of degenerate Monge-Ampere equations"
Date:          Friday, October 12, 2001
Time:         3:30 pm
Room:        BSB/137

Speaker:   Professor Ram Murthy
                 Queen's University
                 Kingston, Ontario
Title:         "Euclidean Rings"
Date:         Friday, September 14, 2001
Time:        3:30 pm
Room:       BSB/137

Speaker:           Dr. Mary Lou Zeeman
                         University of Texas at San Antonio &
                          University of Michigan
Title:                 "TBA"
Date:                 Friday, April 6, 2001
Time:                 3:30 to 4:30 p.m.
Room:                BSB/B103

Speaker:    Dr. F. Larusson
Title:         "Abstract homotopy theory and Gromov's Oka principle "
Date:          Friday February 16, 2001
Time:         3:30 pm

Abstract:
Abstract homotopy theory takes place in a category satisfying a list of axioms due to Quillen, giving a reasonable notion of two arrows being homotopic. This encompasses ordinary homotopy theory of topological spaces and simplicial sets, homological algebra, and much more. Recent applications in arithmetic geometry have attracted much attention. The Oka Principle is an important theme in complex analysis with a long history. Roughly speaking, it states that on a complex submanifold of Euclidean space, analytic problems of a cohomological nature have only topological obstructions. A famous theorem of Gromov is an instance of this, giving sufficient conditions for any continuous map between two complex manifolds to be homotopic to a holomorphic map. The talk will give a non-technical overview of these two topics and describe how Gromov's Oka Principle can be placed in an abstract homotopy-theoretic context and interpreted in purely holomorphic terms, without reference to continuous maps. To this end, we embed the category of complex manifolds into a Quillen category, where we can then do homotopy theory with them. The conclusion of Gromov's theorem turns out to be equivalent to a property called excision, which is familiar from topology and appears nowadays in algebraic geometry.

********************

Title:          "Interacting particles, large deviations, and degenerate diffusions"
Date:          Friday January 12, 2001
Time:         3:30-4:30 pm

Abstract:
We describe how the solution of a problem in degenerate diffusions arose in the study of large deviations for interacting random walks. (Background in probability will not be assumed.)

AMALGAMATED APPLIED MATHEMATICS AND COLLOQUIUM

Title: "Reduced equations for hyperbolic problems on thin domains"
Date: Friday January 5, 2001

I will describe how, using ideas from Hamiltonian mechanics, one can derive equations on two-dimensional spatial domains whose solutions provide accurate approximations to the solutions of equations on thin, three-dimensional domains. Such problems arise frequently in the study of the dynamics of thin elastic structures like plates or shells.

SPECIAL COLLOQUIUM
Speaker:    Alexander I. Suciu
                  Northeastern University, Boston

Title:          "Topology of Complex Line Arrangements"
Date:          Friday December 15, 2000
Time:         3:30 pm
Room:        BSB/108

Speaker:    Professor Arturo Pianzola
                  Fields Institute, Toronto

Date:          Friday December 1, 2000
Time:         3:30-4:30 pm
Room:       BSB/108

Speaker:   Professor Paul Kirk, Indiana University

Date:         Friday, November 17, 2000
Time:        3:30-4:30 pm
Room:       BSB/108

Abstract:
I will explain what the equivalence relation of knot concordance is, and sketch the history of developments in this topic, with an emphasis on examples and pictures. The talk will be accessible to graduate students.

Speaker: Professor Yuly Billig
               School of Mathematics
               Carleton University & the Fields Institute

Date:       Friday November 3, 2000
Time:      3:30-4:30 pm
Room:    BSB/108

Title: "Representations of Kac-Moody algebras, the double KdV
hierarchy and the sine-Gordon equation"

Abstract:
The existence of soliton solutions for several important PDEs can be explained by the fact that these equations possess infinite_dimensional groups of (hidden) symmetries. Two of such equations are the Korteweg_de Vries equation and the sine_Gordon equation. In 1981 Date, Jimbo, Kashiwara and Miwa discovered that starting with the basic highest weight module for the affine Kac_Moody algebra A_1^(1), one can obtain a hierarchy of non_linear PDEs in which the first term is the KdV equation.

In this talk I will show that the sine_Gordon equation arises in the context of a non_highest weight representation of A_1^(1). The corresponding hierarchy contains two copies of the KdV hierarchy, as well as the sine_Gordon equation. Applying the group action to the constant function we obtain soliton solutions for both equations simultaneously.

Refreshments will be served at 3:00 pm in the Math Lounge, BSB/135

Speaker:    Professor B. Wilking
                  University of Pennsylvania
Time:         3:30-4:30 pm
Room:       Burke Science Building 108
Title:         "New Examples of Manifolds with Positive Sectional Curvature Almost Everywhere"

Abstract:
There are only very few examples of Riemannian manifolds with positive sectional curvature known. In fact in dimensions larger 24 the known examples are diffeomorphic to locally rank 1 symmetric spaces, i.e., quotients of the spheres, complex and quaternionic projective spaces, and the Cayley plane.

We will construct metrics with positive sectional curvature on a open and dense set of points on the projective tangent bundles of RP^n, CP^n and HP^n.

The so called deformation conjecture says that these kind of metrics can be deformed into metrics with positive sectional curvature everywhere. However, the simplest new example within our class, the projective tangent bundle of RP^3, is diffeomorphic to the product RP^3 x RP^2. This non-oriented manifold is known not to admit a metric with positive sectional curvature. Thus the construction provides a counterexample to the deformation conjecture.

COLLOQUIUM SPEAKERS FOR 2002

Friday October 18, 2002

Andrew Comech
University of North Carolina

Title: "Purely Nonlinear Instability of Minimal Energy Standing Waves"

Friday September 27, 2002

Huai-Dong Cao
IPAM at UCLA

COLLOQUIUM SPEAKERS FOR 2002

Friday October 18, 2002 Andrew Comech University of North Carolina Title: "Purely Nonlinear Instability of Minimal Energy Standing Waves"

Friday September 27, 2002 Huai-Dong Cao IPAM at UCLA

Friday October 18, 2002

Andrew Comech
University of North Carolina

Title: "Purely Nonlinear Instability of Minimal Energy Standing Waves"

Friday September 27, 2002

Huai-Dong Cao
IPAM at UCLA