Fachrichtung Mathematik

Seminar Algebra, Geometrie und Kombinatorik

Ein Seminar der Institute Algebra und Geometrie.

Do, 23.7.2015, 13:15 Uhr, WIL/C133 tba
Do, 16.7.2015, 13:15 Uhr, WIL/C133 Elementary amenable groups and the space of marked groups
Dr. Phillip Wesolek (UC Louvain)

Abstract: The space of marked groups is a compact totally disconnected space that parameterizes all countable groups. This space allows for tools from descriptive set theory to be applied to study group-theoretic questions. The class of elementary amenable groups is the smallest class that contains the abelian groups and the finite groups and that is closed under group extension, taking subgroups, taking quotients, and taking countable directed unions. In this talk, we first give a characterization of elementary amenable groups in terms of a chain condition. We then show the set of elementary amenable marked groups is not in the Borel sigma algebra of the space of marked groups. This gives a new proof of a theorem of Grigorchuk: There are finitely generated amenable non-elementary amenable groups. We conclude by discussing further questions and possible generalizations of the techniques.

Do, 2.7.2015, 13:15 Uhr, WIL/C133 tba
Do, 25.6.2015, 13:15 Uhr, WIL/C133 tba
Do, 11.6.2015, 13:15 Uhr, WIL/C133 Embedding and packing large graphs into dense and sparse graphs
Prof. Dr. Anusch Taraz (TU Hamburg-Harburg)

Extremal combinatorics is often concerned with the forced appearance of highly organized structures. In this talk, we explain two major research avenues that deal with such situations. On the one hand, density results assert that these substructures must be present in any sufficiently dense host configuration. On the other hand, partition theorems guarantee that, no matter how we partition a sufficiently large object, at least one of the partition classes must contain the desired substructure.
We first survey old and new results of both types, to give a flavour of the field and its methods, and then focus on a sequence of results that generalize the existence of paths and cycles to graphs of sublinear bandwidth.

Do, 21.5.2015, 13:15 Uhr, WIL/C133 Space of actions, ultraproducts and sofic entropy
Alessandro Carderi (ENS Lyon)
Do, 7.5.2015, 13:15 Uhr, WIL/C133 Voronoi game on graphs
Dr. Viola Meszaros (TU Berlin)

The discrete Voronoi game is played on a graph G by two players A and B for a fixed number t of rounds. Player A starts and they alternatingly claim vertices of G in each round. No vertex can be claimed more than once. At the end of the game also the remaining vertices are divided among the players. Each player receives the vertices that are closer to his/her claimed vertices. If a vertex is equidistant to both players then we split it among them. In general it is hard to determine who gets more and therefore wins the game. First Demaine, Teramoto and Uehara dealt with this question. They proved that it is NP-complete to determime the winner of the Voronoi game on a general graph G. The game on trees was studied further by Kiyomi, Saitoh and Uehara. They showed that on a path it always ends in a draw unless the number of vertices is odd and t=1 when A wins by one.
We were interested in knowing when either player could control a large portion of the graph by the end of the game. We proved that there are graphs for which player B gets almost all vertices. The proof is inspired by a geometric construction. On a tree, player A can get at least one quarter of the vertices. If the game lasts for two rounds on a tree, then A can even get one third of all vertices. But A cannot get much more. A construction shows that this result cannot be improved when t is least two. In one round A can always claim half of the vertices of a tree. Inspired by the outcome of the game on a star, we investigated how much B can ensure on graphs with bounded degree. We also made some observations relating the result with many rounds to the one-round game.
It is a joint work with D. Gerbner, D. Palvolgyi, A. Pokrovskiy and G. Rote.

Do, 30.4.2015, 13:15 Uhr, WIL/C133 Reconstructing the topology of polymorphism clones
Dr. Christian Pech

Abstract: Every clone of functions comes naturally equipped with a topology - the topology of pointwise convergence. A clone C is said to have automatic homeomorphicity with respect to a class K of clones, if every clone-isomorphism of C to a member of K is already a homeomorphism (with respect to the topology of pointwise convergence). I am going to talk about automatic homeomorphicity-properties for polymorphism clones of countable homogeneous relational structures. The results base on (and extend) previous results by Bodirsky, Pinsker, and Pongrácz.

Do, 23.4.2015, 13:15 Uhr, WIL/C133 Group laws for finite simple groups
Prof. Dr. Andreas Thom