# 2002 Colloquia

## Spring 2002

The Effect Of Information On A Stochastic Fishery
Greg Cripe
University of Montana

In recent years there has been increasing recognition of the impact of environmental fluctuations on major marine fisheries worldwide. The management of a fish stock is complicated by the resulting uncertainties. The goal of this talk is to provide a better understanding of the impact of knowledge on a model of stochastic fisheries. The intent is to ascertain whether earlier and more accurate information regarding the environmental conditions might allow for better management of the resource. This is indeed the case when a single manager controls the resource, however we will show this is not necessarily true when the stock is harvested by two or more competing interests.

Thursday, 9 May 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Inside Math Excel - A Video Visit
Professor Tom Dick
Oregon State University

High failure rates in introductory college mathematics courses have been of concern for many years, particularly among underrepresented groups of students. One approach to the problem that has experienced some success has been Treisman's Emerging Scholars workshop model. The model involves supplemental workshops where students solve problems in collaborative learning groups. Math Excel is an implementation of Treisman's model now in place at several institutions across the country, including Oregon State University. In this presentation, we'll provide some background on the model and report on research supporting its effectiveness. We will also drop in (by videotape) to visit some actual Math Excel classes and hear from students and instructional leaders about what makes Math Excel work.

Thursday, 2 May 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Asymptotic Analysis of a Fast Reaction Outside a Solid Sphere in a Creeping Flow
Supawan Lertskrai
University of Montana
In partial fulfillment of the requirements for a doctoral degree

A problem that combines mass transfer of an active chemical species through the surface of a solid sphere located in an axially symmetric, incompressible slow flow and a fast reaction outside a sphere is considered. The solute from the interior of the sphere diffuses through the sphere's surface and enters the liquid. Due to the presence of a fast reaction, the problem contains a small parameter which makes the problem singularly perturbed. The asymptotic approximation of the solution is derived using the Boundary Function Method, and compared with numerical solution graphically.

Thursday, 25 April 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

The Instantaneous Pressure-Volume Relationship of the Left Ventricle
Professor Scott A. Stevens
University of Montana

Much information about cardiac output and efficiency may be obtained from the instantaneous pressure-volume curve associated with the left ventricle. In this talk I will describe this curve at the various stages of the cardiac cycle and its relevance in ongoing research at the International Heart Institute of Montana. A major hurdle in such research is the difficulty in measuring and/or accurately approximating ventricular volume. Most techniques are either very intrusive or inaccurate. As a way of testing volume approximation techniques, comparisons are made with very accurate approximations of ventricular volume differences during systole. It turns out that the technique for testing the volume approximations may be more valuable to the medical community than the approximations themselves.

Thursday, 18 April 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Dreamweaver
Jesse Neidigh

Keeping current information on in Internet is vital as more and more people use it to seek various information. Increasing numbers of students are using the Internet to look for information about their instructors' office hours and office locations, as well as information about the classes they are taking and classes they desire to take.

Purpose: To discuss standardization and advancement of the Mathematics Department Faculty web pages. To demonstrate the Dreamweaver web editing software and give brief instructions and examples of its use in web editing and updating web pages. Key focuses will be the demonstration of editing a template for faculty web pages and a collaborative discussion to create a list of instructions and tips for updating web sites, as well as discussing the creation of different templates.

Thursday, 4 April 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Cancelled

Hopf Algebras: An Introduction
Professor Serban Raianu
Algebra Candidate
Syracuse University

Hopf algebras are algebraic structures which were first discovered in topology in the early '40s. Since then, they proved to be one of the most ubiquitous objects in mathematics: they arise in virtually every field, from number theory and algebraic geometry to quantum physics, passing through ring theory, probability theory, representation theory, Lie theory, operator theory, combinatorics, computer science, and many other. Some basic examples of Hopf algebras will be given, along with the description of some problems to be studied in connection with Hopf algebras. Concrete results, some of them very recent, will be used to illustrate the diversity of the subject and some of the techniques used.

Monday, 11 March 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

An Introduction to Noncommutative Geometry
Algebra Candidate
Pomona College

Given a system of polynomial equations (in n unknowns) with real coefficients,

f1 (x1,...,xn)= ... fr(x1,...,xn)=0

can we find all real d x solutions, i.e. can we find all n-tuples of real d x d matrices M1,...,Mn  such that

f1(M1,...,Mn)= ... =fr(M1,...,Mn)=0?

