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Colloquium: 2007-2008

FALL 2007

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September 4

Bourbaki at T and T: A Historical Reflection on the Bourbaki Impact Around 1955-1965 by Sampling the University of Tübingen and Tulane University

Karl H. HofmannTechnische Universität Darmstadt and Tulane University

Abstract: The Bourbaki movement, which started in France in the nineteen thirties and forties, had considerable influence on the style and the direction of mathematics and mathematicians in the second half of the last century. It appears to be interesting to note the difference in the reception of Bourbaki program in various countries. Since I know German universities, notably Tübingen (which was rather prominent in mathematics after WWII), and since I have known Tulane for some time, I venture to offer some observations on the historical plane by looking back at both Tübingen and Tulane around the middle of the last century. Obviously, this is meant to be a non-technical presentation which imposes no prerequisites on the audience.

Note: 4 PM Gibson 310

September 13

Mathematical Modeling of Intracellular Signaling Pathways

Tim ElstonUniversity of North Carolina

Abstract: Cells most respond to a constantly changing environment. Information about their surroundings is transmitted through signaling pathways that receive external cues and initiate the appropriate response. The complexity of these signaling pathways has led to use of mathematical and computational approaches to understand how cells receive and process information. The first part of this talk provides an introduction to intracellular signal transduction and some of the key biological questions in this field. In particular, the idea of pathway specificity is introduced. Pathway specificity refers to the cell's ability to generate appropriate responses to distinct external cues even though the underlying signaling pathways share common components. Next results demonstrating how mathematical modeling has been used to elucidate the mechanisms responsible for pathway specificity are presented. Finally, a developmental decision that is mediated by the yeast mating response pathway is discussed. Depending on the concentration of pheromone yeast either undergo chemotrophic growth toward a mating partner or initiate a mating response. Mathematical modeling is used to provide insight into how information about the pheromone concentration is encoded and transmitted through this pathway to ensure the appropriate transcriptional program is followed.

September 20

Sharp Sobolev Inequalities: From Historic and Geometric View Points

Meijun ZhuUniversity of Oklahoma

Abstract: I shall review the history of the study of sharp Sobolev inequalities on Rⁿ (thus Sⁿ) (back to the early work of Hardy and Littlewood in 1929, including the story of the Bliss's Lemma), and describe their relation to the Yamabe problem and sharp Sobolev inequalities on manifolds. These motivates us to obtain the local sharp inequalities, which yield, among other results, a much simpler proof of the Onofri inequality.

I shall also explain that these sharp inequalities on Sⁿ often encoded in certain geometric flow problems. In fact, the steady state metrics of those flows are usually described by the sharp inequalities. Our current project on the study of low dimensional geometric flows will be described.

September 27

Case-Control Association Testing with Incomplete Genealogy

Timothy Alvin ThorntonUniversity of California, Berkeley

Abstract: We consider the problem of testing for association between a complex trait and a genetic marker in a case-control design in which some individuals are related. Using related individuals in case-control studies has compelling advantages. When related individuals are included in a study, correlations among relatives must be taken into account to ensure validity of the test. We first give an overview of proposed methods when the genealogy of individuals in a study is completely specified. We then consider the case when the genealogy is incomplete and present a new approach to this problem.

October 11

Outside Looking In! Characterizing an Indecomposable Plane Continuum From Its Complement

John MayerUAB

Abstract: It is a long-standing open question whether the Julia set of some rational function is an indecomposable continuum. By definition, a continuum is if it is not the union of two of its proper subcontinua. There are several ways of recognizing intrinsically that a continuum X is indecomposable. For instance, X is indecomposable if and only if every proper subcontinuum of X is nowhere dense in X. We provide a condition for testing whether the Julia set of a rational function is an indecomposable continuum using data from its complement. In the complex dynamics context, that means we are investigating the (possibly topologically and dynamically complicated) Julia set from the point of view of the (always topologically and dynamically simple) Fatou set.

