C5

Abstract: Ricci solitons are genralizations of Einstein metrics which have become subject of much interest over the last decade. In this talk I will give a basic introduction to these metrics and discuss how to reformulate the Ricci soliton equation as a Hamiltonian system assuming some symmetry conditions. Using this approach we will construct explicit solutions to the soliton equation for manifolds of dimension 5.

Interactions between micro-swimmers and their complex flow environments are important in many biological systems, such as sperm cells swimming in cervical mucus or bacteria in biofilm initiation areas. We present a theoretical model describing the dynamics of micro-organisms swimming in a plane Poiseuille flow of a viscoelastic fluid, accounting for hydrodynamic interactions and biological noise. General non-Newtonian effects are investigated, including shear-thinning and normal stress differences that lead to migration of the organisms across the streamlines of the background flow. We show that micro-swimmers are driven towards the centre-line of the channel, even if countered by hydrodynamic interactions with the channel walls that typically lead to boundary accumulation. Furthermore, we demonstrate that the normal stress differences reorient the swimmers at the centre-line in the direction against the flow so that they swim upstream. This suggests a natural sorting mechanism to select swimmers with a given swimming speed larger than the tunable Poiseuille flow velocity. This framework is then extended to study trapping and colony formation of pathogens near surfaces, in corners and crevices. 

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.

Donaldson-Thomas theory was born as a mean to attach to Calabi-Yau 3-manifolds integers, invariant under small deformation of the complex structure. Subsequent evolutions have replaced integers with cohomological invariants, more flexible and with a broader range of applicable cases.

This talk is meant to be a gentle induction to the topic. We start with an introduction on virtual fundamental classes, and how they relate to deformation and obstruction spaces of a moduli space; then we pass on to the Calabi-Yau 3-dimensional case, stressing how some homological conditions are essential and can lead to generalisation. First we describe the global construction using virtual fundamental classes, then the local approach via the Behrend function and the virtual Euler characteristic.
We introduce quivers with potential, which provide a profitable framework in which to build DT-theory, as they are a source of moduli spaces locally presented as degeneracy loci. Finally, we overview the problem of categorification, introducing the DT-sheaf and showing how it relates to the numerical invariants.