# Past Kinderseminar

19 February 2014

10:30

Lukas Buggisch

Abstract

The classical small cancellation theory goes back to the 1950's and 1960's when the geometry of 2-complexes with a unique 0-cell was studied, i.e. the standard 2-complex of a finite presentation. D.T. Wise generalizes the Small Cancellation Theory to 2-complexes with arbitray 0-cells showing that certain classes of Small Cancellation Groups act properly discontinuously and cocompactly on CAT(0) Cube complexes and hence have codimesion 1-subgroups. To be more precise I will introduce "his" version of small Cancellation Theory and go roughly through the main ideas of his construction of the cube complex using Sageeve's famous construction. I'll try to make the ideas intuitively clear by using many pictures. The goal is to show that B(4)-T(4) and B(6)-C(7) groups act properly discontinuously and cocompactly on CAT(0) Cube complexes and if there is time to explain the difficulty of the B(6) case. The talk should be self contained. So don't worry if you have never had heard about "Small Cancellation".

12 February 2014

10:30

Giles Gardam

Abstract

We will introduce some necessary basic notions regarding formal languages, before proceeding to give the classification of groups whose word problem is context-free as the virtually free groups (due to Muller and Schupp (1983) together with Dunwoody's accessibility of finitely presented groups (1985) for full generality). Emphasis will be on the group theoretic aspects of the proof, such as Stalling's theorem on ends of groups, accessibility, and geometry of the Cayley graph (rather than emphasizing details of formal languages).

5 February 2014

10:30

Claudio Llosa Isenrich

Abstract

A Kähler group is a finitely presented group that can be realized as fundamental group of a compact Kähler manifold. It is known that every finitely presented group can be realized as fundamental group of a compact real and even symplectic manifold of dimension greater equal than 4 and of a complex manifold of complex dimension greater equal than 2. In contrast, the question which groups are Kähler groups is surprisingly harder and there are large classes of examples for both, Kähler, and non-Kähler groups. This talk will give a brief introduction to the theory of Kähler manifolds and then discuss some basic examples and properties of Kähler groups. It is aimed at a general audience and no prior knowledge of the field will be required.

29 January 2014

10:30

Robert Laugwitz

Abstract

<p>This talk aims to illustrate how graphical calculus can be used to
reason about Hopf algebras and their modules. The talk will be aimed at a
general audience requiring no previous knowledge of the topic.</p>

22 January 2014

10:30

Simon Rydin-Myerson

Abstract

A major project in number theory runs as follows. Suppose some Diophantine equation has infinitely many integer solutions. One can then ask how common solutions are: roughly how many solutions are there in integers $\in [ -B, \, B ] $? And ideally one wants an answer in terms of the geometry of the original equation.
What if we ask the same question about Diophantine inequalities, instead of equations? This is surely a less deep question, but has the advantage that all the geometry we need is over $\mathbb{R}$. This makes the best-understood examples much easier to state and understand.

4 December 2013

10:30

Giles Gardam

Abstract

Kazhdan introduced property (T) for locally compact topological groups to show that certain lattices in semisimple Lie groups are finitely generated. This talk will give an introduction to property (T) along with some first consequences and examples. We will finish with a classic application of property (T) due to Margulis: the first known construction of expanders.

27 November 2013

10:30

Mark Penney

Abstract

I will discuss what it means to compactify complex Lie groups and introduce the so-called "Wonderful Compactification" of groups having trivial centre. I will then show how the wonderful compactification of PGL(n) can be described in terms of complete collineations. Finally, I will discuss how the new perspective provided by complete collineations provides a way to construct compactifications of arbitrary semisimple groups.

20 November 2013

10:30

Montserrat Casals

Abstract

<div>In this talk I will introduce the class of limit
groups and discuss its characterisations from several different
perspectives: model-theoretic, algebraic and topological. I hope that
everyone will be convinced by at least one of the approaches that this
class of groups is worth studying.</div>

13 November 2013

10:30

Levon Haykazyan

Abstract

<p>(A simplified version of) Ax-Grothendieck Theorem
states that every injective polynomial map from some power of complex
numbers into itself is surjective. I will present a simple
model-theoretical proof of this fact. All the necessary notions from
model theory will be introduced during the talk. The only prerequisite
is basic field theory.</p>