Today, non-commutative projective geometry is an independent area of research, yet it grew from a need to find new tools for understanding algebras arising in quantum physics.  In this talk, the use of non-commutative projective geometry in physics and algebra will be outlined, together with the history of its development.  Some geometry and linear algebra background will be assumed.

Thursday, 11 May 2000
4:10 p.m. in Skaggs 114
Coffee/treats at 3:30 p.m. Math 104 (lounge)

The theory of singularly perturbed differential equations aims to establish the existence of a solution and to determine its asymptotic behavior with respect to the perturbation parameter by means of the corresponding degenerate and associated equation.  The classical approach by N. Tikhonov and N. Levinson and also the geometric theory due to N. Fenichel are based on the crucial assumption that the considered solution of the degenerate equation that represents a family of equilibria of the associated equation does not exhibit an exchange of stability.  This talk represents an extension of the classical approach to the case of exchange of stability where the phenomena of immediate and delayed exchange of stability can be observed.  Our approach is based on the method of asymptotic lower and upper solutions.  We show that this method can be successfully applied to singularly perturbed problems for partial differential equations.

Thursday, 4 May 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (lounge)

To confine a high temperature fusion plasma, magnetic fields must be used since any material wall would be vaporized by contact with a plasma with a temperature greater than 10,000,000 K.  The most promising configurations for a magnetic fusion reactor are toroidal with magnetic field lines mapping out a set of nested toroidal surfaces.  The charged plasma particles, ions and electrons, would ideally stay close to a magnetic surface.  However, the curvature and toroidicity of the magnetic field lines causes the particles to drift away from the surface and some particles are lost from the plasma.  In order to “let the physics design the experiment,” an optimization code has been developed that selects a configuration which minimizes particle losses, maintains plasma stability, and achieves other desired targets.  The task often involves optimizing with 40+ independent variables and thousands of dependent constraints.  This project is a collaboration between researchers at The University of Montana and Oak Ridge National Laboratory.  I will present an overview of this research and some of the surprising and promising results.

Thursday, 13 April 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (lounge)

Matroids are everywhere.  Vector spaces are matroids.  Matroids are useful in situations that are modelled by both graphs and matrices.  Bicircular matroids model generalized network flow problems whose algorithms are more efficient than those available for general linear programming codes.

Let G be a graph (loops and parallel edges allowed) with vertex set V = {1,2, …, n} and edge set E.  In the classical matroid associated with a graph, a set of edges is independent in the matroid if it contains no cycles, and the circuits of the matroid are the single cycles of G.  The bicircular matroid of G is the matroid B(G) defined on E whose circuits are the subgraphs which are subdivisions of one of the following graphs:

The bicircular matroid is known to be a transversal matroid and thus can be represented by a family of sets, called a presentation.  It has been known for some time that the maximal presentation of a transversal matroid is unique.  In general, a transversal matroid has many  minimal presentations.  We show how, given a graph G, one could find a minimal presentation for B(G) and, from that, find all other minimal presentations.

We demonstrate this with the graph

called a wheel on six vertices.  The problem reduces to searching for trees and single cycles.  This example led to the general classification of the minimal presentations of an arbitrary bicircular matroid.

Thursday, 6 April 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (lounge)

When you encounter a quadratic function, do you "see" a parabola?  When you encounter 5² + 12² = 13², do you "see" a right triangle?  In English, "to see" often means "to understand," and I maintain that this is frequently true in mathematics as well.  To make this point, we will examine visually a combinatorial principle (Fubini), and in the process rediscover some interesting facts about sequences of 'figurate" numbers (squares, triangles, hexagons, cubes, etc.) and some of their partial sums.

Monday, 3 April 2000
4:10 p.m. in the Skaggs Building 114
Coffee/treats at 3:30 p.m. Math 104 (lounge)

Sequences (a_n) of complex numbers (usually integers) indexed by lattice points in the positive orthant of R^d are ubiquitous in mathematics, particularly in counting problems and discrete probability.  For many purposes, it is desirable to understand the asymptotic behavior of a_n as n approaches infinity.

Analytic methods involving the study of the sequence's generating function have proven to be very powerful when d=1.  Surprisingly, analogous techniques for several variables are almost entirely missing from the literature.

I will report on a major ongoing project with Robin Pemantle (Ohio State) which aims to create a decent multivariate theory.  I will outline the basic problem, our approach and some results to date.  Several examples will be discussed.

