# Graduiertenkolleg: Methods for Discrete Structures

Deutsche Forschungsgemeinschaft
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# Monday Lecture and Colloquium

Monday, December 4, 2006

Technische Universität Berlin
Straße des 17. Juni 136
10623 Berlin
Math building - Room MA 041
room 005

Lecture - 14:15

### Rewrite methods for algebras related to knots

Abstract:
By Markov, a knot can be viewed as the equivalence class of a braid generated by conjugacy and the equivalence of a braid $b$ on $n$ strings with $bs$ where $s$ intertwines the $n$-th string with a new, $(n+1)$-st, string. This implies that invariants can be seen as functions on the group algebra of the braid group. In this vein, the Jones polynomial can be interpreted as a trace function on the Hecke algebra, and, more generally, the Kauffman polynomial as a trace function on a larger algebra. The latter algebra is known as the Kauffman tangle algebra, defined in terms of partial braids, caleed tangles, and is isomorphic to the BMW algebra, an algebra given by a presentation in terms of generators and relations. This algebra plays role as a centralizer algebra of classical quantum groups. The BMW algebra is the special case of a series of algebras defined by generators and relations in terms of a graph $M$. The case $M = {\rm A}_{n-1}$ coincides with the BMW algebra on $n$ strings. These generalized BMW algebras are finite dimensional if and only if the graph $M$ is a spherical Coxeter diagram. In these cases, we determine their dimensions by means of a correspondence with sets of orthogonal roots of the Coxeter group of type $M$. Also, we present a simplified version (the Brauer algebra) of these algebras, which govern the general structure and have very explicit combinatorial features. We have rewrite rules to solve the word problem for these algebras. Before proving the above results, we conducted experiments with a non-commutative Groebner basis package, which we will also discuss. Focus: noncomm GB + Orthogonal sets of roots + posets

Colloquium - 16:00

### Extremal combinatorics

Abstract:
Questions in extremal combinatorics concern the interplay of global parameters and local restrictions of certain discrete objects such as graphs, hypergraphs, or sets of integers. Turán's theorem in graph theory may be viewed is a prime example of such a result, as it completely determines the structure of (edge)maximal graphs which do not contain a clique on k vertices. In this talk we give a brief introduction in this branch of discrete mathematics. We focus mostly on results and techniques in extremal (hyper)graph theory.

Letzte Aktualisierung: 28.11.2006