Colloquium: 2009-2010

Fall 2009

View Spring 2010 »

(TENTATIVE SCHEDULE)

Time & location: All talks are on Thursday in Gibson 414 at 3:30pm unless otherwise noted. Refreshments in Gibson 426 after the talk.

Comments indicating vacations, special lectures, or change in location or time are made in green.

Organizer: Tai Huy Ha

September 10

Mathematical Models Provide New Insights into Stopping Epidemics

Mac HymanTulane university

Abstract

The pace that new diseases threaten the world is increasing. The current H1N1 flu pandemic follows on the heels of recent epidemics for Herpes-2, Hepatitis C, HIV/AIDS, SARS, and avian flu. Public health workers are reaching out to use all available tools to anticipate the spread of new diseases and evaluate the effectiveness of different approaches for bringing epidemics under control. I will describe how mathematical models, based on the underlying transmission mechanisms, have advanced to help guide these efforts. Today, mathematical scientists are joining with biological, epidemiological, behavioral, and social scientists to fight these emerging epidemics.

The talk will provide an overview, for general audiences, of what type of insights these models can provide. I will also touch briefly on the underlying mathematical advances and theory needed for the next generation of models, and share my personal experiences in controlling the spread of HIV/AIDS, SARS, malaria, foot and mouth disease, and the novel H1N1 (swine) flu.

October 1

Spectral Representation of Solutions of Linear Elliptic Boundary Value Problems

Giles AuchmutyUniversity of Houston

Abstract:

This talk will describe some results about the representation, and approximation, of solutions of Dirichlet problems for second order linear elliptic equations using Steklov eigenfunction expansions.

It is well-known that standard eigenfunction expansions provide representations of solutions for Robin and Neumann boundary value problems with non-zero boundary data. This is not the case for Dirichlet problems with non-zero boundary data. Here we shall describe spectral representations of the solutions of such problems. We first describe Steklov eigenproblems for an elliptic operator and show that the Steklov eigenfunctions provide bases for the space of $H^1-$solutions of the homogeneous elliptic equation. This leads to general representation theorems for the solutions of the Dirichlet problem. It also provides an intrinsic, and constructive, description of the Sobolev trace spaces $H^{1/2}(\partial \Omega)$ and $H^{s}(\partial \Omega)$. Important Hilbert spaces of real harmonic functions on general regions in $R^n$ will be characterized as Reproducing Kernel Hilbert spaces using a kernel defined in terms of the Steklov eigenvalues and eigenfunctions.

October 8

Directed Cohomology

Sanjevi KrishnanIHES

Abstract:

I will present a cohomology theory for "directed spaces," spaces equipped with temporal structure. Examples include spacetime manifolds and classifying spaces. On such spaces, ordinary cohomology groups reveal properties invariant under continuous deformations, while directed cohomology monoids detect finer properties invariant under deformations respecting the temporal structure and thus tease out the "qualitative" structure of time. Directed cohomology extends several well-known properties of its classical analogue: our new invariants admit chain-theoretic constructions, equivalent homotopical descriptions, axiomatic characterizations, and multiplicative structure. After presenting the basic theory of directed homotopy and cohomology, I will sketch real and potential applications to concurrent engineering, string rewriting, and informatics. This talk, aimed at a general audience, assumes no prior experience with directed spaces or cohomology theories.

October 15

Fall Break

October 22

TBA

SpeakerUniversity

Abstract: TBA

October 29

The Diurnal Cycle and the Merodional Extent of the Tropics

Esteban TabakCourant Institute

Abstract

This talk proposes a mathematical theory explaining the sharp transition between tropics and extra-tropics in terms of the diurnal cycle of thermal forcing by the sun. This transition, at a latitude of 30 degrees, coincides with the outer edge of the Hadley cells, and is marked by a steep jump in the height of the troposphere, from fifteen kilometers in the tropics to nine in the mid and high latitudes. The tropics, equatorwards of 30 degrees, are characterized by easterly surface winds -the Trades- and a strong diurnal signal in the wind, pressure and temperature, often marked by regular daily storms in the rainy season.

Polewards of 30 degrees, the winds are westerly, and the weather systems have longer spacio-temporal scales.

This change of behavior can be explained in terms of diurnal waves, created by thermal forcing and trapped equatorwards of 30 degrees by the Coriolis effect. This can be illustrated in simple mathematical models, ranging from forced linear oscillators to nonlinear conservation laws with entraining shock waves, acounting for the entrainment into the troposphere of air from the surface boundary layer.