To state our theorem we need to define some terms. A generalized crosscut of a complementary domain ∪ is an open arc A ⊂ C such that A∖A ⊂ ∂∪. Let U be a plane domain and A a generalized crosscut of ∪. We call each of the two components of ∪\A a crosscut neighborhood. If V is a crosscut neighborhood determined by generalized crosscut A, we call the continuum S = ∂V ∩ ∂∪ a shadow of A. A sequence (∪n)n=₁ of (not necessarily distinct) complementary domains of a continuum X satisfies the double-pass condition iff, for any sequence of generalized crosscuts An of ∪n, there is a sequence of shadows (Sn)n=₁ of (An)n=₁ such that limn → ∞ Sn = Χ we prove the following:

Characterization Theorem: A plane continuum X is indecomposable if X has a sequence (∪n)n=₁ of X complementary domains satisfying the double-pass condition.

Recently, Clinton Curry has extended the Characterization Theorem (with an appropriately modified definition of generalized surface crosscut) to continua in compact surfaces.

Co-Authors: Clinton P. Curry (University of Alabama at Birmingham) and E. D. Tymchatyn (University of Saskatchewan)

October 18

A Solid Angle Theory for Real Polytopes

Sinai RobinsTemple University

Abstract: We extend many theorems from the context of solid angle sums over rational polytopes to the context of solid angle sums over real polytopes. This theory captures a new measure of volume, which is a kind of discrete volume of polytopes. Moreover, we consider any real dilation parameter, as opposed to the traditional integer dilation parameters. One of the main results is an extension of Macdonald's solid angle quasipolynomial for rational polytopes to a real analytic function of the dilation parameter, for any convex polytope whose vertices have arbitrary real coordinates. Some of this work is joint with my student David DeSario. I'll present some computer graphics to illustrate the ideas more clearly.

October 25

The Computer's Role in Mathematical Discovery and Proof

Marc ChamberlandGrinnell College

Abstract: The use of computer packages has brought us to a point where the computer can be used for many tasks: discover new mathematical patterns and relationships, create impressive graphics to expose mathematical structure, falsify conjectures, confirm analytically derived results, and perhaps most impressively for the purist, suggest approaches for formal proofs. This is the thrust of experimental mathematics. This talk will give some examples to discover or prove results concerning geometry, integrals, binomial sums, dynamics and infinite series.

November 1

Tail Inference for Probability Distribution Functions

Javier RojoRice University, Department of Statistics

Abstract: After presenting a review of some concepts of tail-ordering, tail-heaviness, and tail categorization of probability distributions, methodology for testing for medium-tailed distributions against either small- or long-tailed distributions is presented and its operating characteristics examined.

November 8

A Variational Theory for Defects in Patterns

Nick ErcolaniUniversity of Arizona

Abstract: We derive a rigorous scaling law for minimizers in a natural version of the regularized Cross-Newell (phase-diffusion) model for pattern formation far from threshold. The talk will describe the physical, experimental and mathematical motivations for this class of problems as well as recent results.

November 9

Molecular Dynamics Simulations of Live Particles

Joceline LegaUniversity of Arizona

Abstract: I will show results of molecular dynamics simulations of hard spheres with non-classical collision rules. In particular, I will focus on how local interactions at the microscopic level between these particles can lead to large-scale coherent dynamics at the mesoscopic level.

This work is inspired by collective behaviors, in the form of vortices and jets, recently observed in bacterial colonies. I will start with a brief summary of basic experimental facts and review a hydrodynamic model developed in collaboration with Thierry Passot (Observatoire de la Cote d'Azur, Nice, France). I will then motivate the need for a complementary approach that includes microscopic considerations, and describe the principal computational issues that arise in molecular dynamics simulations, as well as the standard ways to address them.

Finally, I will discuss how classical collision rules that conserve energy and momentum may be modified to describe ensembles of live particles, and will show results of numerical simulations in which such rules have been implemented. Randomness, included in the form of random reorientation of the direction of motion of the particles, plays an important role in the type of collective behaviors that are observed.

