Research seminar in Discrete Mathematics and Algebra (DMA)
Organisers: Prof Johannes Carmesin, Dr Jan Kurkofka, Prof Martin Schneider
Time: Wednesdays, 11:3012:30,
Location: MIB1108 or MIB1113
Current semester:

10.7.2024 Various speakers: Workshop on sofic graphs.

3.7.2024 Will J. Turner: Local Separators of Cayley Graphs: Proof Strategy.
Abstract: Stallings' Theorem (1971) states that a finitelygenerated group splits
over a finite subgroup if its Cayley graphs have more than one end. In
2010, Bernhard Krön gave a short proof of Stallings' Theorem using separators of graphs. In this project, we move towards a finite version
of Stallings' Theorem using local separators. We show that a
finitelygenerated group (with bounded nilpotency), having a certain
local separator in one of its Cayley graphs, is necessarily cyclic,
dihedral or decomposes as a direct product of a cycle and an involution.
In particular, we assume the existence of a dlocal cutvertex or a
dlocal 2separator, for d bounded below in terms of the group's
nilpotency. A rlocal cutvertex u in a graph is a vertex which
disconnects the subgraph induced on a ball of diameter d centred at u. A
dlocal 2separator is defined so that it works in a similarly intuitive
way.

5.6.2024 Paula Kahl: Introduction to amenability and related properties of groups.
Abstract: Introduced by Gromov in the late 1990s, the notion of a sofic group turned out to be very fruitful. Within a short period of time several longstanding grouptheoretic conjectures, such as Gottschalk's surjunctivity conjecture or Kaplansky's direct finiteness conjecture (both still open for arbitrary groups), were proved for sofic groups. This suggests to think of soficity as a strong property only few groups satisfy. However, until now, there is not a single group known to be nonsofic. Starting from the older and stronger notion of amenability, the talk provides an introduction to soficity, hyperlinearity and the Haagerup property, as well as the relations between them. In particular we are going to prove that amenability implies soficity.

22.5.2024 Will J. Turner: Local Separators of Cayley Graphs: Towards a StallingsType Theorem for
Finite Groups.
Abstract: Stallings' Theorem (1971) states that a finitelygenerated group splits
over a finite subgroup if its Cayley graphs have more than one end. In
2010, Bernhard Krön gave a short proof of Stallings' Theorem using separators of graphs. In this project, we move towards a finite version
of Stallings' Theorem using local separators. We show that a
finitelygenerated group (with bounded nilpotency), having a certain
local separator in one of its Cayley graphs, is necessarily cyclic,
dihedral or decomposes as a direct product of a cycle and an involution.
In particular, we assume the existence of a dlocal cutvertex or a
dlocal 2separator, for d bounded below in terms of the group's
nilpotency. A rlocal cutvertex u in a graph is a vertex which
disconnects the subgraph induced on a ball of diameter d centred at u. A
dlocal 2separator is defined so that it works in a similarly intuitive
way.

15.5.2024 Tim Planken: Colouring ddimensional triangulations.
Abstract: We consider ddimensional triangulations and show structural equivalences for (d+1)colourability and (d+2)colourability of the vertices of such triangulations. In particular, we show that a ddimensional triangulation admits a proper (d+1)colouring if and only if every (d2)cell is incident with an even number of (d1)cells. Moreover, a ddimensional triangulation C admits a proper (d+2)colouring if and only if there exists a triangulation C' that contains C such that for every (d2)cell in C', the number of incident (d1)cells is divisible by three.

8.5.2024 Josefin Bernhard: Automatic continuity of topological groups.
Abstract: Many concrete groups come equipped with a nontrivial compatible topology. In some natural examples of topological groups, the topology is determined already by the algebraic structure. Such situation is closely linked with the automatic continuity property: a topological group is said to have automatic continuity if every homomorphism from it into any separable topological group is continuous. In the talk, examples and applications of this phenomenon will be given and we will discuss methods of proving this property for a given group.

29.4.2024 Arkady Leiderman (BenGurion, Israel): Embedding the free topological group F(X^n) into F(X).
Previous years
[To be updated]