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.

# Past Kinderseminar

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.