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Mathematics & Statistics

# Colloquia — Spring 1999

## Friday, January 22, 1999

Title
Speaker
Time
Place

The Symbolic Dynamics of Tiling the Integers
Ethan Coven
TBA
TBA
TBA

Abstract

A finite collection of finite sets tiles the integers if and only if the integers can be expressed as a disjoint union of translates of members of the collection. We associate with such a tiling a doubly infinite sequence with symbols the sets in the collection. The set of all such sequences is a sofic system, called a tiling system.

For example, if $$P$$ consists of the sets $$\{0\}$$ and $$\{0,1\}$$, then the tiling system is the collection of all doubly infinite sequences with symbols $$R$$ (red, the “color” of $$\{0\}$$) and $$B$$ (blue, the “color” of $$\{0,1\}$$) such that between any two consecutive appearances of $$R$$, there are an even number of $$B$$, i.e., the “even system”. This sofic system is closely related to the “Golden Mean” shift of finite type. Many transitive shifts of finite type, e.g., the full $$2$$-shift, cannot be realized (up to topological conjugacy) as tiling systems. However, we show that, up to powers of the shift, every shift of finite type can be realized as a tiling system.

## Friday, February 12, 1999

Title
Speaker

Time
Place

An Unusual Way to Generate Conic Sections and two Related Euclidean Constructions
Sam Sakmar
Department of Physics, USF
3:00pm-4:00pm
PHY 130

## Friday, February 26, 1999

Title
Speaker

Time
Place

Braids of surfaces in $$4$$-space
U. of South Alabama (and Osaka City U.)
3:00pm-4:00pm
PHY 130

Abstract

An $$m$$-braid is a collection of $$m$$ strings in a cylinder $$D^2\times I^1$$ satisfying a certain condition. The set of $$m$$-braids forms a group, called the $$m$$-braid group. This group plays an important role in knot theory. Knot theory treats of embedded closed curves in Euclidean $$3$$-space, and $$2$$-dimensional knot theory treats of embedded closed surfaces in $$4$$-space. In this talk, a generalization of $$m$$-braids is introduced, which is called a $$2$$-dimensional $$m$$-braid or a surface braid. That is a surface in a bi-disk $$D^2\times D^2$$ satisfying a certain condition. The set of $$2$$-dimensional $$m$$-braids forms a monoid (a semi-group with identity). $$2$$-dimensional braids are related with $$2$$-dimensional knot theory by the following two theorems.

Generalized Alexander's theorem. Any closed surface in $$4$$-space is described by a closed $$2$$-dimensional braid.

Generalized Markov's theorem. Such a braid description is unique up to braid isotopy, conjugation and stabilization.

## Monday, March 1, 1999

Title
Speaker

Time
Place

The Dangers of Near-Earth Asteroids
Rudy Dvorak
Astronomy Department
University of Vienna
Vienna, Austria
2:00pm-3:00pm
PHY 108

## Thursday, March 4, 1999

Title
Speaker

Time
Place

Some Asymptotic Results and Exponential Approximations in Semi-Markov Models
George Roussas
Professor and Associate Dean
Department of Mathematics & Statistics
University of California, Davis
2:00pm-3:00pm
PHY 118

## Monday, March 15, 1999

Title
Speaker

Time
Place

A Rayleigh-Ritz Method Applied to a Two-Dimensional Inverse Spectral Problem
C. Maeve McCarthy
Murray State University
3:00pm-4:00pm
PHY 120

## Tuesday, March 16, 1999

Title
Speaker

Time
Place

Groups of $$2\times 2$$ Matrices
Ross Geoghegan
SUNY at Binghamton
2:00pm-3:00pm
PHY 108

## Friday, March 19, 1999

Title
Speaker
Time
Place

Estimation With Meyer Types and Wavelets
Marianna Pensky
3:00pm-4:00pm
PHY 118