## Colloquium Abstracts

**Thursday, Oct 10, 2002**

3:10pm, 211 Little Hall

**Prof. Alan Taylor**, *Dept. of Mathematics, Union College
*On Making Honesty the Best Policy (Mathematically)

We will consider a number of procedures that arise in contexts such as auctions, fair division, and voting. For each, we will ask two questions: (1) Is honesty the best policy? (2) If not, can we change the procedure, while preserving the spirit of the original, so that disingenuous behavior is no longer advantageous?

Alan Taylor is a UMaine alumnus and Professor of Mathematics at Union College. This talk is accessible to mathematics undergraduates.

**Thursday, Oct 10, 2002**

7:00pm, 100 Neville Hall

**Prof. Alan Taylor**, *Dept. of Mathematics, Union College
*The Mathematics of Voting

We illustrate mathematical questions and answers that arise from real-world voting systems. Some results suggest that certain election ideals are impossible to attain. Examples include: weights for the US federal system, an equally good alternative to majority rule, fair methods of apportionment on which to base the electoral college, and voting systems for alternatives in which honesty is the best policy.

**Thursday, Oct 24, 2002**

3:00pm, 105 DPC

**Prof. William O. Bray**, *Dept. of Mathematics and Statistics, University of Maine
*A Journey through Classical Partial Differential Equations and related Mathematical Thoughts

An undergraduate course in Partial Differential Equations presents a significant challenge to both student and instructor. In terms of physical phenomena, one has diffusion, waves, and equilibria. Mathematical formulation is constructed through phenomenological principles such as Fick’s diffusion law, conservation laws, and vector methods such as the divergence theorem. From here one is interested in constructing explicit solutions to the simplest problems usually through reduction of the problem via geometrical methods or separation of variables, aka, Fourier analysis. Building the solution from those of the reduced problems, usually through some form of superposition, while intuitively appealing, present intricate challenges in terms of justification and interpretation.

The first part of this talk is a directed journey through part of my course in PDE with discussion of the underlying mathematical ideas involved and supported with computer graphics. The second part has focus the interplay between geometry and analysis in PDE and the role this interplay has in certain contemporary problems in harmonic analysis.

**Thursday, Oct 31, 2002**

3:10pm, 211 Little Hall

**Ali E. Ozluk**, *Dept. of Mathematics and Statistics, University of Maine
*A Stroll Through Primes and Zeros of L-functions

A conjecture of Chebyshev is popularly interpreted as asserting that there are “more” primes of the form 4n+3 than of the form 4n+1. We will discuss some of the senses in which the Chebyshev assertion might be true and how all this relates to the distribution of zeros of Quadratic L-functions.

**Wednesday, Dec 4, 2002**

3:10pm, 211 Little Hall

**Elizabeth S. Allman**, *Dept. of Mathematics and Statistics, University of Southern Maine
*Constructing Phylogenetic Trees using DNA sequences and the Method of Phylogenetic Invariants

An important new application of mathematics to biology is in the construction of phylogenetic trees from DNA sequences. Specifically, if we have a collection of DNA sequences of a certain gene from a number of different species, how can we best determine the evolutionary tree relating the species? For example, which topological tree best relates the aligned sequences of nucleotides CTCG, CACC, ATGG, and AAGC?

In realistic applications we have much longer sequences, coming from more species. Typically there are an enormous number of possible topological trees that could relate the species, and it is not at all clear which one is the most likely to be correct. While a variety of techniques have been developed to attack this problem over the past 15 years, leading to increasingly robust methods, it is far from solved. Researchers from the fields of biology, computer science, mathematics, and statistics are working to find reliable and computationally efficient algorithms for phylogenetic tree construction.

A relatively undeveloped technique which shows theoretical promise is that of phylogenetic invariants. In this talk, I will present an overview of phylogenetic tree construction methods with a focus on the method of invariants. This talk is aimed at a general audience with more technical details given towards the end of the lecture.

**Thursday, Jan 30, 2003**

3:10pm, 421 Neville Hall

**Prof. Henrik Bresinsky**, *Dept. of Mathematics and Statistics, University of Maine
*Macaulay – Groebener Perfect Ideals

Perfect ideals were introduced initially by Macaulay – Groebener. We will discuss why they are, in some sense, “very good” ideals. The talk is aimed at a general audience.