November 15

Statistics for the Genomics Revolution: Some Examples

Rafael IrizarryJohns Hopkins

Abstract: The completion of the human genome project has revolutionized biology and science in general. The information gained from this project along with powerful new technologies has led to de so-called Genomics Revolution and has transformed biology to be a much more quantitative science. Mathematical models have been extremely important in physics, chemistry, and, to some extent, biology. However most biological processes of interest are far too complex for these models to be useful. The impact of Statistics on the Genomics Revolution has been through the development of powerful data analysis techniques, many of which rely on empirically motivated models. In this talk I will give some examples of useful tools developed to aid in the analysis of data produced by one of the most widely used technologies: micro-arrays.

November 29

Max-Stable Processes: Representations, Ergodic Properties and Some Statistical Applications

Stilian A StoevUniversity of Michigan

Abstract: Max-stable stochastic processes arise in the limit of component-wise maxima of independent processes, under appropriate centering and normalization. In this talk, various representations of max-stable processes will be discussed. Then, in terms of these "spectral" representations, necessary and sufficient conditions for the ergodicity and mixing of stationary max-stable processes will be presented.The large classes of moving maxima and mixed moving maxima processes are shown to be mixing. Other examples of ergodic doubly stochastic processes and non-ergodic processes will be given. The developed ergodicity and mixing conditions involve a certain measure of dependence. We will address the statistical problem of estimating this measure of dependence.

December 6

Computational Modeling of Blood Clotting

Aaron FolgesonUniversity of Utah

Abstract: Intravascular hemostasis and thrombosis occur under flow and this can profoundly influence the progress of clot formation. This talk will focus on two different aspects of our efforts to model and probe the interactions of flow and clotting. One involves the biochemistry of the coagulation enzyme network and how the behavior of this system is affected by flow-mediated platelet deposition on an injury and by flow-mediated transport of the enzymes and their precursors. The other involves a continuum model that describes platelet thrombosis initiated by a ruptured atherosclerotic plaque in a coronary-artery-sized vessel. This model includes full treatment of the fluid dynamics, and the aggregation of platelets in response to the plaque rupture and further chemical signals. Among the behaviors seen with this model are the growth of wall-adherent platelet thrombi to occlude the vessel and stop the flow, and the transient growth and subsequent embolization of thrombi leaving behind a passivated injured surface.

SPRING 2008

January 17

Exploiting Symmetries: Alternating Sign Matrices and the Weyl Character Formulas

David M BressoudMacalester College

Abstract: This will be an overview of some of the points of interaction between symmetric functions and representation theory on the one hand, and questions in number theory and combinatorics on the other, culminating in recent work of Okada enumerating alternating sign matrices (aka the six-vertex model of statistical mechanics) through evaluations of Weyl character formulas.

January 24

Highly Generalized Eulerian Calculus on Coxeter Systems

Alessandro ConflittiDMUC Universidade de Coimbra, Portugal

Abstract: Classical Eulerian calculus deals with the distribution of the descent and inversion statistics on the symmetric group, the archetypal example of Coxeter system.

We define a family of statistics over a generic finite Coxeter system indexed by subsets of its reflections set, thus highly generalizing the above--mentioned ones. We study the corresponding generating functions, proving that they have a lot of interesting combinatorial properties.

Specifically, we prove equidistribution results, namely we investigate some conditions in order to have that different subsets have the same associated generating function. In particular, considering the symmetric group $S_{n}$ we prove that any poset of size $n$ corresponds to a subset of traspositions of $S_{n}$, and that the generating function of the corresponding statistic includes partial linear extensions of such a poset. Finally, we explicitly compute the associated generating functions for several classes of subsets.

