Past Kinderseminar

In this talk I will introduce Hilbert's 10th Problem (H10) and the model-theoretic notions necessary to explore this problem from the perspective of mathematical logic. I will give a brief history of its proof, talk a little about its connection to decidability and definability, then close by speaking about generalisations of H10 - what has been proven and what has yet to be discovered.

The “field with one element” is an interesting algebraic object that in some sense relates linear algebra with set theory. In a much deeper vein it is also expected to have a role in algebraic geometry that could potentially “lift" Deligne’s proof of the final Weil Conjecture for varieties over finite fields to a proof of the Riemann hypothesis for the Riemann zeta function. The only problem is that it doesn’t exist. In this highly speculative talk I will discuss some of these concepts, and focus mainly on zeta functions of algebraic varieties over finite fields. I will give a (very) brief sketch of how to interpret various zeta functions in a geometric context, and try to explain what goes wrong for the Riemann zeta function that makes this a difficult problem.

I was speak on the way Newton carries out his calculus in the Principia in the framework of classical geometry rather than with fluxions, his deficiencies, and the relation of this work to inverse-square laws.

Outer Space is an important object in Geometric Group Theory and can be described from two viewpoints: as a space of marked graphs and a space of actions on trees. The latter viewpoint can be used to prove that Outer Space is contractible; and this fact together with some arguments using the first viewpoint enables us to say something about the Outer Automorphism group of a free group - I will sketch both these proofs.

By way of shameless advertising for a TCC course I hope to give next term on the theory of totally disconnected locally compact groups, I will present two interesting and illuminating examples of such groups: the full automorphism group of a regular tree, and Neretin's group of spheromorphisms
 

Classifying line arrangements on the plane is a problem that has been around for a long time. There has been a lot of work from the perspective of incidence geometry, but after a paper of Hirzebruch in in 80's, it has also attracted the attention of algebraic geometers for the applications that it has on classifying complex algebraic surfaces of general type. In this talk I will present various results around this problem, I will show some specific questions that are still open, and I will explain how it relates to complex surfaces of general type. 
 

Modular forms holomorphic functions on the upper half of the complex plane, H, invariant under certain matrix transformations of H. The have a very rich structure - they form a graded algebra over C and come equipped with a family of linear operators called Hecke operators. We can also view them as functions on a Riemann surface, which we refer to as a modular curve. It transpires that the integral homology of this curve is equipped with such a rich structure that we can use it to compute modular forms in an algorithmic way. The modular symbols are a finite presentation for this homology, and we will explore this a little and their connection to modular symbols.

This talk will hopefully highlight the general framework in which Penrose tilings are proved to be aperiodic and in fact a tiling. 

In the game 'Set', players compete to pick out groups of three cards sharing common attributes. But how many cards must be dealt before such a group must appear? 
This is an example of a "cap set problem", a problem in Ramsey theory: how big can a set of objects get before some form of order appears? We will translate the cap set problem into a problem of geometry over finite fields, discussing the current best upper bounds and running through an elementary proof. We will also (very) briefly discuss one or two implications of the cap set problem over F_3 to other questions in Ramsey theory and computational complexity
 

In my talk I will give a basic introduction to the finiteness properties of groups and their relation to subgroups of direct products of groups. I will explain the relation between such subgroups and fibre products of groups, and then proceed with a discussion of the n-(n+1)-(n+2)-Conjecture and the Virtual Surjections Conjecture. While both conjectures are still open in general, they are known to hold in special cases. I will explain how these results can be applied to prove that there are groups with arbitrary (non-)finiteness properties.

There are many natural questions one can ask about presentations of finite groups- for instance, given two presentations of the same group with the same number of generators, must the number of relations also be equal? This question, and closely related ones, are unsolved. However if one asks the same question in the category of profinite groups, surprisingly strong properties hold- including a positive answer to the above question. I will make this statement precise and give the proof of this and similar results due to Alex Lubotzky.

