Colloquium index

Vigre seminars

Undergrad seminar

Clifford lectures

Timo Betcke, Technical University, Braunschweig

"Accurate low and high frequency computations for the Laplace eigenvalue problem"

The Laplace eigenvalue problem has a long and interesting history in the Mathematics and Physics community. Mathematicians were intrigued by Kac's question from 1966 if one can "hear the shape of a drum", which was first solved by Gordon, Webb and Wolpert in 1992. Physicists study the behavior of eigenfunctions associated with high energy eigenvalues in the context of quantum chaos theory. Nevertheless, the accurate numerical computation of eigenvalues and eigenfunctions remains a challenge.

In this talk we review boundary based methods that approximate eigenfunctions from a space of functions that satisfy the eigenvalue equation but not necessarily the zero Dirichlet or Neumann boundary conditions. We discuss the numerical implementation of these methods and give approximation theoretic convergence results. We present accurate eigenvalue computations on many interesting domains and give an outlook on open questions in the design of these methods.

Yan Yan Li, Rutgers University

"Gradient estimates for elliptic equations from composite material"

Abstract:

We present some results on gradient estimates for elliptic equations and systems from composite material. We will also describe some open problems.

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Jerry Bona , University of Illinois, Chicago

"Waves and Sandbar formation "

Abstract: 
We focus upon the use of mathematical analysis in several geophysical contexts.  These include tsunami propagation in the deep ocean, sand bar formation in nearshore zones and possible mechanisms for the generation of rogue waves.

Martin Golubitsky, University of Houston

"Coupled Systems: Theory and Examples "

Abstract:
A coupled cell system is a collection of interacting dynamical systems. Coupled cell models assume that the output from each cell is important And that signals from two or more cells can be compared so that patterns of synchrony can emerge. We ask: How much of the qualitative dynamics observed in coupled cells is the product of network architecture and how much depends on the specific equations?

The ideas will be illustrated through a series of examples and theorems. One example shows how a frequency filter / amplifier can be built from a small three-cell feed forward network; and a second illustrates patterns of synchrony in lattice dynamical systems. One theorem gives ecessary and sufficient conditions for synchrony in terms of network architecture; and a econd shows that synchronous dynamics may itself be viewed as dynamics in a coupled cell system through a quotient construction.

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Brant Jones, University of Washington

"Embedded factor patterns and Deodhar's algorithm for Kazhdan-Lusztig polynomials"

Abstract:

Abstract: The Kazhdan-Lusztig polynomials for finite Weyl groups arise in the geometry of Schubert varieties and representation theory. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no completely combinatorial interpretation for them is known in general. Deodhar has given a framework with a very nice form for computing the Kazhdan-Lusztig polynomials, which generally involves recursion. In this talk, we introduce a new kind of pattern-avoidance defined for general Coxeter groups to characterize when Deodhar's algorithm yields a simple combinatorial formula for the Kazhdan-Lusztig polynomials.
This generalizes results of Billey-Warrington which identified the 321-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups.

This is joint work with Sara Billey.

Jennifer Morse , Drexel

"Refined combinatorics and geometry of Schur functions"

Abstract:

Abstract After a brief introduction to the space of symmetric polynomials we will discuss how one basis, the Schur function basis, connects symmetric function theory to combinatorics. We will then show how our study of an open combinatorial problem in symmetric function theory led us to the discovery of a new family of polynomials.
We will see that our polynomials are a combinatorial analog for the Schur basis, and that they play a central role in the understanding of mysterious objects called Macdonald polynomials.

Time permitting, we will discuss how our polynomials also provide the fundamental analog for Schur functions in a geometric sense and are the natural vehicle to study the ''quantum cohomology" of the Grassmannian.

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Rebecca Lehman , MIT
2:00pm Gibson 414

"Brill-Noether Theorems with a Ramification Condition "

Abstract:

Brill-Noether questions deal with counting the ways a curve can be mapped to projective space under some given set of constraints.  The classical Brill-Noether theorem, first stated by Brill and Noether in  1879 and finally proved in 1980 by Griffiths and Harris, describes the family of maps from a general curve of genus g to a non-degenerate curve of degree d in P^r.

