Colloquium index

Vigre seminars

Undergrad seminar

Clifford lectures

Speaker, Institution

"Title"

Abstract:

Michael Joyce, Tulane University

"A Brief History of Fermat's Last Theorem"

Speaker, Institution

"Title"

Abstract:

Keye Martin, Tulane University

"Physics and Domain Theory "

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.

John Dauns, Tulane University

"Type Submodules"

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.

Gang Bao, Michigan State

"Direct and Inverse Scattering Problems for Maxwell's Equations"

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 planewave incident on medium enclosed by a bounded domain. Given the indicent 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 nonlinearity, ill-posedness, and the large scale computation associated with the inverse scattering problem.

David Bradley, University of Maine

"q-analogues"

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 idenity, 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 claculus at MIT.  An advantage of the q-calculus is that, due to is descrete nature, the concepts of limits and infinitessimals are avoided, yet it reduces to the ordinary infinitessimal calculus of Newton and Leibniz in the limit as q tends to 1.

Amanda Knecht, Rice University

"Interpolation on Rational Surfaces "

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.

Thang Le, Georgia Tech

"The quantum MacMahon's master theorem and knot theorem"

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.

Dexter Kozen, Computer Science Department, Cornell University

"Coinductive Proof Principles for Stochastic Processes"

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.

Ronald Fintushel, Michigan State University

"Smooth structures on 4-manifolds"

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.

Matt Papanikolas, Texas A&M

"Hypergeometric functions over finite fields and modular forms"

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.

Thanksgiving holiday,

Jason Cantarella, University of Georgia

"The Geometric Structure of Tight Knots"

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 "ropelength 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).

Mark Coffey, Colorado School of Mines

"The Riemann Hypothesis: A Central Problem of Modern Mathematics"

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.