**Thursday, Feb 13, 2003** (joint with Computer Science)

3:30pm, 107 DPC

**Dr. David Gautschi
**Strategy Laboratories: the what, the why, and the how

Imagine an executive team of any given enterprise that could work through the intricacies of running the enterprise in a risk-free environment. This would mean practicing analysis of the business, specifying and framing choices, executing, assessing consequences, detecting mistakes and conjuring remedial actions, and so forth. Not only is this technically feasible, some enterprises have been doing this in some way for some time. This talk addresses the concept of the strategy laboratory, why it is of particular use to management teams of various kinds of enterprises, and how a laboratory may be designed and applied. Developing and applying a strategy laboratory draws upon techniques, methodologies, and concepts from mathematics, computer science, microeconomics, and the functional areas of business administration. Because of its cross disciplinary nature, the concept of the strategy laboratory has developed unsystematically in academic institutions. This presentation attempts to provide some avenues for fostering constructive collaboration among the disciplines that would contribute to a fuller flowering of the strategy laboratory concept.

David Gautschi has had a checkered career in academe, industry, and consulting. He has served on the faculties of Cornell, Yale, INSEAD, and the University of Washington. He founded and managed two software development companies, PRISM (Fontainebleau, France) and Janus Enterprise International (New Jersey). Most recently, he served as a firm director at Deloitte & Touche where he held a leadership in the e-business, net markets, and business economics practices. David has published over forty articles and two books in a broad range of applied economics areas spanning choice models, optimization models, the economics of service institutions, and the economics of technology. He has developed seven industry specific simulation models and delivered over 50 laboratories to enterprises in North America, Europe, and Asia. David holds a B.A. in mathematics from the University of Maine, an M.B.A. (applied statistics concentration) from the University of Oregon, and a PhD in business administration from the University of California, Berkeley. He and his wife, Adelle, reside in Seattle, Washington.

**Thursday, Feb 20, 2003**

3:10pm, 421 Neville Hall

**Prof. Jim Page**, *Philosophy Dept., University of Maine
*Are Equinumerosity Principles Analytic (and why should we care)? (Part 1)

It was long thought that logicism – the program to reduce arithmetic to logic – foundered on Russell’s Paradox. Work in the 1980s and 90s, however, showed how to derive Peano arithmetic from a single equinumerosity principle together with standard second-order logic. The question of whether these results constitute a resurgence of logicism thus depends on whether this equinumerosity principle is analytic.

In part I, I will outline the first logicist program as developed by Frege and Russell with an emphasis on the philosophical as well as the mathematical motivations, and show how this program ran aground on Russell’s discovery of his now-famous paradox. In Part II, I will outline the neo-logicist results and rehearse arguments for and against the claim that the crucial equinumerosity principle is analytic.

Dr. Page completed his masters at the University of St. Andrews, Scotland, on issues in the interpretation of the quantum formalism, and his doctorate at MIT on the logical and philosophical foundations of mathematics. Following teaching and research residencies at a number of academic institutions, he returned to his native Maine where he is now CEO of James W. Sewall Company and adjunct Associate Professor at in the Philosophy dept. at the University of Maine.

**Thursday, Feb 27, 2003**

3:10pm, 421 Neville Hall

**Prof. Jim Page**, *Philosophy Dept., University of Maine
*Are Equinumerosity Principles Analytic (and why should we care)? (Part 2)

It was long thought that logicism – the program to reduce arithmetic to logic – foundered on Russell’s Paradox. Work in the 1980s and 90s, however, showed how to derive Peano arithmetic from a single equinumerosity principle together with standard second-order logic. The question of whether these results constitute a resurgence of logicism thus depends on whether this equinumerosity principle is analytic.

In part I, I will outline the first logicist program as developed by Frege and Russell with an emphasis on the philosophical as well as the mathematical motivations, and show how this program ran aground on Russell’s discovery of his now-famous paradox. In Part II, I will outline the neo-logicist results and rehearse arguments for and against the claim that the crucial equinumerosity principle is analytic.

Dr. Page completed his masters at the University of St. Andrews, Scotland, on issues in the interpretation of the quantum formalism, and his doctorate at MIT on the logical and philosophical foundations of mathematics. Following teaching and research residencies at a number of academic institutions, he returned to his native Maine where he is now CEO of James W. Sewall Company and adjunct Associate Professor at in the Philosophy dept. at the University of Maine.

**Monday, March 24, 2003**

3:30pm, 105 DPC

**Dr. Tamas Wiandt**, *Dept. of Mathematics, Rice University
*Conley Decomposition for Closed Relations

We present a theory of dynamics of closed relations on compact Hausdorf spaces. We establish generalizations for some topological aspects of dynamical systems theory, including recurrence, attractor-repeller structure and the Conley Decomposition Theorem. A short description of Liapunov functions for relations and intensity of attraction will follow.

**Thursday, March 27, 2003**

3:40pm, 107 DPC

**Dr. Jianqiang Zhao**, *Dept. of Mathematics, University of Pennsylvania
*Partial Sums of Multiple Zeta Value Series

In this talk I will study the p-divisibility of partial sums of multiple zeta value series. First I consider generalizations of the classical Wolstenholme’s Theorem. Then I provide some evidence for the following conjecture: for any prime p and any MZV series there is always some N such that if n > N then p does not divide the numerator of the n-th partial sum of the MZV series. This generalizes a conjecture of Eswarathasan and Levine and Boyd for harmonic series.

**Monday, March 31, 2003**

3:10pm, 105 DPC

**Eric Rowell**, *Dept. of Mathematics, University of California, San Diego
*Quantum Groups and Braided Categories

Quantum groups were first discovered around 1985 by Drinfeld and Jimbo in connection with finding solutions to the celebrated Yang-Baxter equation. As q-deformations of Lie algebras, their representation theory can be studied in a similar fashion as that of the classical groups in the work of Weyl, Brauer and others in the early 20th century. Whereas in the classical theory the symmetric group is used to describe the decomposition of tensor product representations, for quantum groups it is the braid group that plays the central role. With the finite dimensional representations as objects, a menagerie of braided tensor categories can be constructed which have applications in low-dimensional topology, mathematical physics and operator algebra. In this talk I will outline some of the techniques used in studying quantum groups and discuss some recent results and applications.

**Wednesday, April 2, 2003**

3:10pm, 105 DPC

**Dr. Erik Talvila**, *Dept. of Mathematical and Statistical Sciences, University of Alberta
*Henstock-Kurzweil Fourier Transforms

The Fourier transform of a function $f:mathbb{R} rightarrow mathbb{R}$ is $hat{f}(s) = int_{-infty}^{infty} e^{-isx} f(x) dx$. Typically these are considered for $f in L^p$ for $1 leq p leq 2$. Because of the oscillatory exponential kernel, the Fourier transform may exist as a conditionally convergent integral without being absolutely convergent. This case can be handled by the Henstock-Kurzweil integral. This is an integral with a simple definition in terms of Riemann sums and yet it includes the Lebesgue and improper Riemann integrals. This talk will be an introduction to the Henstock-Kurzweil integral and its application to Fourier transforms.

**Monday, April 7, 2003**

2:10pm, 107 DPC

**Dr. Antun Milas**, *Dept. of Mathematics, University of Arizona
*Conformal Field Theories and Vertex Operator Algebras

Two-dimensional conformal field theories have been studied extensively over the past 20 years. These theories stem from statistical physics and string theory and have been used in many areas of mathematics such as finite group theory, infinite–dimensional Lie theory, mathematical physics and topology.

In the first part of my talk I will introduce a notion of vertex operator algebra and explain the relationship between vertex operator algebras and meromorphic conformal field theories. Afterward, I will discuss non-meromorphic generalizations and the categorical setup.

In the second part I will discuss recent progress in this area and present several applications.

This talk is meant to be elementary and expository.

**Thursday, April 10, 2003**

3:10pm, 421 Neville Hall

**Prof. Bill Bray**, *Dept. of Mathematics and Statistics, University of Maine
*Hadamard meets Harmonic Analysis

Classically, certain problems in partial differential equations are easier to solve when the underlying manifold is odd dimensional. For the wave equation (and other hyperbolic equations) Hadamard introduced a technique which bears his name in order to shift the problem from even to odd dimensions. In this talk we will introduce certain partial Radon transforms which effectively intertwine the Laplacian between manifolds of different dimension. In the realm of harmonic analysis, such a property leads to a transplantation formula for the basic quantity of interest, the spherical function, between even and odd dimensional manifolds. In effect, this allows shifting local problems of harmonic analysis from even dimensions to odd dimensions where they are more easily solved.

**Thursday, May 15, 2003**

3:10pm, 421 Neville Hall

**Rebecca Rozario**, *Dept. of Mathematics and Statistics, University of Maine
*The Distribution of The Irreducibles in an Algebraic Number Field

The first part of the talk will be a survey of primes and irreducibles in an algebraic number field. In the second part we are going to focus on asymptotic results for the distribution of the number of principal ideals generated by an irreducible element in an algebraic number field.

Rebecca Rozario is a graduate student in the Dept. of Mathematics and Statistics at the University of Maine.