# Research

## Discrete Mathematics

### Friday, November 21, 2003

Title: $$3$$-coloring and other elementary invariants of knots
Speaker: Kheira Ameur
Time: 1:00pm‐2:00pm
Place: PHY 120

#### Abstract

Classical Knot theory studies the position of a circle (knot) or of several circles (link) in $$R^3$$ or $$s^3$$. The fundamental problem of classical knot theory is the classification of links (including knots) up to natural movement in space which is called an ambient isotopy. To distinguish knots or links we look for invariants of links which are unchanged under ambient isotopy. The tricoloring invariant is the simplest invariant which distinguish between the trefoil knot and the trivial knot. The idea of tricoloring was introduced by R. Fox around 1960 and has been extensively used and popularized by J. Montesinos and L. Kauffman. It turns out that the $$3$$-coloring invariant is nicely related to other invariants like the Alexander polynomial and the Kauffman polynomial.

### Friday, November 14, 2003

Title: Evolution and observation: A new way to look at computation
Speaker: Matteo Cavaliere, University of Rovira i Virgili
Tarragona, Spain
Time: 1:00pm‐2:00pm
Place: PHY 120

#### Abstract

In biology and chemistry a standard proceeding is to conduct an experiment, observe its progress, and then take the result of this observation as the final output. Inspired by this, we have introduced a new framework where computation is obtained by observing the “evolution” of a system. The approach has been applied to classical formal language theory (a derivation of a context-free grammar is observed by a finite automaton) and to membrane computing (a membrane system is observed by a multiset automaton). In both cases (surprising) universality results have been obtained.

### Friday, November 7, 2003

Title: Modular Leonard Triples as $$q$$-Analogs of the Pauli Matrices
Speaker: Brian Curtin
Time: 1:00pm‐2:00pm
Place: PHY 120

### Friday, October 24, 2003

Title: Kambi Kolam and Circular DNA Splicing
Speaker: Rani Siromoney, Madras Christian College/
Chennai Mathematical Institute, India
Time: 1:00pm‐2:00pm
Place: PHY 120

#### Abstract

Kolam patterns have motivated the Madras group to define formal grammars for picture languages. Of special interest are the patterns which have rotational symmetry. Kambi Kolam (literally meaning wire decoration) provide us with interesting classes of cycle languages. Gift Siromoney had conducted several experiments to find out how the women folk memorise, store and retrieve from their memory, the rules for complicated Kolam patterns. His findings were that this memorisation process was similar to the turtle moves. Yet another study involved the complexity of the patterns by the number of threads (cycles) in it — whether a kambi kolam with more number of cycles is easier to remember than a single kambi (cycle) kolam. He had introduced operations which turned out to be closely related to the splicing rules in Circular DNA Splicing.

### Friday, October 17, 2003

Title: Solution to a Problem of S. Payne, Part II
Speaker: Xiang-Dong Hou
Time: 1:00pm‐2:00pm
Place: PHY 120

### Friday, October 10, 2003

Title: Solution to a Problem of S. Payne
Speaker: Xiang-Dong Hou
Time: 1:00pm‐2:00pm
Place: PHY 120

#### Abstract

A problem posed by S. Payne calls for determination of all linearized polynomials $$f(x)\in\mathbb{F}_{2^n}[x]$$ such that $$f(x)$$ and $$f(x)/x$$ are permutations of $$\mathbb{F}_{2^n}$$ and $$\mathbb{F}_{2^n}^*$$, respectively. We show that such polynomials are exactly of the form $$f(x)=ax^{2^k}$$ with $$a\in\mathbb{F}_{2^n}^*$$ and $$(k,n)=1$$. In fact, we solve a $$q$$-ary version of Payne's problem.

### Friday, October 3, 2003

Title: Characteristic Vectors of Unimodular Lattices, Part II
Speaker: Mark Gaulter
Time: 1:00pm‐2:00pm
Place: PHY 120

### Friday, September 26, 2003

Title: Characteristic Vectors of Unimodular Lattices
Speaker: Mark Gaulter
Time: 1:00pm‐2:00pm
Place: PHY 120

#### Abstract

In this talk, we define and give examples of positive definite unimodular $$Z$$-lattices. We describe applications to crystallography, and to codes. We discuss tools that can be used to construct new unimodular lattices from old ones, including the neighbo[u]r lattice process and glue vectors. We also introduce the theta series of a lattice. To conclude the lecture, we introduce the notion of a characteristic vector in a lattice, and prove a theorem of Elkies; that the only unimodular lattice in $$R^n$$ whose shortest characteristic vectors have norm $$\ge n$$ is $$Z^n$$.

### Friday, September 19, 2003

Title: Minimal Combinatorial Models for Maps of an Interval With a Given Set of Periods, Part II
Speaker: David Kephart
Time: 1:00pm‐2:00pm
Place: PHY 120

### Friday, September 12, 2003

Title: Minimal Combinatorial Models for Maps of an Interval With a Given Set of Periods
Speaker: David Kephart
Time: 1:00pm‐2:00pm
Place: PHY 120

#### Abstract

Continuous self-maps of the interval reveal more subtle properties than can be anticipated, even — and especially the simplest such functions, i.e., piecewise weakly monotonic functions. In Presentation I we give the background and outline of one proof of the Sharkovskii Theorem. This theorem is the starting point for the combinatorial examination of the dynamics of iterations of such functions. In Presentation II we will show how, in recently published work, (“Minimal combinatorial models for maps of an interval with a given set of periods” by Block, Coven, Geller, and Hubner, published this year in Ergodic Theory and Dynamical Systems, this approach is applied to the problem of establishing minimal combinatorial models for certain families of continuous self-maps.

NOTE: There will be an organizational meeting prior to the talk.