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Colloquium: 2006-2007

FALL 2006

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September 7

A Brief History of Fermat's Last Theorem

Michael JoyceTulane University

Abstract: We will give an overview of the main ideas behind two attempts to prove Fermat's Last Theorem. The first attempt was developed in the 19th century and the primary name associated to it is Kummer. Though this attempt was not ultimately successful, it brought forth a vast development of interesting mathematics, not least of which was the development of ideal theory that we all learn in a first abstract algebra course. The second attempt took place during the end of the 20th century and the primary name associated to it is Wiles. As you hopefully know already, this attempt was ultimately successful. Like Kummer's attempt, it used and introduced modern mathematical ideas to solve this classical problem.

Not only did Wiles' work prove FLT, it connected two very interesting but seemingly different mathematical objects—elliptic curves and modular forms. During the upcoming year, I plan to give a series of talks for graduate students and undergraduates (only prerequisite: a first-semester course in abstract algebra) which will go into more detail about the mathematics that is involved in Fermat's Last Theorem and more carefully explain the terms that come up in this talk.

September 21

Physics and Domain Theory

Keye MartinTulane University

Abstract: Since its inception, domain theory has played an important role in modeling various forms of computational phenomena. This talk is about the recent discovery of its importance to physics, especially in quantum mechanics and general relativity.

In quantum mechanics, domains have been used to solve physics problems in the quantum information literature, even one originally considered by Schrodinger himself. They have been used to calculate the complexity of certain quantum algorithms, in such a way that their relation to their classical counterparts becomes clear. They have been used to characterize various forms of entanglement transformation.

In general relativity, they have been used to explain how spacetime may be topologically reconstructed from a discrete set of events from a purely causal viewpoint. They have been used to prove essential results like the compactness of the space of causal curves, which plays a important role in establishing the existence of maximum length geodesics, certain positive mass theorems and in the singularity theorems.

In thermodynamics, they have been used to explain the relationship between algorithmic complexity and entropy, and to provide algorithms for computing the maximum entropy state.

The fact that domains arise in these areas of physics makes one wonder if these subjects have something in common. The answer that we give to this question in this talk comes in the form a single theorem, and is an essential part of a new mathematical model of secure communication.

The purpose of this model is to allow for the accurate assessment of threats and capabilities posed by communication schemes that are based on current and emerging technologies. This requires one to consider not only both relativistic and quantum effects, but also to understand how channel capacity in the sense of Shannon is ultimately determined by physical parameters. It also requires a model of computation. The only area of mathematics that we are aware of which can support such a rich diversity of ideas in a single consistent framework is domain theory.

September 28

Type Submodules

John DaunsTulane University

Abstract: Type submodules can be defined by two independent or parallel descriptions. One of these uses natural classes of R-modules, which are of independent interest.

For a fixed ring R, a class of right modules F is a natural class, or a type, if it is closed under isomorphic copies, (i) submodules, (ii) arbitrary direct sums, and (iii) injective hulls. The class of all natural classes forms a complete Boolean lattice N(R). This lattice defines various intrinsic new module classes. For example, an R-module A is atomic if the natural class it generates is an atom in the lattice N(R). Or, a module is bottomless if it does not contain any atomic modules. These modules, the atomic ones, generalize the uniform modules, and one of their applications is to define a dimension similar to the finite Goldie dimension based on uniform modules. This opens up the study of rings and modules satisfying finiteness conditions based on this new dimension, but not satisfying ordinary finiteness conditions: ascending or descending chain conditions, or finite Goldie dimension. Also, N(R) can be made into a functor N( -). A submodule K of a module right R-module M is a type submodule if there exists some natural class F as above such that K belongs to F, and among the submodules of M , K is maximal with respect to belonging to F.

