N3.12

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
 

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.

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.

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.