January 31

Quasi-Isometric Classification of 3-Manifold Groups

Jason BehrstockColumbia University

Abstract: Any finitely generated group can be endowed with a metric which is unique up to maps of bounded distortion (quasi-isometries). A fundamental question is to classify finitely generated groups up to quasi-isometry. Considered from this point of view, fundamental groups of 3-manifolds provide a rich source of examples. We will survey the world of 3-manifold groups from classical results to the recent resolution of some long standing questions. This talk is intended for a broad audience of mathematicians.

February 7

Symplectic Geometry and Invariants for Low-Dimensional Topology

Matthew HeddenMIT

Abstract: Over the past few years, ideas from symplectic geometry have had a major impact on low-dimensional topology. Some of the most impressive results stem from a set of invariants developed by Ozsvath and Szabo. Though defined using symplectic geometry, they turn out to be surprisingly powerful invariants of low-dimensional objects e.g. knots, and three- and four-manifolds. In this talk, I will survey these invariants and discuss how I have used them to prove results related to knot theory, complex curves, surgery theory in dimension three, and the theory of foliations and contact structures on three-manifolds.

February 8

The Characteristic Polynomial of A Multi-arrangement

Max WakefieldHokkaido University-Japan

Abstract: One of the most fundamental invariants of a hyperplane arrangement is its characteristic polynomial. For a hyperplane arrangement this polynomial encodes combinatorial, topological, and even algebraic information. In this talk we will review this characteristic polynomial and then discuss a generalization to multi-arrangements of hyperplanes.

Note: 3:30pm G310

February 12

The Core of Ideals

Louiza FouliUT Austin

Abstract: Let R be a Noetherian local ring with infinite residue field k and I an R-ideal. The ideal J is a reduction of I if JI and Ir+1 = JIr for some positive integer r. A reduction can be thought of as a simplification of the ideal I. The notion of a reduction for an ideal was introduced by D.

Northcott and D. Rees in order to study multiplicities. Reductions are connected to the study of blowup algebras such as the Rees ring R( I ) = R [ It ] of I, and the associated graded ring gr_{I} ( R ) = R [ It ] / IR [ It ] of I.

In general minimal reductions are not unique. To remedy this lack of uniqueness, one considers the intersection of all reductions, namely the core of the ideal, core ( I ). This object, that appears naturally in the context of the Brian\c con-Skoda theorem, encodes information about all possible reductions. We present some recent work on the shape of the core of ideals.

Note: 3:30pm G310

February 14

On Nice Geometric and Homological Properties of Algebraic Sets

Hailong DaoUniversity of Utah

Abstract: In 1890, Hilbert proved that any graded module over a polynomial ring over a field (the coordinate ring of an affine space) has a finite free resolution. This was later extended to all regular local rings by Auslander-Buchsbaum-Serre. Such results hint at a broader pattern: varieties with nice geometric properties also enjoy nice homological properties (and vice versa). This point of view motivates questions and conjectures which have been studied in Commutative Algebra and Algebraic Geometry over the last 50 years. In this talk we will survey the history of some of these questions, as well as recent developments.

February 21

Arithmetic Dynamics and Galois Groups of Iterated Maps

Rafe JonesUniversity of Wisconsin-Madison

Abstract: I'll begin with a brief overview of arithmetic dynamics, which seeks to understand the arithmetic of orbits in discrete dynamical systems. I have considered a question about the set of primes dividing certain orbits, and I will explain how this question reduces to one involving properties of Galois groups of iterated maps. These groups remain mysterious in many cases, although they have many applications, some of which I will discuss. Finally, I'll talk about some of the techniques involved in understanding these groups, and give directions for future projects.

March 13

Serre Curves and Their Division Fields

Nathan JonesCentre de Recherches Mathématiques Université de Montréal

Abstract: The cyclotomic fields comprise one of the most classical families of number fields. The N-th cyclotomic field is a Galois extension of the rational numbers whose Galois group is isomorphic to (Z/NZ)*, the unit group of the integers modulo N. In this talk, I will discuss a two-dimensional analogue of the N-th cyclotomic field, namely the N-th division field of an elliptic curve, and discuss a theorem that for "almost all" elliptic curves, the N-th division field has Galois group isomorphic to GL_2(Z/NZ), the unit group of the ring of 2 by 2 matrices modulo N. If time permits, I will discuss applications of this theorem to the problem of averaging constants appearing in various conjectural prime-counting asymptotics attached to elliptic curves.

