Past Kinderseminar

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.

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.

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.

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.

(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.

A large class of links in $S^3$ has the property that the complement admits a complete hyperbolic metric of finite volume. But is this volume understandable from the link itself, or maybe from some nice diagram of it? Marc Lackenby in the early 2000s gave a positive answer for a class of diagrams, the alternating ones. The proof of this result involves an analysis of the JSJ decomposition of the link complement: in particular of how does it appear on the link diagram. I will tell you an outline of this proof, forgetting its most technical aspects and explaining the underlying ideas in an accessible way.

I will look at the classical constructions that can be made using a straight edge and compass, I will then look at the limits of these constructions. I will then show how much further we can get with Origami, explaining how it is possible to trisect an angle or double a cube. Compasses not supplied.

I shall outline a procedure for efficiently approximating arbitrary elements of certain topological groups by words in a finite set. The method is suprisingly general and is based upon the assumption that close to the identity, group elements may be easily expressible as commutators. Time permitting, I shall discuss some applications to uniform diameter bounds for finite groups and to quantum computation.

Symplectic reflection algebras are an important class of algebras related to an incredibly high number of different topics such as combinatorics, noncommutative geometry and resolutions of singularities and have themselves a rich representation theory. We will recall their definition and classification coming from symplectic reflection groups and outline some of the results that have characterised their representation theory over the last decade, focusing on the link with representations of quivers.

Saturated fusion systems are both a convenient language in which to formulate p-local finite simple group theory and interesting structures in their own right. In this talk, we will start by explaining what is meant by a 'tree of fusion systems' and give conditions on such an object for there to exist a saturated completion. We then describe how this theory can be used to understand a class of fusion systems first considered by Bob Oliver, which are determined by modular representations of finite groups. If time permits, we will discuss joint work with David Craven towards a complete classification of such fusion systems. The talk is aimed at a general mathematical audience with some background in algebra.

Concepts such as infinitesimal numbers and fluxions have been used by Leibnitz and Newton for the initial development of calculus. However, their non-rigorous nature has caused a lot of controversy and they have eventually been phased out by epsilon-delta definitions. In early 60s Abraham Robinson realised that methods of mathematical logic can be used to provide rigorous meaning to such concepts. This talk is a gentle introduction to some of Robinson's ideas.

I will give a gentle introduction to uniformly finite homology. The highlight application will be showing existence of aperiodic tilings of the hyperbolic plane.

In this talk, I will discuss homotopy limits: The basics, and why you should care about them if you are a topologist, an algebraic geometer, or an algebraist (have I missed anyone?).

Categorification is a fancy word for a process that is pretty ubiquitous in mathematics, though it is usually not referred to as "a thing". With the advent of higher category theory it has, however, become "a thing". I will explain what people mean by this "thing" (sneak preview: it involves replacing sets by categories) and hopefully convince you it is not quite as alien as it may seem and maybe even tempt you to let it infect some of your maths. I'll then explain how this fits into the context of higher categories.

I shall talk about Subgroup Membership Problem for amalgamated products of finite rank free groups. I'm going to show how one can solve different versions of this problem in amalgams of free groups and give an estimate of the complexity of some algorithms involved.  This talk is based on a joint paper with A. J. Duncan.

Following the recent paper of Ogasa, we attempt to construct Boy's surface using only paper and tape. If this is successful we hope to address such questions as:

Is that really Boy's surface?

Why should we care?

Do we have any more biscuits?

We'll provide some motivation for the appearance of factorization algebras in physics, before discussing the definition of a factorization monoid. We'll then review the definition of a principal G-bundle and show how a factorization monoid can help us understand the moduli stack Bun_G of principal G-bundles.

I will give an introduction to The McKay Correspondence, relating the irreducible representations of a finite subgroup Γ ≤ SL2 (C), minimal resolutions of the orbit space C2 /Γ, and affine Dynkin diagrams.

A number is called transcendental if it is not algebraic, that is it does not satisfy a polynomial equation with rational coefficients. It is easy to see that the algebraic numbers are countable, hence the transcendental numbers are uncountable. Despite this fact, it turns out to be very difficult to determine whether a given number is transcendental. In this talk I will discuss some famous examples and the theorems which allow one to construct many different transcendental numbers. I will also give an outline of some of the many open problems in the field.

In the Greek mythology the hydra is a many-headed poisonous beast. When cutting one of its heads off, it will grow two more. Inspired by how Hercules defeated the hydra, Dison and Riley constructed a family of groups defined by two generators and one relator, which is an Engel  word: the hydra groups. I will talk about its remarkably wild subgroup distortion and its hyperbolic cousin. Very recent discussions of Baumslag and Mikhailov show that those groups are residually torsion-free nilpotent and they introduce generalised hydra groups.

