Aberdeen

Droplet impacts form an important part of many processes and a detailed
understanding of the impact dynamics is critical in determining any
subsequent splashing behaviour. Prior to touchdown a gas squeeze film is
set-up between the substrate and the approaching droplet. The pressure
build-up in this squeeze film deforms the droplet free-surface, trapping
a pocket of gas and delaying touchdown. In this talk I will discuss two
extensions of existing models of pre-impact gas-cushioned droplet
behaviour, to model droplet impacts with textured substrates and droplet
impacts with surfaces hot enough to induce pre-impact phase change.

In the first case the substrate will be modelled as a thin porous layer.
This produces additional pathways for some of the gas to escape and
results in less delayed touchdown compared to a flat plate. In the
second case ideas related to the evaporation of heated thin viscous
films will be used to model the phase change. The vapour produced from
the droplet is added to the gas film enhancing the existing cushioning
mechanism by generating larger trapped gas pockets, which may ultimately
prevent touchdown altogether once the temperature enters the Leidenfrost
regime.

Inspired largely by the fact that commutative C*-algebras correspond to

(locally compact Hausdorff) topological spaces, C*-algebras are often

viewed as noncommutative topological spaces. In particular, this

perspective has inspired notions of noncommutative dimension: numerical

isomorphism invariants for C*-algebras, whose value at C(X) is equal to

the dimension of X. This talk will focus on certain recent notions of

dimension, especially decomposition rank as defined by Kirchberg and Winter.

A particularly interesting part of the dimension theory of C*-algebras

is occurrences of dimension reduction, where the act of tensoring

certain canonical C*-algebras (e.g. UHF algebras, Cuntz' algebras O_2

and O_infinity) can have the effect of (drastically) lowering the

dimension. This is in sharp contrast to the commutative case, where

taking a tensor product always increases the dimension.

I will discuss some results of this nature, in particular comparing the

dimension of C(X,A) to the dimension of X, for various C*-algebras A. I

will explain a relationship between dimension reduction in C(X,A) and

the well-known topological fact that S^n is not a retract of D^{n+1}.

This talk is about some recent joint work with Sarah Witherspoon. The representations of some finite dimensional Hopf algebras have curious behaviour: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. I shall describe a family of examples of such Hopf algebras and their modules, and the classification of left, right, and two-sided ideals in their stable module categories.

Chataur and Menichi showed that the homology of the free loop space of the classifying space of a compact Lie group admits a rich algebraic structure: It is part of a homological field theory, and so admits operations parametrised by the homology of mapping class groups.  I will present a new construction of this field theory that improves on the original in several ways: It enlarges the family of admissible Lie groups.  It extends the field theory to an open-closed one.  And most importantly, it allows for the construction of co-units in the theory.  This is joint work with Anssi Lahtinen.

De Concini-Kac-Procesi conjecture gives a good estimate for the dimensions of finite--dimensional non-restricted representations of quantum groups at m-th root of unity. According to De Concini, Kac and Procesi such representations can be split into families parametrized by conjugacy classes in an algebraic group G, and the dimensions of representations corresponding to a conjugacy class O are divisible by m^{dim O/2}. The talk will consist of two parts. In the first part I shall present an approach to the proof of De Concini-Kac-Procesi conjecture based on the use of q-W algebras and Bruhat decomposition in G. It turns out that properties of representations corresponding to a conjugacy class O depend on the properties of intersection of O with certain Bruhat cells. In the second part, which is more technical, I shall discuss q-W algebras and some related results in detail.