Colloquium

Research seminars

VIGRE Working Groups

Clifford lectures

Robert Krasny, University of Michigan

"Computing Vortex Sheet Roll-Up"

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.

Speaker, University

"Title"

TBA

Zhijian Wu, University of Alabama

"A Class of Matrix Operators"

Jason Cantarella, University of Georgia

"The Square Peg Problem"

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.

Christoph Koutschan, Tulane University

"Proof of the q-TSPP Conjecture"

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.

Karl H. Hofmann, Darmstadt and Tulane University

"The Dauns-Hofmann Theorem Revisited"

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.

Alina Chertock, North Carolina State

"Particle Methods for Nonlinear Time-Dependent PDEs "

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.

Clifford Lectures

SPRING BREAK

Paul Melvin, Bryn Mawr College

"Asteroids, triple linking, and bicycle:"

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.

Nicole Lemire, University of Western Ontario

"Essential Dimension"

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.

Ron Fintushel, Michigan State University

"The Voodoo of 4-Manifolds" 

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.