For the ring R=Z of the integers, and an abelian group M, and prime p, the p-torsion component is an example of a type submodule. It is well known that every torsion abelian group is a direct sum of its p-torsion subgroups. This is only a very special case of a decomposition of a certain modules ( or even von Neumann algebras ) as a direct sum of type submodules (or algebras of types I, II, and III).

Historically, first module structure and decomposition theorems were proved by placing restrictive assumptions such as the descending and ascending chain conditions on all submodules (or right ideals in case M=R). Now it seems that a new generation of ring and module theory could be developed in the future by putting restrictive assumptions only on the type submodules of a module.

October 5

Direct and Inverse Scattering Problems for Maxwell's Equations

Gang BaoMichigan State

Abstract: The inverse medium scattering problem arises in diverse areas of industrial and military applications, such as nondestructive testing, seismic imaging, submarine detections, near-field or subsurface imaging, and medical imaging. The problem is concerned with a time-harmonic electromagnetic plane wave incident on medium enclosed by a bounded domain. Given the indecent field, the direct problem is to determine the scattered field for the known scatterer. The inverse medium scattering problem is to deterine the scatterer from the boundary measurements of near filed currents densities. Although this is a classical problem in inverse scattering theory, little is known on reconstruction methods, especially in the three dimensional case, due to the non-linearity, ill-posedness, and the large scale computation associated with the inverse scattering problem.

In this talk, our recent progress in mathematical analysis and computation of time harmonic Maxwell's equations in complicated media will be discussed. For the direct problems, recent regularity results will be introduced. Various types of boundary conditions will be discussed to reduce the scattering problem into a bounded domain. The first convergence analysis of the recent Perfect Matched Layer (PML) approach for Maxwell's equations will be presented. For the inverse medium scattering, a continuation approach based on uncertainty principle will be presented for both multiple and fixed frequency boundary data. Issues on convergence will be addressed. Our on-going research on related topics and multi-scale modeling of nano optics will be highlighted.

October 12

Q-Analogues

David BradleyUniversity of Maine

Abstract: The study of q-series began in 1748 when Euler considered the generating function for the number of partitions of a positive integer. But the subject itself did not really come into its own until some 100 years later when Heine developed a theory of basic hypergeometric series that contains the theory of the Gauss hypergeometric series as a limiting case. The latter is obtained from the former in the limit as q tend to 1. Although the work stemming from heine is highly analytic in nature, many of the most beautiful results have combinatorial or number-theoretic significance as well: the q-binomial theorem, Jacobi's triple product identity, and Ramanujan's 1-Psi-1 identity come to mind.

In a productive mathematical life spanning the latter part of the nineteenth century and the first half of the twentieth century, F. H. Jackson developed a systematic theory of q-analogues, including the operations of q-differentiation and q-integration, which have recently been made the basis of an undergraduate course in quantum calculus at MIT. An advantage of the q-calculus is that, due to is discrete nature, the concepts of limits and infinitesimals are avoided, yet it reduces to the ordinary infinitesimal calculus of Newton and Leibniz in the limit as q tends to 1.

The purpose of this talk is to provide an accessible introduction to the vast subject of q-analogues. My intention is to give an idea of what the subject is about, why people are interested in it, and how I became involved.

October 19

Interpolation on Rational Surfaces

Amanda KnechtRice University

Abstract: Tsen's theorem is a classical result which states that over the function field of a complex projective curve, a homogeneous polynomial has a nontrivial solution provided the degree of the polynomial is less than than the number of variables. In 2001 Graber, Harris, and Starr generalized this result by proving that every rationally connected variety over the function field of a curve has a rational point. A proper variety over an algebraically closed field is rationally connected if any two points can be connected by a rational curve. The GHS result is a generalization of Tsen because a smooth hypersurface in CPⁿ is rationally connected if its degree is not greater than n.