March 25

Recent Advances on theRrook Monoid

Mahir CanUniversity of Pennsylvania

Abstract: Shellability is a combinatorial property of a cell complex (of a poset) with important topological and algebraic consequences. For example, a shellable complex has the homotopy type of a wedge of r-spheres and its Stanley-Reisner ring is Cohen-Macaulay. Among the interesting classes of shellable posets is the symmetric group with respect to the Bruhat-Chevalley ordering. The rook monoid is a finite (inverse) monoid having symmetric group as its group of invertible elements. There is a natural extension of the Bruhat-Chevalley ordering on the rook monoid (originating from the Bruhat decomposition of the nxn matrices).

In this talk, we shall first give an overview of the symmetric group and the rook monoid, then we shall work-out some examples showing that the rook monoid is shellable. No prior knowledge is required.

Note: 3:30pm G310

March 27

Classification of Algebraic Varieties

Bangere PurnaprajnaUniversity of Kansas

Abstract: Classification problems are similar to taxonomy in botany or zoology. It is necessary to give a good division and hierarchy to algebraic varieties before we can study them. One wants also to find crucial characteristics to tell us when a given specimen, that is, a given variety, belongs to a certain family.

To separate varieties by dimension of a variety is an obvious first step. The second step is to classify varieties of a given dimension. There is a rough division for varieties of a given dimension based on a quantity called the Kodaira dimension denoted by k, which is a number between -1 and the dimension of the variety. This is a very coarse classification and is like assigning a given vertebrate into fish, amphibians, reptiles, birds and mammals and not being able to say more. For instance, not being able to distinguish between a whale and a bat.

Geometers like their counterparts in zoology want a much finer classification accounting many other complexities. Even for varieties of dimension two (also called surfaces), a finer classification is far from complete. We will concentrate on the finer classification of algebraic surfaces for this talk.

April 3

Uniform Bounds in Noetherian Rings

Ian AberbachUniversity of Missouri, Columbia

Abstract: Let R be a commutative ring (with unit). Then R is Noetherian if every ideal of R is finitely generated. In the words of Craig Huneke, "behind the obvious finiteness condition in Noetherian rings,... , there lie many deeper and hidden types of finiteness which come to light in terms of uniform behavior. . . . [by that] we mean statements which give some bounds (usually numerical) not just for one ideal, but for all ideals simultaneously."

The talk will try to give a flavor of such results, touching on, for instance, uniform Artin-Rees theorems, uniform annihilation of local cohomology and the connection to uniform annihilators of homology in classes of free complexes, tight closure and its uniform annihilation (i.e., test elements), Briancon-Skoda type theorems, and uniform degrees of nilpotency for parameter ideals.

April 10

New Approaches to Predicting the Spread of Epidemics

Mac HymanLos Alamos National Laboratory

Abstract: Mathematical models based on the underlying transmission mechanisms of a disease can help the medical/scientific community anticipate the spread of an epidemic and evaluate the potential effectiveness of different approaches for bringing the epidemic under control. I will describe how these models can aid in understanding of the underlying transmission pathways of an epidemic and be used to estimate the benefits and the costs of possible interventions. I will describe how the early epidemic models have evolved to account for variations in the infectiousness and transmissibility of different diseases, behavior changes in response to an epidemic, the impact of biased mixing and variations in behavior among people in a susceptible population, and to quantify the uncertainty in the predictions. The lecture will describe the broad classes of epidemic models, from simple ordinary differential equations to massive stochastic agent-based simulations for understanding the spread of a disease within a major city. I will describe how these new models are creating a mathematical foundation to facilitate collaborations among the biological, public health, behavioral, social, and mathematical science communities.