November 5

The Challenge of Simularing Fluid Flow Accurately in the Presence of Boundaries

Jian-Guo LiuDuke University

Abstract

The physical world has a rich diversity of fluid dynamics, ranging from the micron scale to the galactic scale, and varying from high Mach number compressible flows to low Mach number incompressible flows. Large variations in scales, flow properties, and surrounding environments pose many challenges for computations. These issues are particularly important in domains with boundaries. Much of the scientific and technological impact of the Navier-Stokes equations derives from the effect of no-slip boundary conditions in creating physical phenomena such as lift, drag, boundary-layer separation and vortex shedding, for which the behavior of the pressure near boundaries is of great significance. In this talk, I will present new equivalent formulations of Navier-Stokes equations (NSE) better suited for numerical computations. The emphasis will be on the enforcement of imcompressibility and the discovery of intrinsic stability properties that lead to accurate, efficient and practical computations of three dimensional problems. I will also present some efficient methods for more completed flows such as low-mach flow, MHD, kinetic equations with different scales.

November 12

TBA

SpeakerUniversity

Abstract: TBA

November 19

Self-Similar Gaussian Random Fields and Their Fractal Properties

Yimin XiaoMichigan State University

Abstract

Self-Similar Gaussian random fields are useful as stochastic models in many applied areas and their sample functions are often random fractals. In this talk we present some results on construction of Gaussian random fields and on their geometric and asymptotic properties.

November 26

December 3

Analyticity of solutions of first-order fully nonlinear pdes

Shiferaw BerhanuTemple University

Abstract: TBA

February 4

Computing Vortez Sheet Roll-Up

Robert KrasnyUniversity of Michigan

Abstract

Vortex sheets are used in fluid dynamics to represent thin shear layers in slightly viscous flow. Some of the earliest simulations in computational fluid dynamics used the point vortex approximation to compute vortex sheet roll-up, but later simulations encountered difficulty because the initial value problem is ill-posed and singularities form from general smooth initial data. I'll review the fundamental results on vortex sheet roll-up by Louis Rosenhead, Garrett Birkhoff, and Derek Moore, and then discuss recent developments concerning regularized point vortex simulations, spiral roll-up in the Kelvin-Helmholtz problem, and chaotic dynamics in vortex cores. Finally I'll describe a new panel/particle method for vortex sheet roll-up in 3D flow which uses a treecode algorithm to advect the particles. An application to vortex ring dynamics will be presented.

February 11

TBA

SpeakerUniversity

Abstract: TBA

February 18

A Class of Matrix Operators

Zhijian WuUniversity of Alabama

Abstract: TBA

February 25

The Square Peg Problem

Jason CantarellaUniversity of Georgia

Abstract

Suppose we have a closed curve in the plane. Is it always true that there are four points on the curve that form a perfect square? Or three points that form an equilateral triangle? In this talk we present some new results on this old problem, which was posed by Toeplitz almost a century ago. In addition to giving a new proof of the theorem (when the curve is sufficiently regular), we'll talk about connections to the Fabricius-Bjerre theorem for the plane curves. The talk will include plenty of pictures and animations, and much of it will be appropriate for undergraduates.

March 4

Proof of the q-TSPP Conjecture

Christoph KoutschanTulane University

Abstract:

In the early 1980s, George Andrews and Dave Robbins independently conjectured a nice product formula for the orbit-counting generating function of totally symmetric plane partitions (TSPPs). This conjecture, being part of Richard Stanley's famous collection "A baker's dozen of conjectures concerning plane partitions", has attracted a lot of interest among enumerative combinatorialists: it is the only open problem from this article that so far resisted all efforts. We present a proof of this long-standing conjecture. It is based on Soichi Okada's reduction to a certain determinant evaluation and Doron Zeilberger's holonomic ansatz for such determinants. We extensively employ computer algebra methods, and in particular, our software package HolonomicFunctions.

March 9

The Dauns-Hofmann Theorem Revisited

Karl H. HofmannDarmstadt and Tulane University

Abstract

In memoriam John Dauns

JOHN DAUNS died on June 4, 2009 of cancer in New Orleans, aged 73. His work on rings and modules is well-known in the algebra community. However, functional analysts working in the area of C*-algebras are likely to know his name from one theorem that is a corollary of results JOHN DAUNS obtained in the mid-sixties of the last century when I was collaborating with him [1], [3], and which became known in papers and books devoted to C*-algebra theory as the Dauns-Hofmann Theorem [2], [4]. The problem with the historical record of the DHT is that it used to be somewhat obscure how it originated and that the full weight of what was proved was not precisely understood. As JOHN DAUNS was deeply, if not subbornly involved in the development of the early phases of the representation of rings, algebras, C*-algebras by continuous sections in bundles (sometimes called continuous fields) and since his contributions were substantial I feel that it is justified to attempt a clarification. In the lecture I shall attempt to refrain from technicalities and to explain and exemplify what is being asserted.