Thursday, 30 March 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (lounge)

Evolutionary computation algorithms are based on the simulated evolution of a population of potential solutions to a problem.  They have been applied to a very wide variety of practical problems.  The behavior of these algorithms can be modeled both as dynamical systems and as Markov chains.

These algorithms can be categorized into generational, where the entire population is replaced at each time step, and steady state, where only a single individual is replaced per time step.  Previous dynamical system models have been discrete-time models of generational evolutionary algorithms.  I will describe how a continuous-time dynamical system model of a steady state evolutionary algorithm can be derived by letting the population size go to infinity while the time step goes to zero.  It turns out that the discrete-time system is the Euler approximation to the continuous-time system.  Then it is shown how the continuous-time system might be stable while the discrete-time system is unstable.

Thursday, 16 March 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (lounge)

Compartmental models provide an attractive means of studying biological fluid dynamics.  In this modeling approach, fluid and matter within the system being investigated are subdivided into a number of interacting subunits (compartments).  The dynamics in each compartment are specified by spatially-averaged, time-dependent functions.  This spatial averaging results in a system of ordinary, differential equations describing the temporal evolution of compartmental properties such as pressure, volume, and fluid discharge.  The history of this technique, in the study of intracranial pressure dynamics, dates back to the late 1700's. The first half of this talk will give a general overview of this approach with some examples of the types of problems I have been involved with over the past few years.  The second half will be devoted to a specific example of a relatively simple model describing the pressure-volume relationship of the intracranial cerebrospinal fluid.

Wednesday, 15 March 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (lounge)

The theory of operator spaces and completely bounded maps is receiving a great deal of attention at present as a new category for the study of Functional Analysis.  This area, whose key ideas come from the theory of operator algebras, allows for the extension of many results known for bilinear maps to multilinear maps of the appropriate type.  We will present they key concepts and then apply them to a problem in Harmonic Analysis.  Namely, given a finite collection of locally compact groups G, H, . . . , K, we will describe how they Banach space of completely bounded, complex-valued multilinear forms on the product C_O(G) x . . . x C_O(K) can be given the structure of a convolution Banach algebra, extending the classical convolution of measures on G x . . . x K.

Wednesday, 15 March 2000
3:10 p.m. in Math 311
Coffee/treats at 2:30 p.m. Math 104 (lounge)

The set of three reaction-diffusion equations describing the time-space behavior of the intermediate chemical species in the Oregonator model of the Belousov-Zhabotinsky reaction is investigated in an open, gel-disk reactor in one and two spatial dimensions.  Numerical simulations using equal values of the three diffusion coefficients indicate the presence of solutions corresponding to large-amplitude, apparently stable, stationary concentration patterns.  The requirement of differential transport rates of chemical activator and inhibitor species for the development of stable patterns is apparently met in this system by differential exchange rates with the reservoir(s) rather than by differential diffusion rates within the gel reactor.  The characteristics of these patterns as well as their stability and bifurcation properties are investigated and suggest that their appearance is dependent upon the existence of bistability in the homogeneous reaction kinetics.  The patterns have an intrinsic wavelength, and one of a particular wave-number destabilizes via a Hopf bifurcation as the length of the gel-reactor is varied, giving rise to oscillatory breather solutions past the bifurcation but before decomposition into a spatially homogeneous state occurs.  The relationship of these results to experimental systems, as well as an analogy to biological membranes, is discussed.

Thursday, 9 March 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (lounge)

There are many problems in engineering and scientific disciplines that can be described by convection-diffusion equations.  In a convection-diffusion equation, if the convection term is very dominant, the linear system of equations that result from either finite-differencing or finite elementing will not have a diagonally dominant coefficient matrix.  So, if one tries a convectional iteration method (Jacobi or Gauss-Seidel) to solve the linear system of equations, the iteration matrix may not satisfy the spectral radius condition for convergence and hence no converging solution may be obtained.  The problem can be overcome, under certain conditions, if one uses a two-step iteration procedure involving the spectral enveloping ellipse for the iteration matrix.  The talk will present such a two step method that combines an Arnoldi-Chebyshev approach for convection-diffusion computations, that generate faster and better solutions.   A domain decomposition method for solving convection diffusion problems will be discussed.  Finally, a finite difference singular perturbation technique for solving problems with boundary layers, will be presented.

Monday, 28 February 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (lounge)

The gas diffusion electrode is a critical component of the proton exchange membrane fuel cell.  Electrodes are composed of a highly porous material that serves to distribute reactant gases uniformly to the active catalyst sites.  We develop a mathematical model for flow of a binary gas mixture in a porous medium which consists of a coupled system of nonlinear parabolic differential equations: a porous medium equation for the evolution of the gas mixture; and a singularly perturbed convection-diffusion equation for interspecies mass transfer within the mixture.  The two equations are supplemented by a set of nonlinear boundary conditions that describe consumption of reactants and generation of end products at the catalyst layer.