We can restate the theorems of Tsen and Graber, Harris, Starr in terms of the existence of sections of fibrations. Once we know that a section of our fibration exists, we can ask interpolation questions about the sections: Can we find a section through a prescribed number of points? Can we prescribe a Taylor series for the section at a finite number of points? I will give some examples of varieties for which we know the answers to such questions. We will discuss in more detail the case where the general fiber is a degree-two del Pezzo surface.

October 26

The Quantum MacMahon's Master Theorem and Knot Theorem

Thang LeGeorgia Tech

Abstract: MacMahon's Master Theorem (MMT) is a matrix generalization of the identity: 1 + x + x2 + ... = 1/(1-x).

MMT has played an important role in combinatorics. Motivated by knot theory, we state and prove a quantum-generaliztion of MMT. This answers G. Andrews' long standing problem of finding a natural q-analog of MMT.

This is joint work with S. Garoufalidis and D. Zeilberger.

November 2

Coinductive Proof Principles for Stochastic Processes

Dexter KozenComputer Science Department, Cornell University

Abstract: Coinduction (or "baseless induction") has been shown to be a useful tool in type theory and functional programming. Streams, automata, concurrent and stochastic processes, and recursive types have been successfully analyzed using coinductive methods.

Most approaches apply coinduction to infinite recursively-defined objects such as streams and recursive types. There the coinduction principle that states that under certain conditions, two bisimilar processes must be equal. For example, to prove the equality of infinite streams s=merge(split(s)), where merge and split satisfy the familiar coinductive definitions:

merge(a::s,t) = a::merge(t,s)
#1(split(a::b::r)) = a::#1(split(r))
#2(split(a::b::r)) = b::#2(split(r)),

it suffices to show that the two streams are bisimilar.

In this talk I will describe an explicit coinduction principle for recursively-defined stochastic processes. The principle applies to any closed property, not just equality, and works even when solutions are not unique. The rule encapsulates low-level analytic arguments, allowing reasoning about such processes at a higher algebraic level. I will illustrate the use of the rule in deriving properties of a simple coin-flip process.

I will provide the necessary background, so the talk will be accessible to students and faculty who are unfamiliar with bisimulation and coinduction.

November 9

Smooth Structures on 4-Manifolds

Ronald FintushelMichigan State University

Abstract: In dimensions less than 4, each closed topological manifold has a unique smooth structure, and in dimensions greater than 4, a topological manifold has at most finitely many smooth structures. By contrast many (and perhaps all) topological 4-manifolds which admit at least one smooth structure admit infinitely many. I will discuss ways to change the smooth structure of a 4-manifold with the goal of finding a "dial" inside the manifold which one can turn to change its smooth structure much as one changes (or used to) channels on a TV set.

November 16

Hypergeometric Functions over Finite Fields and Modular Forms

Matt PapanikolasTexas A&M

Abstract: First studied by Greene and Stanton in the 1980's, finite field hypergeometric functions are constructed as certain sums of products of Jacobi sums. Work of Ahlgren, Koike, Ono, and others have shown in certain examples that values of these hypergeometric functions are closely related to counting points on some Calabi-Yau manifolds over finite fields as well as to Fourier coefficients of modular forms. Our overall goal is to explain these phenomena, and we consider additional examples of values of 4F3-hypergeometric functions and investigate how they count points on families of varieties whose Picard-Fuchs equations are essentially hypergeometric. Joint work with S. Frechette.

November 30

The Geometric Structure of Tight Knots

Jason CantarellaUniversity of Georgia

Abstract: Knotting and linking are an important process in scientific applications ranging from the subatomic scale (glueballs) to the astrophysical scale (linked flux tubes in solar flares). In many of these problems, geometry plays a role as well as topology, as the materials being knotted have some width or "thickness". In this talk we discuss an abstracted form of these geometric constraints: the "rope length problem", which considers the shapes of minimum length knots tied in tubes of uniform circular cross-section.

