**Analysis and Combinatorics Session**

*Talks will be in MR3, except for Friday 11.00-11.30 (MR2).*

**Thursday**

- 2.30-3.00: Michael Doré - Non-linear analysis in finite dimensions
- 3.00-3.30: Iryna Schlackow - C(K) spaces with few operators
- 3.30-4.00: Sam Elliott - Hardy spaces and Composition Operators
*Tea/Coffee*- 4.30-5.00: Nicholas Korpelainen - How to tell a friend from a foe: what every algorithmic graph problem should know about graph classes
- 5.00-5.30: Daniele Garrisi - On the spectral flow for Banach spaces
- 5.30-6.00: Mette Iversen - Minimisers of Dirichlet Eigenvalues with Geometric Constraints

**Friday**

*Plenary lecture, Tea/Coffee*- 11.00-11.30: Vicky Neale - Bracket quadratics as asymptotic bases for the integers (accessible talk)
- 11.30-12.00: David Conlon - Combinatorial theorems in sparse random sets
- 12.00-12.45: Ben Green (keynote speaker) - Approximate structure
*Lunch, Panel discussion, Tea/Coffee*- 4.00-4.30: Matthew White - Two-factors in hamiltonian graphs
- 4.30-5.00: Andrew Treglown - Perfect packings in graphs
- 5.00-5.30: Richard Mycroft - Hamilton cycles in hypergraphs

*Non-linear analysis in finite dimensions* - Michael Doré

I shall discuss some recent results in classical analysis inspired by methods used in Banach spaces. The focus will be on differentiability.

*Hardy spaces and Composition Operators* - Sam Elliott

An interesting class of Banach spaces is that of the Hardy spaces. We introduce these spaces, consisting as they do of analytic functions, and look at composition operators, which are those operators derived from pre-composition with some map, called the symbol. In particular, we look at characterisations of various important properties of these operators in terms of their symbol.

*How to tell a friend from a foe: what every algorithmic graph problem should know about graph classes * - Nicholas Korpelainen

Why are some graphs easier to deal with than others for difficult algorithmic problems? Indeed, many NP-complete graph problems (such as finding a maximum independent vertex set) can be made polynomial-time solvable by restricting to an interesting subclass of all graphs. We will discuss strategies for determining whether a graph class is `friendly' in this respect. We will also describe how to find minimal `unfriendly' classes of graphs. The talk will concentrate on examples and graph-theoretical proofs, and it will be extremely light on the technical details of algorithms.

*On the spectral flow for Banach spaces* - Daniele Garrisi

Given a Banach space E, we present the definition of the spectral flow, an application which associates an integer with a continuous path of linear, bounded applications of E. Such integer counts, with multiplicities, how many eigenvalues change their sign from negative to positive.

We illustrate some properties of the spectral flow in finite-dimensional spaces. We show that the spectral flow of a loop of matrices is zero, and outline the argument used by Weyl, in 1925, to prove that the special unitary group SU(n) is simply-connected.

We show a simple argument which proves that a loop with a non-vanishing spectral flow always exists in an infinite-dimensional Hilbert space, where the spectral flow induces an isomorphism of the fundamental group of the space of Fredholm and self-adjoint operators with the integers Z.

We conclude with the definition of the spectral flow for infinite-dimensional Banach spaces and how the existence of loops with a non-vanishing spectral flow is related to the existence of complemented subspaces connected to proper subspaces with finite co-dimension.

*Minimisers of Dirichlet Eigenvalues with Geometric Constraints* - Mette Iversen

This talk considers the problem in shape optimisation of minimising the eigenvalues of the Dirichlet Laplacian subject to geometric constraints on the minimising set. Specifically, the talk will consider properties of minimising sets subject to a constraint on the Hausdorff measure of the boundary. It will, in particular, look at cases where the minimiser for the kth eigenvalue of the Dirichlet Laplacian in Rm can be seen to be connected.

*Approximate structure* - Ben Green (keynote speaker)

Lately people have been asking questions of the sort "How can one define an approximate X, to what extent does it resemble an actual X, and is this useful?" In my talk I will touch on such questions when X stands for "group", "polynomial" or "homomorphism". I will attempt to say what this has to do with finding prime numbers in particular patterns, for example arithmetic progressions such as 5,11,17,23,29.

*Two-factors in hamiltonian graphs* - Matthew White

A well-known theorem of Dirac states that a graph with minimal degree at least n/2 is hamiltonian, this result was extended by Brandt et al to show that a graph with minimal degree at least n/2 has a two-factor with k components for 4k<=n. Both results are tight, as shown by the complete bipartite graph with classes of order (n-1)/2 and (n+1)/2 for n odd. However, this graph has no two-factors at all, and so a natural question is what minimal degree is required in a hamiltonian graph to guarantee a two-factor with k components. This talk will look at this problem, with particular emphasis on the case k=2.