When d = 1 , solutions are elements of Rn. The set of all solutions is a geometric object called a variety. Algebraic geometry is the study of the interplay between the geometry of the variety and the nature of the polynomials f1,...,fr

When d >1, it is often true that MN ≠ NM  for d x d matrices M and N, so in this case, our equations are "noncommutative". Is there still a bridge between the worlds of algebra and geometry? We describe recent efforts to make sense of the notion "noncommutative variety". We shall see that, while some important noncommutative varieties don't have any points, they can be embedded in slightly larger spaces which have enough points so that they can be understood geometrically.

Friday, 8 March 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Analysis of the Distribution of Grades from the Fall 2001 Calculus 152 Final Examination and
Nonparametric Estimation of a Linear Relationship for Bivariate Data
Professor Rudy Gideon
The University of Montana

The Correlation Principle for the estimation of a parameter theta: A statistical inference or procedure should be consistent with the assumption that any explanation of a set of data should be accompanied by theta-hat, a value of theta, that makes some correlation coefficient zero.

Although extensive methods have been developed for linear models, generalized linear models, non-linear models, and time series models, as well as estimation of parameters for a particular distribution, we only have time to give one result. A nonparametric correlation coefficient measures monotonicity rather than linearity. It will be shown how to measure linearity with any nonparametric correlation coefficient. The Greatest Deviation correlation coefficient (GD) will be used and its robustness demonstrated.

Thursday, 28 February 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

On Universal Cycles of k-sets of an n-set
Professor Brett Stevens
Carlton University

In 1992 Chung, Diaconis and Graham wrote the readable and thoroughly enjoyable "Universal Cycles for Combinatorial Structures". In it they generalize both the definition and construction of de Bruijn cycles to other families of combinatorial objects: permutations, partitions and subset systems. These generalizations resonate with generalizations of Gray Code, being gray codes that are compatible with queue data structures. Hurlbert and Jackson have continued this work solving, among other families, universal cycles for k-sets of n-sets for k = 2, 3, 6 and partially for other k. In their empirical work it was noted and conjectured that a universal cycle for the n – 2-sets of an n-set never exists, even thought the standard necessary conditions are satisfied for all odd n. This was recently proved by Stevens et al. This talk will review the past work, this recent result and the future look at Gray codes or de Bruijn generalizations compatible with different data structures.

Thursday, 14 February 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Is There a Shoe Polish with Fluoride?
Johnny W. Lott
President Elect of the National Council of Teachers of Mathematics

The President Elect of NCTM is thrust into national issues almost from the time of the election announcement. This year the issues below have risen as challenges for NCTM and me.

Mathematics in Doctoral Education Programs?
Quantitative Literacy?
National Testing Programs?
Articulation Programs?
Scores of Minority Students?

How does one develop a response to these issues? Some of them could have implications for The University of Montana Department of Mathematical Sciences. What are the issues, and how will we respond?

Thursday, 7 February 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Graphs in the Theory of Location of Facilities
K. B. Reid
California State University San Marcos

The theory of location of facilities in networks combines tools from graph theory, basic analysis, optimization, and complexity theory. The central issue is the study of the optimal location(s) of a facility such as emergency installation, a supply depot, a switching center, a pumping station, an obnoxious dump, a communications center, or the like in a network such as a street or road network, an electrical network, a network of channels or pipes, a communications network or the like. Optimality depends on criteria usually involving some idea of distance and varies according to the application. Weighted graphs, often referred to as networks, provide a context for studying these types of problems, where vertices and edges are assigned weights representing certain parameters according to the application. Usually, special sets of points in the network are sought that are either "central" or "peripheral." Results range from the descriptions of optimal locations to the computational difficulty in actually determining these optimal locations. Considerable study has been focused on weighted trees. These issues have motivated graph theorists to probe many different notions of centrality and notions of the "outer fringes" in ordinary (unweighted) graphs, particularly trees. In such models, users and facility locations are thought to be restricted to vertices. However, the graph theoretical origins of centrality precede the advent of modern location theory as C. Jordan introduced the concepts of the center of a tree and the branch weight centroid of a tree in 1869.

We will survey results concerning the structure of several "central sets" of vertices in trees, beginning with the center, the branch-weight centroid, and the median. We will discuss the (distance) balanced vertices, several one-parameter families of central sets, a two-parameter family of central sets, the cutting center, and the security centroid.

Thursday, 31 January 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

## Fall 2002

Amalgams or A Tale About Two Groups Who Wish To Become One
Dr. Arturo Magidin
University of Montana

Webster's Ninth Collegiate Dictionary defines an amalgam as a mixture of mercury and another metal, or more generally as a mixture of two different elements. Dental fillings, for example, are made of amalgams.

