Oxford University

Modularity is a quality function on partitions of a network which aims to identify highly clustered components. Given a graph G, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the modularity q(G) of G is the maximum modularity of a partition of V(G). Knowledge of the maximum modularity of the corresponding random graph is important to determine the statistical significance of a partition in a real network. We provide bounds for the modularity of random regular graphs. Modularity is related to the Hamiltonian of the Potts model from statistical physics. This leads to interest in the modularity of lattices, which we will discuss. This is joint work with Colin McDiarmid.

The curve graph of a surface has a vertex for each curve on the surface and an edge for each pair of disjoint curves. Although it deals with very simple objects, it has connections with questions in low-dimensional topology, and some properties that encourage people to study it. Yet it is more complicated than it may look from its definition: in particular, what happens if we start following a 'diverging' path along this graph? It turns out that the curves we hit get so complicated that eventually give rise to a lamination filling up the surface. This can be understood by drawing some train track-like pictures on the surface. During the talk I will keep away from any issue that I considered too technical.

Cluster algebras are commutative algebras generated by a set S, obtained by an iterated mutation process of an initial seed. They were introduced by S. Fomin and A. Zelevinski in connection with canonical bases in Lie theory. Since then, many connections between cluster algebras and other areas have arisen.
This talk will focus on cluster algebras for which the set S is finite. These are called cluster algebras of finite type and are classified by Dynkin diagrams, in a similar way to many other objects.

Stallings' folding technique lets us factor a map of graphs as a sequence of "folds" (edge identifications) followed by an immersion. We will show how this technique gives an algorithm to express a free-group automorphism as the product of Whitehead automorphisms (and hence Nielsen transformations), as well as proving finite generation for some subgroups of the automorphism group of a free group.

Cubature is the business of describing a probability measure in terms of an empirical measure sharing its support with the original measure, of small support, and with identical integrals for a class of functions (eg polynomials with degree less than k). 

Applying cubature to already discrete sets of scenarios provides a powerful tool for scenario management and summarising data.  We refer to this process as recombination. It is a feasible operation in real time and has lead to high accuracy pde solvers.

The practical complexity of this operation has changed! By a factor corresponding to the dimension of the space of polynomials. 

We discuss the algorithm and give home computed examples of nested sparse grids with only positive weights in moderate dimensions (eg degree 1-8 in dimension 7).  Positive weights have significant advantage over signed ones when available.
 

Lagrangian Floer cohomology categorifies the intersection number of (half-dimensional) Lagrangian submanifolds of a symplectic manifold. In this talk I will describe how and when can we define Lagrangian Floer cohomology. In the case when Floer cohomology cannot be defined I will describe an alternative invariant known as the Fukaya (A-infinity) algebra.

In this talk we want to discuss results by Dimca, Papadima, and Suciu about the finiteness properties of Kähler groups. Namely, we will sketch their proof that for every $2\leq n\leq \infty$ there is a Kähler group with finiteness property $\mathcal{F}_n$, but not $FP_{n+1}$. Their proof is by explicit construction of examples. These examples all arise as subgroups of finite products of surface groups and they are the first known examples of Kähler groups with arbitrary finiteness properties. The talk does not require any prior knowledge of finiteness properties or of Kähler groups.

Historically, the study of positively curved manifolds has always been challenging. There are many reasons for this, but among them is the fact that the existence of a metric of positive curvature on a manifold imposes strong topological restrictions. In this talk we will discuss some of these topological implications and we will introduce the main results in this area. We will also present some recent results that relate positive curvature to the smooth structure of the manifold.