# Colloquium: 2005-2006

## FALL 2005

The Fall 2005 colloquiua were canceled due to hurricane Katrina.

## SPRING 2006

View Lagniappe 2006 »

January 26

## On Writing Numbers: Multidimensional Continued Fractions and the Hermite Problem

Thomas A. GarrityWillians University

Abstract: What is the best way of writing numbers? In particular, are there ways of writing real numbers as sequences of integers so that special properties of the real numbers can be easily seen from the sequence? For example, most of us usually express real numbers in terms of their decimal expansions. Here, a real number will be rational precisely when its decimal expansion is eventually periodic. One can also write a real number in terms of its continued fraction expansion, associating to each real number another sequence of integers. A numbers continued fraction expansion will be eventually periodic precisely when the number is a quadratic irrational (i.e., a number involving a square root). But what about other types of numbers, such as cube roots, fourth roots and other algebraic numbers? Is there some way of expressing real numbers as a sequence of integers such that periodicity is equivalent to being a cubic irrational or some other type of algebraic number? This is the Hermite Problem, which, in 1848, Hermite posed to Jacobi. While there have been many attempts to develop algorithms to solve this problem over the years, it is still quite open, and is, in fact, related to a number of different areas of mathematics. This talk will be about some recent geometric attempts to solve the Hermite Problem.

February 2

## Parametric Sets and Virtual Classes

Dana S. ScottCarnegie Mellon University

Abstract: In an axiomatic development of geometry, there is much convenience to be found in treating various loci as sets. Thus, a line corresponds to the set of all points lying on the line; a circle, to the set of all points on the circumference. Moreover, sets of sets are natural, say in considering pencils of lines or circles or conics. And families of pencils are used as well. Does geometry need a full set theory, therefore? In giving a negative answer, we shall consider higher-type sets introduced by parametric definitions with just finite lists of points as parameters. We will show how to formulate a simple axiomatization for such sets together with a notation for virtual classes. The objective is to have the USE of set-theoretical notations without the ONTOLOGY of higher-type logic or Zermelo- Fraenkel set theory.

February 9

## Some Models for Ecological Invasions

Hans WeinbergerUniversity of Minnesota

Abstract: This lecture will introduce some properties of the solutions of a class of multispecies systems which model ecological invasions. The basic form of the system is a discrete-time recursion un + 1 = Q [un].

Here Q is a translation-invariant monotone nonlinear operator on the class of continuous nonnegative vector-valued functions u(x) on an infinite habitat. If there is only one space dimension, then under a few conditions on Q it is shown that such a system has two asymptotic spreading speeds c∗ ≤ c₊∗ with the property that for any solution un whose initial data u₀ vanish for all sufficiently large x no component spreads more slowly than c∗, no component spreads more rapidly than c₊∗, and these bounds are sharp. The existence of traveling wave solutions w( x-nc ) for every cc∗ and the extension to higher space dimensions will also be discussed.

February 15

## Mathematical Models and the Transmission Dynamics of Tuberculosis

Carlos Castillo-ChavezArizona State University

Abstract: This lecture focuses on the joint research that I have carried out on the transmission dynamics of tuberculosis (TB) over the past 15 years. First a brief review on the use of models in the evaluation of epidemiological factors on the transmission dynamics (at the population level) of TB is provided. Efforts to explain the observed high levels of of latent TB and relatively low levels of active TB (using mathematical models) are presented. The presentation concludes with a quick glance at optimal vaccination strategies in the context of the study of communicable diseases like TB.

Note: Wednesday special day!

March 13

## Approximating Pi

Michael JoyceTulane University

Abstract: People are often fascinated by pi. Learning the digits of pi is an obsession for some. But just how do we figure out the digits in pi's decimal expansion? And what about other ways to approximate pi? In this talk we will go over just some of the many ways that mathematicians have thought about pi and tried to wrap their minds around this enigmatic transcendental number.

Note: Special "Pi Days" Colloquium

March 16

## The Accuracy of Finite Difference Methods for Problems with Interfaces

Tom BealeDuke University

Abstract: For problems formulated with differential equations, it is convenient to find numerical solutions at points on a regular rectangular grid. However, often the region of interest is irregular or changing. Some methods try to retain the simplicity of rectangular grids by making special corrections at a boundary. We will describe certain methods of this type, introduced by A. Mayo, R. LeVeque, Z. Li and others, for problems with interfaces, e.g., transmission through different media or a boundary separating two fluids. We will describe error analysis of these methods using an approach based on a discrete version of basic ideas in the theory of elliptic partial differential equations. We will review these ideas as needed. With grid spacing h, the error in the solution can be uniformly of order h2, even if the error in the problem near the interface is of order h. We will see where this gain comes from. A few applications will be discussed.

March 23

## Classical Stone Duality for Classical Spaces

Achim JungThe University of Birmingham

Abstract: In 1936/37 Marshall Stone published his now-famous papers on the representation of Boolean algebras as fields of sets. In order to capture algebra homomorphisms as well, he equipped the representing set with a topology, thus demonstrating that point-set topology had applications outside analysis.

