I will present a definition of symplectic homology groups for pairs of Liouville cobordisms with fillings, and explain how these fit into a formalism of homology theory similar to that of Eilenberg and Steenrod. This construction allows to understand form a unified point of view many structural results involving Floer homology groups, and yields new applications. Joint work with Kai Cieliebak.

Let X be a complex manifold with divisor D. I will describe a construction, which is due to Deligne, whereby given a choice of a branch of the logarithm one can canonically extend a holomorphic flat connection on the complement of the divisor X\D to a flat logarithmic connection on X.

T cells are important white blood cells that continually circulate in the body in search of the molecular signatures ('antigens') of infection and cancer. We (and many other labs) are trying to construct models of the T cell signalling network that can be used to predict how ligand binding (at the surface of the cell) controls gene express (in the nucleus). To do this, we stimulate T cells with various ligands (input) and measure products of gene expression (output) and then try to determine which model must be invoked to explain the data. The challenge that we face is finding 1) unique models and 2) scaling the method to many different input and outputs.

We aim to determine how cells faithfully complete genome replication. Accurate and complete genome replication is essential for all life. A single DNA replication error in a single cell division can give rise to a genomic disorder. However, almost all experimental data are ensemble; collected from millions of cells. We used a combination of high-resolution, genomic-wide DNA replication data, mathematical modelling and single cell experiments to demonstrate that ensemble data mask the significant heterogeneity present within a cell population; see [1-4]. Therefore, the pattern of replication origin usage and dynamics of genome replication in individual cells remains largely unknown. We are now developing cutting-edge single molecule methods and allied mathematical models to determine the dynamics of genome replication at the DNA sequence level in normal and perturbed human cells.

[1] de Moura et al., 2010, Nucleic Acids Research, 38: 5623-5633

[2] Retkute et al, 2011, PRL, 107:068103

[3] Retkute et al, 2012, PRE, 86:031916

[4] Hawkins et al., 2013, Cell Reports, 5:1132-41

Recently, there has been some progress in examining mirror symmetry beyond Calabi-Yau threefolds. I will discuss how this is related to flux vacua of type II supergravity on eight-dimensional manifolds equipped with SU(4)-structure. It will be shown that the natural framework to describe such vacua is generalized complex geometry. Two classes of type IIB solutions will be given, one of which is complex, the other symplectic, and I will describe in what sense these are mirror to one another.