 We impose the additional condition of a ramification point of given type. For instance, how many ways can a general curve of genus four map to a plane sextic with a cusp?  (Answer: A three-dimensional family of them.)  How many have a higher-order singularity of type y^3=x^5?  (Answer: Exactly twenty-four.) We shall discuss enumerative methods to test for existence, and degeneration methods to bound the dimension.  We focus on the most elegant case where r=2

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Christian Klingenberg , Mathematisches Institut, Universitaet Wuerzburg
3:30pm Gibson 414

"Relating an atomistic view of fluid flow to their description by partial differential equations"

Abstract:

In this talk we try to build a bridge between statistical physics and the theory of conservation laws. Following Boltzmann's paradigm, where a microscopic particle view of a fluid flow is related to a macroscopic PDE description, we shall study this for a particular class of flow models, namely sedimentation of grains in water.
Surprisingly this approach
leads to a better understanding of the theory of conservation laws. This is joint work with Gui-Qiang Chen.

Spring break

Minerva Cordero, University of Texas at Arlington

"Semifield planes with a transitive autotopism group"

Abstract:
A semifield is a non-associative division ring; finite projective planes coordinatized by semifields that are not fields are called semifield planes. If π is a non-desarguesian semifield plane with an autotopism group transitive on the non-vertex points of a line L , then π is a generalized twisted field plane (with a few exceptions on the order). In this presentation we discuss this result and some of its corollaries.

Easter break

Shi Jin ,University of Wisconsin, Madison

"Hamiltonian Systems and Liouville Equations with Discontinuous Hamiltonians:Computation of High Frequency Waves in Heterogeneous MediaTitle"

Abstract:

We introduce Eulerian methods that are efficient in computing high frequency waves through heterogeneous media. The method is based on the classical Liouville equation in phase space, with discontinuous Hamiltonians(or singular coefficients) due to the barriers or material interfaces. We provide physically relevant interface conditions consistent with the correct transmissions and reflections, and then build the interface conditions into the numerical fluxes. This method allows the resolution of high frequency waves without

numerically resolving the small wave lengths, and capture the correct transmissions and reflections at the interface. Moreover, we extend the method to include diffraction, and quantum barriers. Applications to semiclassical limit of linear Schrodinger equation, geometrical optics, elastic waves, and semiconductor device modeling, will be discussed.

Daniel B. Szyld, Temple University

"An overview of Krylov subspace methods and their application to scientific computing"

Abstract:

Krylov subspace methods such as GMRES are extensively used for the solution of (preconditioned) linear equations, especially those arising from discretizations of differential equations.
In this talk we first present an overview of these methods, and then discuss some recent advances which make their applicability more attractive for certain class of large or difficult problems.

Qingbo Huang, Wright State University

"Reflector Problem in geometric optics "

Abstract:
Recently there have been a lot of interests in mathematical study of reflector problem arising in engineering.  The PDE governing this problem is an equation of Monge-Ampere type.  In this talk, we will descuss some important progress on this problem.

Steven Krantz, American Institute of Mathematics

"Analysis on the Worm Domain"

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Robert Lipton , Louisiana State University

"Local Field Assessment Inside Multi-scale Composite Architectures"

Abstract:

We introduce an asymptotic theory for numerically resolving the local field behavior inside multi-scale pre-stressed composite architectures. The ability to numerically recover local field information is crucial for the design of advanced composite architectures used in the next generation of commercial aircraft. The asymptotic theory applies to zones containing abrupt changes in the composite micro geometry. Such zones occur in multi-ply fiber reinforced laminated composites. The asymptotic expansions are used to develop a numerical algorithm that is able to extract local field information inside a prescribed subdomain without having to resort to a full numerical simulation. For regions of homogeneous micro structure, the analysis delivers rigorous upper bounds on the magnitude of the local stress and strain fields inside the composite. Numerical examples are provided to demonstrate the utility of the new asymptotic theory for quickly assessing the location and magnitude of local field concentrations inside complex composite architectures.
(Research supported by Boeing Aircraft Company, AFOSR, and NSF).

note time and location change: 3:00pm, Gibson 414