Using a multi-scale asymptotic expansion, we obtain a reduced system of equations that captures the slow, diffusively-driven, adiabatic relaxation to the steady state at each electrode.  The asymptotic results are compared with computations of the full system.  We also present numerical simulations that show how fuel cell performance can be optimized by varying electrode geometry and material parameters.

Tuesday, 22 February 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (lounge)

We consider a uniform class of information transformers as a family of morphisms of a category that satisfies a certain set of axioms.  The talk defines basic concepts for information transformers and studies their main properties in terms of categories of information transformers.  In particular it generalizes the Bayesian approach to decision-making problems.  It also introduces two different approaches to comparison of informativeness of information transformers and investigates their interrelations.  This topic incorporates a unified algebraic approach to statistics, fuzzy decision making, etc., in a variety of applied problems.  Several examples of concrete categories of information transformers are presented.

Thursday, 10 February 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (lounge)

A simple theorem turns out to have some surprising consequences.

This talk is sponsored by and presented as an element of The University of Montana President’s Lecture Series.  The Big Sky Conference is also sponsored by the National Science Foundation and the Department of Mathematical Sciences.

Thursday, 7 September 2000
4:10 p.m. in James E Todd Building (Cont. Ed.) 203-204
Refreshments at 3:30 p.m. in CE 204

Historically, one of the most profound influences on the development of matroid theory has been the interaction between graphs and matroids. Indeed, a powerful statement of the strength of this link is contained in the following words of Tutte: “If a theorem about graphs can be expressed in terms of edges and circuits alone it probably exemplifies a more general theorem about matroids.” This talk will survey some recent extremal results for  graphs and matroids that illustrate how the subjects of graph theory and matroid theory are mutually enriching.

This talk is sponsored by The Big Sky Conference. The Big Sky Conference is sponsored by the National Science Foundation and the Department of Mathematical Sciences.

Friday, 8 September 2000
4:10 p.m. in Skaggs 117
Refreshments at 3:30 p.m. in foyer of Skaggs

For this distribution the moments or integrals do not exist that allow classical estimation of parameters. So one can revert to reliable nonparametric methods that are distribution free over the whole class of bivariate distributions that are  elliptically symmetric; this includes the normal distribution and all the Student t distributions. The Greatest Deviation Correlation Coefficient will be explained and used. Most real data has a number of questionable data points and a method of regression that works on the Cauchy Regression can then be reliably used on real data without the data analyst worrying so much about the effect of “outliers”. The robustness of the method will be demonstrated by an example. PowerPoint will be used to demonstrate the geometric method of defining the Greatest Deviation Correlation Coefficient. Quantile plots are explained because they are a necessary tool for this method allowing scale factors to be estimated.

Thursday, 14 September 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Mathematics is found in leisure reading whether you read comic strips, romance novels, science fiction, or detective/spy novels. What mathematics is used and how that mathematics is used is a reflection of the current culture and helps to define student and adult attitudes in today’s world. The view of mathematics in popular literature is not necessarily the view of the mathematician. Do we need to try and bridge the gap, or do we view this bit of ethnography as unimportant? Just because they all speak the same language, are mathematicians and non-mathematicians able to discuss mathematical topics with understanding?

Thursday, 21 September 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Motivation:  In order to develop advanced techniques of repair and more biologically correct instruments of replacement, further study of the aortic valve structure and function is essential.  The analysis focuses on the left ventricular myocardium, the aortic valve and the ascending aorta working together as a unit.

Method:  Digital sonomicrometry yields data which illustrates the cardiac cycle in two and three-dimensional manners.  The key benefit of the method is a high frequency sampling rate (200 Hz) which provides measurements that are yet even closer to the continuous nature of the cycle.  Fifteen crystals are placed at specific points in the structure.  These crystals report values of distance between each other, which are used to form geometric shapes that are continually changing in size.

Goals:  Analysis of the data will include both summary statistics and continuous models that will describe the behavior of various locations within the structure.  Relationships between location behavior and pressure (LV and AA) will be analyzed.  Ideally, a more advanced understanding of the structure's behavior will lead to successful surgical techniques.