Surprising and fruitful as this discovery proved to be, one cannot deny that the topological spaces that arise as duals of Boolean algebras are rather "unusual", to be precise, they are totally disconnected (and compact Hausdorff). If one is interested in linking more common topological spaces with algebra then locale theory is the answer but the algebraic structures ("frames") are infinitary.

Against this well-established background, it is perhaps surprising that it is possible to link all compact Hausdorff spaces to finitary algebraic structures, very similar to Stone's original work. Even more surprising, perhaps, is the fact that this was discovered by working on the denotaional semantics of programming languages.

In this talk, I will re-trace the line of research that led to this and other Stone-type dualities, and also explain the link to locale theory, logic, and computer science. Time permitting, I will report on more recent joint work with Drew Moshier on dualizing objects for these dualities which suggests that the correct general setting ought to be that of bitopological spaces.

March 30

## Cantor and Serpinski, Julia and Fatou: Crazy Topology and Complex Dynamics

Bob DeVaneyBoston University

Abstract: Topologists are used to looking at such complicated planar sets as Cantor bouquets, indecomposable continua, and Sierpinski curves. Each of these spaces has some very interesting and almost counter-intuitive properties. Often these spaces are considered as ``exceptions'' or counterexamples to specific topological constructions.

In this talk we will describe how each of these sets arise naturally and very often as the Julia sets of complex dynamical systems. We give specific examples of how infinitely many types of each of these sets arise in certain families of complex exponentials and rational maps. We also describe the rich dynamics that occur on each of these sets.

April 6

## Nonlinear Finite Element Approximation for First-Order PDEs in L1¹

Jean-Luc GuermondTexas A & M

Abstract: An approximation technique for solving & first-order PDE’s in Lp,1 ≤ p < + ∞ is proposed. The method is a generalization of the Least-Squares method to non-Hilbertian settings. A priori and a posteriori error estimates are proven for well-posed linear problems. The results are extended to linear problems equipped with ill-posed boundary conditions.

The method extends to Hamilton-Jacobi as well. It is shown that for convex a Hamiltonian, the L1-approximate solution converges to the unique viscosity solution. This result holds on regular finite element meshs in dimension two using piecewise polynomials of arbitrary degree.

Numerical tests on nonlinear transport equations in L1 show that this type of technique can handle discontinuities without resorting to limiting procedures or upwinding.

April 20

## The Writing Gambit in Math: A Small Success Story

Gerard EastSouthwestern Oklahoma State University

Abstract: In a Technology in Math class at a regional university in western Oklahoma, the use of TeX had some unanticipated results in improving students' math writing skills. This example serves to illustrate several ideas in working with students in the setting of a regional university.

Note: Special Colloquium

April 24

## Creating a Curriculum in a Vacuum: The Strange and Terrible Saga of an Innovative, New Minority Mathematics Program Doomed to Failure in the Land of Enchantment

Gregg TurnerBoise State University

Abstract: In the fall of 2000, I was hired as an assistant professor in the mathematics department at New Mexico Highlands University, a small, regional public Hispanic-serving university in Las Vegas, New Mexico. Despite two competing offers from two other colleges at that time, I was attracted to the challenge of devising a rigorous, new high-level mathematics program at NMHU with the intent of recruiting talented Hispanic students in California and throughout the southwest (the local public school system in San Miguel County serving Highlands inadequately prepared its students for any semblance of continued study of science and math at the college level; when I arrived at Highlands we had only 4 math majors on board). I have a background in mathematics education as well as specialization in curriculum development in support of students of under-represented populations. Four years of arduous planning, creation of numerous new course syllabi, uncountable meetings with the school’s presidents (there were four, if you can believe, over a five year period !) and the Board of Regents, as well as close collaboration with a valued mentor, Professor Uri Treisman (Director of the Charles A. Dana Center in Austin, TX and one of the most respected mathematics education authorities in the country), paved the way for an auspicious debut, targeted for the Fall of 2005. We called the program, Computer and Mathematical Modeling. It was designed as a five-year interdisciplinary course of study culminating in a Bachelor of Science degree in both computer science and mathematics. I cemented commitment from the Los Alamos National Laboratory to sign on as an active partner; they enthusiastically agreed to sponsor paid student summer internships. And the NSF had indicated approval of a \$2 million grant in support of the program. I recruited affiliation from over 15 community colleges throughout the southwest and (at last count) 120 Hispanic students with mathematical and/or computer science aptitude in California had expressed interest in jumping ship from suddenly pricey UC campuses and giving our curriculum a try.

And then Governor Bill Richardson nudged the appointment of a 30-year political crony in the state legislature with no academic background in his past to become the school’s president in the summer of 2004.