For every $\epsilon>0$ does there exist some $n\in\mathbb{N}$ and a bijection $f:\mathbb{Z}_n\to\mathbb{Z}_n$ such that $f(x+1)=2f(x)$ for at least $(1-\epsilon)n$ elements of $\mathbb{Z}_n$ and $f(f(f(f(x))))=(x)$ for all $x\in\mathbb{Z}_n$? I will discuss this question and its relation to an important open problem in the theory of countable discrete groups.

The Robin-Schensted-Knuth insertion algorithm provides a bijection between non-negative integer matrices and pairs of semistandard Young tableau. However, by relaxing the conditions on the correspondence, it allows us to define the Poirer-Reutenauer bialgebra, which exactly describes the algebra of symmetric functions viewed as generated by the Schur polynomials. This gives an interesting combinatorial decomposition of symmetric products of Schur polynomials, called a Littlewood Richardson rule, which we will discuss. We will then power through as many generalisations as I have time for: Hecke insertion and stable Grothendieck polynomials, shifted insertion and Schur P-functions, and shifted Hecke insertion and weak shifted stable Grothendieck polynomials

Deficiency is a measure of how complicated the presentations of a particular group need to be; it is defined as the maximum of the number of generators minus the number of relators (over all finite presentations of the group). This talk will introduce the basics of deficiency, give a deft example of Swan which illustrates why our understanding of deficiency is deficient, and conclude with some new examples that defy this defeatism: finite $p$-groups can have any deficiency you could (reasonably) wish for.

A variety of groups is an equationally defined class of groups, namely the class of groups in which each of a set of "laws" (or "identical relations") holds. Examples include the abelian groups (defined by the law $xy = yx$), the groups of exponent dividing $d$ (defined by the law $x^d$), the nilpotent groups of class at most some fixed integer, and the solvable groups of derived length at most some fixed integer. This talk will give an introduction to varieties of groups, and then conclude with recent work on determining for certain varieties whether, for fixed coprime $m$ and $n$, a group $G$ is in the variety if and only if the power subgroups $G^m$ and $G^n$ (generated by the $m$-th and $n$-th powers) are in the variety.

In this talk I will describe another strong link between the behaviour of a 3-manifold and the behaviour of its fundamental group- specifically the theorem that the group splits as a free product if and only if the 3-manifold may be divided into two parts using a 2-sphere inducing this splitting. This theorem is for some reason known as Kneser's conjecture despite having been proved half a century ago by Stallings.

An expander is a family of finite graphs of uniformly bounded degree, increasing number of vertices and Cheeger constant bounded away from zero. They occur throughout mathematics and computer science; the most famous constructions of expanders rely on powerful results in geometric group theory and number theory, while expanders are used in everything from error-correcting codes, through disproving the strongest version of the Baum-Connes conjecture, to affine sieve theory and the twin prime, Mersenne prime and Hardy-Littlewood conjectures.

However, very little was known about how different the geometry of two expanders could be. This question was raised by Ostrovskii in 2013, and a year later Mendel and Naor gave the first example of two 'distinct' expanders.

In this talk I will construct a continuum of expanders which are, in a certain sense, geometrically incomparable. Once the existence of a single expander is accepted, the remainder of the proof is a heady mix of counting, addition, multiplication, and just for the experts, a little bit of division. Two very different - and very interesting - continuums of 'distinct' expanders have since been constructed by Khukhro-Valette and Das.

I'll be talking about my masters' research in Quantum Gravity in a way that is accessible to mathematicians.

This talk will be on general amalgamation theory, covering ground from the 1950s to original research, with applications and examples from many different areas of mathematics and ranging from classical results to open problems.

I will try to give a brief description of the use of group theory and character theory in chemistry, specifically vibrational spectroscopy. Defining the group associated to a molecule, how one would construct a representation corresponding to such a molecule and the character table associated to this. Then, time permitting, I will go in to the deconstruction of the data from spectroscopy; finding such a group and hence molecule structure.