We will cover some recent progress in developing a a criticality condition describing these tight knots. This condition can sometimes be used to determine the shape of length-critical configurations for knots. For example, the new logo for the International Mathematical Union shows a tight configuration of the Borromean rings obtained in this way. We will discuss some of these shapes and give an overview of some techniques used in the proof of our criticality theorem. The talk will conclude with some computer animations showing simulations of the tightening process and some open problems. Joint work with: J. Fu, R. Kusner, J. Sullivan, N. Wrinkle (theory), and T. Ashton, M. Piatek, E. Rawdon (computations).

December 7

The Riemann Hypothesis: A Central Problem of Modern Mathematics

Mark CoffeyColorado School of Mines

Abstract: The Riemann Hypothesis is generally recognized as the most significant open problem of mathematics, and investigators have been seeking this holy grail for nearly a century and a half. The problem itself arises in complex analysis, but it has meaning throughout the body of mathematics—quite possibly thousands of theorems are now conditional upon it. Riemann conjectured as to the precise location of the complex zeros of a certain meromorphic function, the zeta function. This talk presents an overview of the life of Bernhard Riemann, the content of the Hypothesis, the context of the Millenium Problems, and of the status of recent computational work on the complex zeta zeros. In addition, two equivalences of the Riemann hypothesis for which there has been much active recent research will be discussed.

SPRING 2007

January 4

Accurate Low and High Frequency Computations for the Laplace Eigenvalue Problem

Timo BetckeTechnical University, Braunschweig

Abstract: 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.

January 11

Gradient Estimates for Elliptic Equations from Composite Material

Yan Yan LiRutgers University

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

February 1

Waves and Sandbar Formation

Jerry BonaUniversity of Illinois, Chicago

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 near shore zones and possible mechanisms for the generation of rogue waves.

February 8

Coupled Systems: Theory and Examples

Martin GolubitskyUniversity of Houston

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 necessary and sufficient conditions for synchrony in terms of network architecture; and a second shows that synchronous dynamics may itself be viewed as dynamics in a coupled cell system through a quotient construction.

February 22

Embedded Factor Patterns and Deodhar's Algorithm for Kazhdan-Lusztig Polynomials

Brant JonesUniversity of Washington

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 non-negative 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.

March 1

Refined combinatorics and geometry of Schur functions

Jennifer MorseDrexel

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.

March 15

Brill-Noether Theorems with a Ramification Condition

Rebecca LehmanMIT

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 Pr.

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 y³=x⁵? (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

Note: 2pm Gibson 414


Relating an Atomistic View of Fluid Flow to Their Description by Partial Differential Equations

Christian KlingenbergMathematisches Institut, Universitaet Wuerzburg

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.

Note: 3:30pm Gibson 414

March 29

Semi-field Planes with a Transitive Autotopism Group

Minerva CorderoUniversity of Texas at Arlington

Abstract: A semi-field is a non-associative division ring; finite projective planes coordinatized by semi-fields that are not fields are called semi-field planes. If π is a non-desarguesian semi-field 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.

April 11

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

Shi JinUniversity of Wisconsin, Madison

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 semi-classical limit of linear Schrodinger equation, geometrical optics, elastic waves, and semiconductor device modeling, will be discussed.

April 12

An Overview of Krylov Subspace Methods and Their Application to Scientific Computing

Daniel B. SzyldTemple University

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.

April 19

Reflector Problem in Geometric Optics

Qingbo HuangWright State University

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 discuss some important progress on this problem.

April 26

Analysis on the Worm Domain

Steven KrantzAmerican Institute of Mathematics

Abstract: The Bergman kernel is an important biholomorphic invariant of complex analysis. We explain the provenance of the Bergman kernel, and its applications in several complex variables. Of particular interest are instances in partial differential equations and mapping theory. At the end we indicate some techniques for calculating the Bergman kernel on particular domains.

May 9

Local Field Assessment Inside Multi-Scale Composite Architectures

Robert LiptonLouisiana State University

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 will be 3:00pm, Gibson 414

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