[1] Dauns, J. and K. H. Hofmann. Representation of Rings by Sections, Memoirs

of the Amer. Math. Soc. 83, 1968, 180pp.

[2] Dixmier, J. Ideal center of a C*-algebra, Duke Math. J. 35 (1968), 375-382.

[3] Dupré, M. J. and H. M. Gillette, "Banach Bundles, Banach Modules and Auto-

morphisms of C*-Algebras," Research Notes in Math., Pitman, London, 1983.

[4] Hofmann, K. H. Gelfand-Naimark theorems for non-commutative topological

rings, in: Second Symposium on General Topology and its Relations to Modern

Algebra and Analysis in Prague, 1966. Prague, 1967, 184-189.

March 11

March 18

Particle Methods for Nonlinear Time-Dependent PDEs

Alina ChertockNorth Carolina state

Abstract:

In recent years, particle methods have become one of the most useful and widespread tools for approximating solutions of partial differential equations in a variety of fields. In these methods, the solution is sought as a linear combination of Dirac distributions, whose positions and coefficients represent locations and weights of the particles, respectively. The solution is then found by following the time evolution of the locations and the weights of the particles according to a system of ODEs, obtained by considering a weak formation of the problem. The main advantage of the particle methods is their low numerical diffusion that allows to capture a variety of nonlinear waves with a high resolution. Even though the most "natural" application of the particle methods is linear transport equations, over the years, the range of these methods has been extended for approximating solutions of nonlinear equations including degenerate parabolic, convection-diffusion and dispersive equations.

In this talk, I will review different aspects of a practical implementation of particle methods such as recovering an approximate solution from the particle distribution and investigation of various particle redistribution algorithms. I will also present new numerical techniques for nonlinear PDEs, with particular reference to problems that admit nonsmooth (discontinuous) solutions and on problems that involve multiple scales, and therefore, are difficult to solve numerically by traditional finite-difference methods. The new techniques are based on the particle methods and their hybridization with Eulerian (finite-volume) methods. I will demonstrate the performance of the new methods in a number of numerical examples, among which are the Euler-Poincaré equation, models of transport of pollutant in shallow water, reactive Euler equations describing stiff detonation waves, pressureless gas dynamics, and others.

March 22-26

April 1

April 8

Asteroids, triple linking, and bicycle

Paul MelvinBryn Mawr College

Abstract:

The linking number of a pair of closed curves in 3-space can be expressed as the degree of a map from the 2-torus to the 2-sphere, by means of the linking integral Gauss wrote down in 1833. In the early 1950's, John Milnor introduced a family of higher order linking numbers (the "mu-bar invariants"). In this talk I will describe a formula for Milnor's triple linking number as the "degree"

of a map from the 3-torus to the 2-sphere; asteroids and bicycles will come into play along the way. This is joint work with DeTurck, Gluck, Komendarczyk, Shonkwiler and Vela-Vick.

April 15

Essential Dimension

Nicole LemireUniversity of western ontario

Abstract

Essential Dimension is, roughly speaking, a measure of the degree of complexity of an algebraic or geometric object defined over a base field k. Given an algebraic or geometric object X over an extension eld K of k, the essential dimension of X is the least transcendence degree of a field of definition of X over the base field k. It determines how many independent parameters are required to define X. Essential dimension was first introduced by Buhler and Reichstein as a numerical invariant of finite and then by Reichstein for algebraic groups. It was then generalised by Merkurjev into functorial language. We will give a survey of results on essential dimension focusing on nite and algebraic groups and algebraic stacks. We will end with a discussion of recent joint work with Indranil Biswas and Ajneet Dhillon on the essential dimension of the moduli stack of vector bundles over a curve.

April 22

The Voodoo of 4-Manifolds

Ron Fintushelmichigan state University

Abstract

In every dimension but 4 the problem of classifying smooth manifolds up to diffeomorphism has (more or less) been solved. I will explain why techniques from other dimensions fail in dimension 4 and then describe an approach which Ron Stern and I call 'Santeria surgery' for studying smooth structures of 4-dimensional manifolds. As an example, I will then focus on the complex projective plane to see how Santeria surgery can be used to produce manifolds which are homeomorphic but not diffeomorphic to rational surfaces. This will be a nontechnical lecture suitable for a general mathematical audience.

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu