**Emily Stone**

Email

# 1997 Colloquia

## Fall 1997

### September 4 - Generic Polynomials

*Generic Polynomials*The talk will focus on the classical question of finding a "canonical form" for degree n polynomials in one variable. For example, the quadratic polynomial x^{2} + ax + b can be transformed into one of the form x^{2} + c by completing the square. The latter can be viewed as a canonical form which depends on only one parameter (namely, c) rather than the two we started out with (namely, a and b). In general, we will show that any canonical form for degree n polynomials depends on at least [n/2] parameters. This result (obtained in collaboration with J. Buhler) generalizes a theorem of Felix Klein on quintic polynomials and is related to Hilbert's 13-th problem.

Thursday, September 4, 1997

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### September 12 - Escher's Combinatorial Patterns

*Escher's Combinatorial Patterns***Abstract**: It is a little-known fact that M.C. Escher posed and answered some combinatorial questions about patterns produced in an algorithmic way. We report on his explorations, indicate how close he came to the correct solutions, and pose an analogous problem in three dimensions.

**Biographical Information**: Doris Schattschneider received an M.A. and Ph.D. in mathematics from Yale University and is Professor of Mathematics at Moravian College in Bethlehem, Pennsylvania. Her dual interest in geometry and art led naturally to the study of tiling problems and the work of the Dutch artist M. C. Escher. She has authored many scholarly articles on plane tiling and has acted as "Boswell" to reveal to the professional world the mathematical investigations of homemaker Marjorie Rice and M. C. Escher. She is co-author of a book and collection of geometry models: "M. C. Escher Kaleidocycles", Pomegranate Artbooks, 1987, that has been translated into 16 European languages. Her book on Escher, "Visions of Symmetry: Notebooks, Periodic Drawings and Related Work of M.C. Escher," W. H. Freeman, 1990, was supported by the National Endowment for the Humanities. Her articles about Escher's work include "Escher's Metaphors," in Scientific American, (November 1994) "Escher's Combinatorial Patterns," in The Electronic Journal of Combinatorics, v.4, no.2 (1997), #R17, and "Automating Escher's Combinatorial Patterns," (with Rick Mabry and Stan Wagon), in Mathematica in Education and Research, v. 5, no. 4 (1997).

This talk is given as part of the 1997 Big Sky Conference on Geometry, Discrete Mathematics, and Algorithms.

Friday, September 12, 1997

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### October 10 - Professor Catherine Kriloff, Idaho State University

*The representation theory of graded Hecke algebras*Graded Hecke algebras are infinite-dimensional noncommutative algebras which can be defined using reflection groups, such as the symmetries of a 2*n*-gon or an icosahedron. Not only did certain of these algebras arise originally in the work of G. Lusztig on the representation theory of *p*-adic groups, they are also of interest for their own sake, in part due to recent connections to Yang's *n*-particle problem in physics. This talk will include results on the classification of representations of graded Hecke algebras, as well as an unexpected result related to a number theoretic conjecture on the existence of Whittaker models.

Friday, October 10, 1997

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### October 16 -The representation theory of graded Hecke algebras

**Cycle Decompositions and Double Covers of Graphs**Informally, an *Euler tour* of a connected graph *G* is a tracing of the edges of *G*, with the conditions that you must start and finish at the same vertex, you must trace over each edge exactly once, and you must not lift your pencil from the page. A characterization of graphs that have Euler tours was given by Leonhard Euler in 1736: a graph has an Euler tour if and only if it is connected and all its vertices have even degree. The paper in which Euler gives this characterization is considered to be the first paper in the area of mathematics that we now know as graph theory.

Another way to characterize graphs with an Euler tour is the following: a graph *G* has an Euler tour if and only if *G* is connected and admits a cycle decomposition. A *cycle decomposition* of a graph *G* is simply a partition of the edge set of *G* into cycles. There are a number of ways to generalize the notion of a cycle decomposition of a graph; the one that we will concern ourselves with is cycle double covers. 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

**. The one obvious necessary condition that is required for a graph to have a cycle double cover is that the graph be bridgeless, and in his well known Cycle Double Cover Conjecture, P.D. Seymour asserts that this condition is also sufficient.**

*C*This talk will report on the status of the cycle double cover conjecture, as well as provide a survey of results and conjectures concerning cycle double covers and cycle decompositions.