I will look at some tools for proving expansion in the Cayley graphs of finite quotients of a given infinite group, with particular emphasis on Bourgain-Gamburd’s work on expansion in Zariski-dense subgroups of SL_2(Z), and speculate to what extent such expansion may be said to be “uniform”.

There appears to be no universally accepted rigorous definition of a "flexagon" (although I will try to give a reasonable one in the talk). Examples of flexagons were most likely discovered and rediscovered many times in the past - but they were "officially" discovered in 1939, a serendipitous consequence of the discrepancy between US paper sizes and sensible paper sizes.* I'll describe a couple of the most famous examples of flexagons (with actual models to play with of course), and also introduce some of the more abstract theory of flexagons which has been developed. Feel free to bring your own models of flexagons!

* The views expressed herein are solely those of the speaker, and do not reflect the official position of the Kinderseminar w.r.t. international paper standards. 

I will give a brief report on some the topics discussed at the workshop "Golod-Shafarevich groups and rank gradient" that took place this August in Vienna. I will focus on results involving rank gradient.

Saturated fusion systems are a relatively new class of objects that are often described as the correct 'axiomatisation' of certain p-local phenomena in algebraic topology. Despite these geometric beginnings however, their structure is sufficiently rigid to afford its own local theory which in some sense mimics the local theory of finite groups. In this talk, I will briefly motivate the definition of a saturated fusion system and discuss a remarkable result of Michael Aschbacher which proves that centralisers of normal subsystems of a saturated fusion system, F, exist and are themselves saturated. I will then attempt to justify his definition in the case where F is non-exotic by appealing to some classical group theoretic results. If time permits I will speculate about a topological characterisation of the centraliser as the set of homotopy fixed points of a certain action on the classifying space of F.

I will present a history of the problem, relate it to other conjectures, and, with time permitting, indicate recent developments. The focus will primarily be group-theoretic and intended for the non-specialist.

It is known that the minimum number of generators d(G^n) of the n-th direct power G^n of a non-trivial finite group G tends to infinity with n. This prompts the question: in which ways can the sequence {d(G^n)} tend to infinity? This question was first asked by Wiegold who almost completely answered it for finitely generated groups during the 70's. The question can then be generalised to any algebraic structure and this is still an open problem currently being researched. I will talk about some of the results obtained so far and will try to explain some of the methods used to obtain them, both for groups and for the more general algebraic structure setting.

We explore methods (deterministic and otherwise) of composing music using mathematical models. Musical examples will be provided throughout and the audience (with the speakers assistance) will compose a brand new piece.

We will start off with a crash course in General relativity, and then I'll describe a 'recipe' for a time machine. This will lead us to the question whether or not the topology of the universe can change. We will see that, in some sense, this is topologically allowed. However, the Einstein equation gives a certain condition on the Ricci tensor (which is violated by certain quantum effects) and meeting this condition is a more delicate problem.

I will begin by defining the notion of a characteristic class of surface bundles, and constructing the MMM (Miller-Morita-Mumford) classes as examples. I will then talk about a recent theorem of Church, Farb, and Thibault which shows that the characteristic numbers associated to certain MMM-classes do not depend on how the total space is fibred as a surface bundle - they depend only on the topology of the total space itself. In particular they don't even depend on the genus of the fibre. Hence there are many 'coincidences' between the characteristic numbers of very different-looking surface bundles.

A corollary of this is an obstruction to low-genus fiberings: given a smooth manifold E, the non-vanishing of a certain invariant of E implies that any surface bundle with E as its total space must have a fibre with genus greater than a certain lower bound.

Also, following the paper of Church-Farb-Thibault, I will sketch how to construct examples of 4-manifolds which fibre in two distinct ways as a surface bundle over another surface, thus giving concrete examples to which the theorem applies.

The Hanna Neumann Conjecture provides a bound on the rank of the intersection of finitely generated subgroups of a free group. We will follow Mineyev's recent elementary and beautiful proof of this longstanding conjecture.

This talk will summarize some of the problems and conjectures that I haven't managed to solve (although I have tried to) while spending my three years in this job. It will cover the areas of group theory, representation theory, both of general finite groups and of symmetric groups, and fusion systems.

The talk will start with the definition of amenable groups. I will discuss various properties and interesting facts about them. Those will be underlined with representative examples. Based on this I will give the definition and some basic properties of sofic groups, which only emerged quite recently. Those groups are particularly interesting as it is not know whether every group is sofic.