Mathematically, an amalgam is obtained when we take two structures (two topological spaces, two graphs, two rings), and mix them by identifying some subparts of them; The challenge then is to find a bigger structure where this mixture can live.

For example, if we have two groups G and K, and we have a subgroup H which is common to both, we want to think of G and K as being "glued together along H;" this is an amalgam of the two groups. It is not a group, because there is no way to multiply an element of G which is not in H by an element of K which is not in H. What we want is to find some larger group M, which contains G and K as subgroups in such a way that their intersection is still H. Similar problems exist if we replace "group" and "subgroup" with "topological space" and "topological subspace"; or with "graph" and "subgraph"; or with "manifold" and "submanifold", etc.

This simple question leads very easily to some very powerful mathematics, and of course to even more questions, many of which we cannot yet answer. I will give a tour of amalgams of groups, assuming nothing more than a basic knowledge of them. As opposed to getting to know what a dental amalgam is, it will not hurt at all.

Thursday, 12 December 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Toida's Conjecture is True and Other Results on the Cayley Isomorphism Problem
Dr. Joy Morris
University of Lethbridge

A Cayley graph X(G;S) where G is a group and S ⊂ G , where S is inverse-closed, is defined by taking the vertices of the graph to be the elements of G, with an edge between vertices g and g' iff g-1g'∈ S. A graph has the Cayley Isomorphism property if whenever it can be isomorphically represented as both the Cayley graph X(G;S) and X(G;S'), there is an automorphism of the group G that takes S to S'. A group has the Cayley Isomorphism property if all Cayley graphs on that group have the Cayley isomorphism property.

The Cayley Isomorphism problem is the question of which groups, and which graphs, have the Cayley Isomorphism property. This talk will provide an overview of results in this problem, leading up to the solution of Toida's conjecture: that if G is a cyclic group, and S is a subset of the units of G, then X(G;S) has the Cayley Isomorphism property.

Thursday, 5 December 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Mathematics in Estonia
Dr. Mati Abel
University of Tartu - Estonia
&

President of the Estonian Mathematical Society

Short introduction of the University of Tartu and of the Estonian Mathematical Society.

Main research areas of Mathematics in Tartu and in Tallin (the capital of Estonia).

Competitions in Mathematics for Estonian students.

Tuesday, 26 November 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

How Can the "I-Like-Ike Teacher" Motivate the
"Be-Like-Mike Student" in Middle Grades Mathematics
Dr. Rick Billstein
University of Montana

This session will discuss how mathematics education has changed over the years, what problems we are facing today, and what might be done about it in the future. The history of the development of a standards-based mathematics curriculum will be discussed and examples of some new ideas used in this curriculum will be shown. There will be my typical use of off-the-wall cartoons to illustrate points to try to keep everyone awake.

Did you know that ...

• 99 percent of lawyers give the rest a bad name
• 42.7 percent of all statistics are made up on the spot
• Half the people you know are below average
• Monday is an awful way to spend 1/7 of your life
• Black holes are where God divided by zero
• The ratio of an igloo's circumference to its diameter is Eskimo Pi
• 1 million microphones is equal to 1 megaphone

Thursday, 21 November 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

The 24th International Congress of Mathematicians (ICM 2002 - Beijing)
Dr. Thomas Tonev
University of Montana

The 24th International Congress of Mathematicians (ICM2002) was held this Summer in Beijing, China. About 5000 people from all over the world took part in this first major gathering of mathematicians for the new millennium. Twenty plenary talks, nineteen specialized sections, special talks and meetings, and forty six Satellite Conferences in various cities in China and in neighboring countries were the major features of this congress. The latest Fields medals (L. Lafforgue and V. Voevodsky) and the Nevanlinna prize (M. Sudan) were awarded. The next ICM will take place in Madrid, Spain in 2006.

In this talk I will share my impressions from the ICM2002-Beijing and from the Satellite Conference on Mathematics Education in Lhasa, Tibet.

Thursday, 10 October 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Graphs & Digraphs with Large Chromatic Numbers & Long Shortest Cycles
Dr. Mark Kayll
University of Montana

To get an idea of how extensive is the chromatic theory of graphs, one can open any graph theory book and read the chapters on coloring. To get an idea of how limited is the same theory, one can simply open a particular book, Graph Coloring Problems, a 300-page monograph listing primarily open problems in the subject. Thus we observe a dichotomy: graph coloring enjoys a rich, yet largely incomplete, theory. Since the problems are out there and the foundations are strong, interesting new theorems appear virtually every year.