Thursday, October 16, 1997

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### October 17 - Homomorphic Compactness of Infinite Graphs

*Homomorphic Compactness of Infinite Graphs*The University of CalgaryIn 1951 de Bruijn and Erdös proved that an infinite graph is *n*-colourable if and only if each of its finite subgraphs is n- colourable. This is often referred to as 'compactness of *n*-colouring'. Using the fact that *n*-colouring is essentially identical to finding a graph homomorphism to a complete graph on *n* vertices, we say that a graph *G* is homomorphically compact if each infinite graph *H* admits a homomorphism to *G* exactly when all of its finite subgraphs admit such a homomorphism.

We will show that (really) infinite compact graphs exist and explore various other problems related to them.

Friday, October 17, 1997

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### October 23 - Finite fields, codes and quasirandomness

*Finite fields, codes and quasirandomness*A classical problem considered by Davenport asks for the number of occurrences of a certain pattern of quadratic residues and nonresidues among the set of integers modulo a prime *p*. We will take this problem as a starting point of an itinerary revealing the interplay between quasirandom structures over finite fields, definable subsets of finite fields, character sums and coding theory.

Thursday, October 23, 1997

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### November 6 - Toward an Elementary Axiomatic Theory of a Category of Matroids

*Toward an Elementary Axiomatic Theory of a Category of Matroids*Graphs, point arrangements and sets of vectors can all be described in terms of their closed sets. In each case, the collection of closed sets satisfies certain properties. We define a matroid to be an object whose closed sets satisfy these properties. The matroids in which we are interested are those arising from finite graphs with a single loop. These matroids are called loopless pointed matroids. We consider these objects and special maps, called strong maps, between them. This collection of objects and maps is called the category of loopless pointed matroids and strong maps.

In our research we find a set of axioms satisfied by the category of loopless pointed matroids and strong maps. Our goal is to show any other category satisfying these axioms is equivalent to the category of loopless pointed matroids and strong maps. We use as a model for our research the work done by D. I. Schlomiuk in 1971. Schlomiuk studied the category of topological spaces and continuous mappings and found twelve axioms satisfied by this category. In addition, she proved that any category satisfying these twelve axioms is equivalent to the category of topological spaces and continuous mappings. We begin our research by examining these twelve axioms to determine which hold in our category. Moreover, we describe many matroid notions purely in term of strong maps.

This talk will provide an overview of our research. No prior knowledge of matroid or category theory is assumed!

Thursday, November 6, 1997

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### November 13 - COMPASS: Computerized Adaptive Placement Assessment and Support System

*COMPASS*

*Computerized Adaptive Placement Assessment and Support System*COMPASS is a comprehensive new computerized placement and diagnostic testing system from ACT that helps place students into appropriate courses.

More than a series of tests on a network of microcomputers, COMPASS is a course placement and diagnostic assessment system. The integrated software package provides retention, advising, demographic, reporting, research, and data management capabilities.

COMPASS measures students' mathematics, reading, and writing skills on demand and reports results immediately; this can be a real benefit in placement and advising situations. Tests and reporting features can be customized.

Thursday, November 13, 1997

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### November 20 -A duality for the category of directed multigraphs

**A duality for the category of directed multigraphs**The well known duality between the category of sets and the category of ccd (complete completely distributive) Boolean algebras given by the contravariant "subsets" functor, is extended to a duality between the category of graphs (directed multigraphs) and a category of ccd "graphic" algebras. Graphic algebras are Heyting algebras with one further unary operation, satisfying (in addition to the identities for Heyting algebras) one further identity. A Boolean algebra is a graphic algebra satisfying an obvious identity, and a set is construed as a graph having no arrows. The dualizing functor is the extension of the subsets functor to the contravariant subgraphs functor.

Thursday, November 20, 1997

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### December 11 - Regression Without Calculus

*Regression Without Calculus*It is possible that the overuse of optimization techniques brought on by Calculus has prevented a general development of statistics. Correlation Coefficients induce an "orthogonality" that can be used to develop statistical methods. This talk will show how the use of correlation allows a general definition of regression estimation in simple linear regression. The three correlation coefficients Pearson, Kendall, and Greatest Deviation will be used to illustrate an example of the general framework of the method without Calculus.

If two vectors of bivariate data (*x,y*) of size *n* are looked at in *n*-space, it becomes easy to define "natural" correlation coefficients. An *n*-dimensional interpretation of Pearson's *r* as the difference in the standardized L_{2} norms of *x*+*y* and *x*-*y* leads to correlation coefficients based on other measures of distance such as L_{1}. This "natural" definition has been missing in statistics at least since 1906 when Charles Spearman published an incomplete attempt at an absolute value rank correlation coefficient. However, this "natural" definition has been available in analysis since the time of Hilbert.

Thursday, December 11, 1997

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)