Thursday, 28 September 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Kazan State University is the third largest University in today’s Russia (after Moscow State and St. Petersburg State Universities). It is also one of the oldest universities in Russia, founded by a special order of Emperor Alexander I of Russia in 1804. Kazan State University has a glorious past. It was the place where Lobachevsky developed his non-Euclidean geometry. It was also the Alma Mater of many famous people (Lenin including). Many fundamental results were obtained, and many scientific discoveries were made in this university throughout the years.

How is Kazan State University doing now? What is the life of mathematical science like there? What are Kazan mathematicians interested in now?

These are some of the questions to be considered during this talk. Based on several particular results of Kazan mathematicians from mathematical logics, probability, function theory, modern algebra, algebraic geometry and C*-algebras, a survey on current mathematics developments at Kazan State University will be presented.

Thursday, 5 October 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

The emptying time of a reservoir of arbitrary shape is computed as a function of the initial volume and height of liquid. Various reservoir forms are compared in terms of their emptying efficiency. It is shown that there is no reservoir with the minimal emptying time.

Thursday, 12 October 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Sedimentation velocity experiment is a widely used method to analyze biochemical systems using the analytical ultracentrifuge. This instrument subjects a small volume of solution to a precisely controlled centrifugal force, and records by means of self-contained and photographic systems the concentration distribution produced in the solution. The problem of estimating the parameters associated with the solution is an inverse problem where the (unknown) parameters are determined from the experimental concentration data of the solution (Protein or DNA) generated by the ultracentrifuge. Those parameters are used to analyze the system in obtaining critical information such as composition, molecular weight, identification of multiple components, and determination of binding coefficients of interacting macromolecules.

The problem, its numerical solution, and the developed software will be discussed.

Thursday, 19 October 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

The theme of this talk is a new hybrid of statistical classification rules used for land cover mapping.  A land cover type map identifies the dominant vegetation, or land surface types for a region.  A Landsat TM land cover map is produced from a base map consisting of as many as 1 million polygons, each of which belongs to an unknown land cover type.  Usually, land cover type is assigned to polygons using a classification rule constructed from remotely sensed variables and a training set obtained from ground sampling.   Spatial classifiers are classification rules that use spatial information extracted from the training set in addition to remotely sensed variables.  Optimally, the distribution of sample locations should cover the map region; however, good coverage often is not possible.  With incomplete coverage, accuracy is compromised and estimates of map accuracy are highly suspect.  This talk discusses the statistical analysis of spatial classifiers for a Landsat TM mapping problem with grossly poor spatial coverage.

Thursday, 2 November 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Presentation will give a historical investigation of the various methods used to determine the circumference of the earth.

Is the shape of the earth a sphere, an ellipsoid, or more like a lump of coal and how do we know for sure?

Thursday, 9 November 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

This talk will report on my adventures to The 9th International Congress on Mathematics Education (ICME 9) held July 31-August 6, 2000 in Tokyo, Japan. At this conference, nations from throughout the world presented their mathematics programs to an international audience of mathematics educators and mathematicians. Because the Asian countries scored the highest of all the countries on the Third International Mathematics and Science Study (TIMSS), I chose to focus on the programs of the Asian countries, especially Japan. The Japanese were very concerned that their students did not like mathematics and also that their students were good in computation but weak in reasoning and communication. There was also concern for Japanese students that were suffering from "examination hell".

I wanted to study the present curricula of the Asian countries and also explore the directions they might go in the future. In this presentation I hope to share my insights and make some suggestions for mathematics education in the United States.

Thursday, 9 November 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

As more information becomes available about nearly every phase of life, it has become more important than ever to prepare students to manage data and to use it to make intelligent decisions.  How does this affect the role of statistics at the university level and school levels?  There has been a shift in both statistical content and in how statistics is taught. This shift in part reflects the increasing importance of statistics and the need to have a statistics program that is relevant to the things that graduates need to know and be able to do with data.   In addition, an increasing number of students are taking Advanced Placement Statistics, which signals a different entry level for incoming students.  What does this mean for university programs in both mathematics and statistics?  We don't have the answers to these questions, but there are some indicators that might provide useful direction.

Thursday, 7 December 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

A model of a `flexible manufacturing system' under such scheduling policies as `clearing' or `first come, first served' gives an interesting class of dynamical systems. Formerly, it was conjectured that a simple capacity condition sufficed to ensure stability [boundedness, uniform in t, for the state for any initial conditions]. However, it has been possible to construct examples, for each of these policies, which are unstable even though the natural capacity condition is satisfied. On the other hand, for simple flow geometries it is possible to show stability (and estimate performance) for a variety of simple control policies.

Thursday, 14 December 2000
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)