In this colloquium, I'll introduce some basic graph coloring notions and explore a famous paradoxical theorem of Paul Erdös from 1959. This will set the stage for a natural extension of the chromatic number concept to directed graphs, also known as digraphs. Finally, I'll present a digraph analogue of Erdös' theorem. The proof, discovered in Slovenia last November, uses probabilistic ideas and a surprising application of a fact from basic algebra. Which fact shall remain top secret until the key moment in the talk when it is needed.

Thursday, 26 September 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

A TEXnology Tour
Dr. Karel Stroethoff
University of Montana

I will show some aspects related to type-setting mathematics, and provide a tour through the wonderful world of TEX. The TEX language, developed by Donald Knuth some twenty years ago, represents the state-of-the art in computer typesetting. I will show how TEX works and give a variety of examples of what TEX and its derivative LATEX can do. The presentation will assume no familiarity with TEX. In addition to elementary examples I will demonstrate some advanced features and show some of the latest applications of TEX, such as creating interactive documents for the world wide web and making colorful slides for presentations.

Thursday, 19 September 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Graph Products with Small Cycle Double Covers
Dr. Karen Seyffarth
University of Calgary

A cycle double cover of a graph G is a collection of cycles, C, such that every edge of G lies in precisely two cycles of C. The Small Cycle Double Cover (SCDC) Conjecture, proposed by J.A. Bondy, asserts that every simple bridgeless graph on n vertices has a cycle double cover with at most n-1 cycles, and is a strengthening of the well-known Cycle Double Cover (CDC) Conjecture.

Both the CDC Conjecture and the SCDC Conjecture have been verified for various classes of graphs, but remain open in general. The graphs for which that SCDC Conjecture has been verified all have well defined structural properties that play an important role. The structure that is inherent in graph products makes such graphs ideal special cases for which to verify the SCDC Conjecture. There are various graph products that can be considered, and in this talk I will describe some results and techniques for proving the SCDC Conjecture for certain graph products.

This talk will be accessible to a general mathematics audience: all relevant terms will be defined, and proofs will be illustrated with examples. This is joint work with R.J. Nowakowski (Dalhousie University).

Thursday, 12 September 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

## Fall 2002 Big sky Conference on Discrete Mathematics Events

Colouring Problems and Transversals in Graphs
Dr. Penny Haxell

Let G be a graph whose vertex set is partitioned into classes V1 ∪... ∪ Vi. An independent transversal of G with respect to the given classes is an independent set {v1,...,vi}  in G such that viVi for each i. We give conditions that guarantee the existence of an independent transversal in a graph with specified vertex classes, and we show how various colouring and matching problems can be addressed using these results.

Friday, 6 September 2002
4:10 p.m. in James E Todd Building 203-204
Reception at 3:30 p.m. 204

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

The Shape of Space
Dr. Jeffery Weeks

When we look out on a clear night, the universe seems infinite. Yet this infinity might be an illusion. During the first half of this presentation, computer games will introduce the concept of a "multiconnected universe". Interactive 3D graphics will then take the viewer on a tour of several possible shapes for space.

Finally, we'll see how data from a small NASA satellite could soon reveal the true shape of our universe. The only prerequisites for this talk are curiosity and imagination. For middle school and high school students, people interested in astronomy, and all members of the Missoula community.

Thursday, 5 September 2002
8:00 p.m. in the Music Recital Hall
Reception following

This talk is part of The Presidents Lecture Series & The Big Sky Conference, and is sponsored in part by the National Science Foundation and the Department of Mathematical Sciences.

The Curvature of Space
Dr. Jeffery Weeks

The talk will begin with an elementary introduction to curved space, using physical models and interactive 3D graphics to build intuition and demonstrate some surprising visual effects. We'll then see how physicists' understanding of a curved, expanding universe evolved over the 20th century, leading to measurements of the microwave background radiation which are now revealing the curvature of the observable universe. But even as these measurements answer old questions about the curvature of space, they raise new questions about the matter and energy it contains.

For mathematics faculty, graduate students, and undergraduate math and physics majors. Note: The other half of the story, namely the topology of space, will be the subject of the evening lecture.

Thursday, 5 September 2002
4:10 p.m. in James E Todd Building 203-204
Reception at 3:30 p.m. 204

This talk is part of The Presidents Lecture Series & The Big Sky Conference, and is sponsored in part by the National Science Foundation and the Department of Mathematical Sciences.