Note: Special Colloquium

April 27

## Deriving Secrecy and Authentication in Key Establishment Protocols

Abstract: A key establishment protocol is a communication protocol by which principals obtain cryptographic keys in order to communicate securely. They are difficult to design correctly; one of the reasons is the mutually dependent nature of secrecy and authentication. Every authentication is based on some secrets, and every secret must be authenticated. This interdependency is a significant source of complexity in reasoning about security. We describe a method to simplify it, by encapsulating the authenticity assumptions needed in the proofs of secrecy and the secrecy assumptions needed in the proofs of authentication. While logically straightforward, this idea of encapsulation in general, and the present treatment of secrecy and authentication in particular, allow formulating scalable and reusable reasoning patterns about the families of protocols of practical interest. The approach came about as a design detail in an ongoing development effort towards Protocol Derivation Assistant (Pda), a semantically based environment and toolkit for derivational approach to security.

Note: Math Month Colloqiuim Talk

June 8

## Mechanical and Geometrical Aspects of Uterine Contractions

Angela GallegosOccidental College

Abstract: Uterine contractions play a critical role in labor and parturition and are an important factor in female reproductive health. However, how uterine contractions are regulated and controlled is not well understood. I analyze how structural components such as muscle layer arrangement and muscle wall geometry affect uterine contractions, utilizing a theoretical framework that makes the force-balance assumption of quasi-static equilibrium. Linear elasticity equations of passive deformation are also considered in order to investigate both non-pregnant and pregnant cases. Numerical analysis indicates that the existent muscle layer is necessary to achieve wall movement in the non-pregnant uterus with realistic forces as well as to minimize movement outside of the uterus organ. Model results indicate that a cylinder is a reasonable approximation to the uterus in the non-pregnant state. Other simulations indicate that the ellipsoid is likely a better approximation than the sphere to the term pregnant uterus and that the process of cervical dilation during labor is mechanically self reinforcing. I discuss biological implications of these results and propose future extensions to the work presented.

Note: Located in Gibson 308

June 15

## Theoretical, Analytical and Computational Studies of Convection and Turbulence in a Fluid Saturated Porous Layer

Charles DoeringUniversity of Michigan

Abstract: Thermal convection is the buoyancy driven flow resulting when a fluid is heated from below and cooled from above. Convection in porous media is relevant to a variety of phenomena ranging from geothermal energy transport to fiberglass insulation. In this presentation we provide some general background as well as describing some current theoretical and computational research where the Darcy-Boussinesq equations are used to study convection and heat transport over a broad range of heating levels as measured by the nondimensional Rayleigh number. High resolution direct numerical simulations are performed to explore the modes of convection and measure the heat transport as quantified by the Nusselt number, the enhancement factor of total heat flux over pure conduction alone. Rigorous upper estimates on the high Rayleigh number heat transport have been derived and at high Rayleigh numbers they are of the scaling form predicted by Howard's classical marginally stable boundary layer argument. The bounds are compared directly to the results of the simulations as well as to real laboratory experiments. This is joint work with (former) students Jesse Otero and Lubomira A. Dontcheva, and (former) postdocs Hans Johnston, Rodney A. Worthing, Alexander Kurganov and Guergana Petrova. It is the content of a paper published in Journal of Fluid Mechanics.

Note: Located in Gibson 308

June 29

## Local Post-Processing for the Discontinuous Galerkin Method and Applications

Jennifer K. RyanVirginia Tech

Abstract: In this presentation an overview of post-processing for discontinuous Galerkin methods will be given. This accuracy enhancement technique has been shown to improve discontinuous Galerkin approximations from order k+1 to order 2k+1 for linear hyperbolic equations, where k is the highest polynomial used in the approximation. This talk will focus on extensions of this technique to include nonuniform meshes, post-processing near a discontinuity or boundary, and post-processing for derivatives. The specific technique that we will examine was introduced by Cockburn, Luskin, Shu, and Süli. Using a negative norm estimate along with previous results of Bramble and Schatz as well as Mock and Lax, Cockburn et al. are able to show that the order of accuracy of the approximation can almost be doubled. Additionally, a uniform mesh assumption allows for simple implementation via small matrix-vector multiplications making the application of this post-processor attractive. A discussion of implementation issues surrounding previously used assumptions of the post-processor as well as extension to a variety of problems will be given. Further, applications in aero-acoustics as well as visualization of streamlines will be explored.

Note: Located in Gibson 308

July 13

## Why Eisenstein Proved the Eisenstein Critereon and Why Schoenemann Discovered It First

David CoxAmherst University

Abstract: Although we all learn the Eisenstein Irreducibility Criterion in abstract algebra, Eisenstein was not the first person to discover this wonderful result. My lecture will explore topics from 19th century number theory, beginning with Gauss's version of Gauss's Lemma and culminating with the two very different problems that led Eisenstein and Schoenemann to discover their irreducibility criterion. Along the way, we will discuss topics ranging from finite fields to straightedge and compass constructions on the lemniscate.

Note: Located in Gibson 308

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