April 17

Controlling the Shapes of Micro- and Nano-Scale Structures

John LowengrubUniversity of California, Irvine

Abstract: The ultimate goal of materials design is to start by a specifying a set of desirable properties and then to follow-up by fabricating a material that meets these specifications optimally. By controlling and patterning the micro- and nano-structures, this dream is growing ever closer to reality. Materials that can be uniquely targeted to specific applications have the potential to make an enormous technological impact. In this talk, we present mathematical theory that can be used for controlling the shapes of growing crystals at the microscale and controlling the spatial orientation of nano-structures (quantum dots) during epitaxial growth of thin films.

At the microscale, we demonstrate that there exist critical conditions of growth such that the Mullins-Sekerka instability may be suppressed and instead universal limiting shapes exist. That is, we find that the morphologies of the non-linearly evolving crystals tend to limiting shapes that evolve self-similarly and depend only on the far-field conditions. We then design protocols by which the compact growth of crystals with desired symmetries can be achieved. We present both 2D and 3D results using adaptive boundary integral methods. Preliminary experimental results are presented that suggest the confirmation of the theory.

The theory at the microscale is then extended to nanoscale studies of monolayer, epitaxially growing islands. Here, the control variables are the deposition flux and a far-field flux that can be manipulated so as to control the shape of the island. We conclude with a study of strained epitaxial thin films. In this case, the relaxation of strain provides a mechanism for influencing the self-organization of quantum dot structures. Using newly-developed, adaptive phase-field methods, we demonstrate that strain patterning, as well as control of the deposition flux, may result in ordered self-organized arrays of nanostructures (quantum dots).

April 18

On the Two-Dimensional Boussinesq Solitons

Christo ChristovUniversity of Louisiana at Lafayette

Abstract: The quasi-particle behavior of solitons is very important for physical applications. While the mechanics of soliton interactions is well understood in 1D, the works in multidimension are scant. Even the shapes of the steady propagating, noninteracting solitons are not known for most of the main soliton-supporting models. This is mainly due to the fact that no analytical solutions are available in more than one spatial dimension. For this reason, it is very important to develop numerical approaches that can adequately handle the infinite domain and the bifurcation nature of the problem.

We consider here the steady-propagating single soliton of the so-called Proper Boussinesq equation in two spatial dimensions. We develop three different techniques: a finite difference scheme with nonuniform grid; Galerkin spectral technique base on a special complete orthonormal system of functions on the infinite interval; and a perturbation expansion for small phase speeds, c, of propagation of the soliton up to order O(c⁴), including. We demonstrate that the three techniques are in excellent quantitative agreement, which validates them.

Our results show that there is a dramatic change of the asymptotic behavior of the soliton shape when for phase speeds c ≠ 0, i.e. the decay at infinity of the shape of a moving soliton is qualitatively different from the standing soliton. The latter decays super-exponentially with the radial coordinate, while the former decays as O (r ⁻ 2). This makes the moving solitons spread wider than the standing soliton. In addition, small depressions (of order of Q(c²)) appear in the forerunner and the trail of the profile. This appears to be a novel result and can have a profound impact on the way the interaction of two (or more) two-dimensional solitons is modeled. The potential of interaction of two quasi-particles identified with the respective solitons depends predominantly on the asymptotic behavior of soliton's shapes. Thus the algebraically decaying shapes yield an algebraically-decaying attraction potential, which is qualitatively different from the 1D case, where the attraction potential always decay exponentially. The 2D solitons 'feel' each other at much farther distances than the 1D ones.

Note: 3pm G325

April 24

Applications of Hensel's Lemma to the Representation Theory of SL(2)

Carlos Julio MorenoCUNY

Abstract: The Representation theory of the p-adic group SL(2) differs significantly in the two cases: p=2 and p odd. We shall examine some of the number theoretic reasons for this from the optic of Hensel's Lemma.

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