Past

In this talk our aim is to explain why there exist hyperkähler metrics on the cotangent bundles and on coadjoint orbits of complex Lie groups. The key observation is that both the cotangent bundle of $G^\mathbb C$ and complex coadjoint orbits can be constructed as hyperkähler quotients in an infinite-dimensional setting: They may be identified with certain moduli spaces of solutions to Nahm's equations, which is a system of non-linear ODEs arising in gauge theory. 

In the first half we will describe the hyperkähler quotient construction, which can be viewed as a version of the Marsden-Weinstein symplectic quotient for complex symplectic manifolds. We will then introduce Nahm's equations and explain how their moduli spaces of solutions may be related to the above Lie theoretic objects.

In recent years there has been extensive interest in word maps on groups, and various results were obtained, with emphasis on simple groups. We shall focus on some new results on word maps for more general families of finite and infinite groups.

I revisit the identification of Nekrasov's K-theoretic partition function, counting instantons on $R^4$, and the (refined) Donaldson-Thomas partition function of the associated local Calabi-Yau threefold. The main example will be the case of the resolved conifold, corresponding to the gauge group $U(1)$. I will show how recent mathematical results about refined DT theory confirm this identification, and speculate on how one could lift the equality of partition functions to a structural result about vector spaces.

 The use of tissue engineered implants could facilitate unions in situations where there is loss of bone or non-union, thereby increasing healing time, reducing the risk of infections and hence reducing morbidity. Currently engineered bone tissue is not of sufficient quality to be used in widespread clinical practice.  In order to improve experimental design, and thereby the quality of the tissue-constructs, the underlying biological processes involved need to be better understood. In conjunction with experimentalists, we consider the effect hydrodynamic pressure has on the development and regulation of bone, in a bioreactor designed specifically for this purpose. To answer the experimentalists’ specific questions, we have developed two separate models; in this talk I will present one of these, a multiphase partial differential equation model to describe the evolution of the cells, extracellular matrix that they deposit, the culture medium and the scaffold.  The model is then solved using the finite element method using the deal.II library.

This is a report of joint work with T. Koppe, P. Majumdar, and K.

 Ray.

I will define new partition functions for theories with targets on toric

singularities via

products of old partition functions on  crepant resolutions. I will

present explicit examples 

and show that the  new partition functions turn out to be homogeneous on

MacMahon factors.

Pathwise Holder convergence with optimal rates is proved for the implicit Euler scheme associated with semilinear stochastic evolution equations with multiplicative noise. The results are applied to a class of second order parabolic SPDEs driven by space-time white noise. This is joint work with Sonja Cox.

We describe how one can recover the Mori--Mukai

classification of smooth 3-dimensional Fano manifolds using mirror

symmetry, and indicate how the same ideas might apply to the

classification of smooth 4-dimensional Fano manifolds. This is joint

work in progress with Corti, Galkin, Golyshev, and Kasprzyk.

"The model of Vertex Reinforced Random Walk (VRRW) on Z goes back to Pemantle & Volkov, '99, who proved a result of localization on 5 sites with positive probability. They also conjectured that this was the a.s. behavior of the walk. In 2004, Tarrès managed to prove this conjecture. Then in 2006, inspired by Davis'paper '90 on the edge reinforced version of the model, Volkov studied VRRW with weight on Z. 

He proved that in the strongly reinforced case, i.e. when the weight sequence is reciprocally summable, the walk localizes a.s. on 2 sites, as expected. He also proved that localization is a.s. not possible for weights growing sublinearly, but like a power of n. However, the question of localization remained open for other weights, like n*log n or n/log n, for instance. In the talk I will first review these results and formulate more precisely the open questions. Then I will present some recent results giving partial answers. This is based on joint (partly still on-going) work with Anne-Laure Basdevant and Arvind Singh."

We establish a correspondence between generalized quiver gauge theories in

four dimensions and congruence subgroups of the modular group, hinging upon

the trivalent graphs which arise in both. The gauge theories and the graphs

are enumerated and their numbers are compared. The correspondence is

particularly striking for genus zero torsion-free congruence subgroups as

exemplified by those which arise in Moonshine. We analyze in detail the

case of index 24, where modular elliptic K3 surfaces emerge: here, the

elliptic j-invariants can be recast as dessins d'enfant which dictate the

Seiberg-Witten curves.

We present a general valuation framework for commodity storage facilities, for non-perishable commodities. Modeling commodity prices with a mean reverting process we provide analytical expressions for the value obtainable from the storage for any admissible injection/withdrawal policy. Then we present an iterative numerical algorithm to find the optimal injection and withdrawal policies, along with the necessary theoretical guarantees for convergence. Together, the analytical expressions and the numerical algorithm present an extremely efficient way of solving not only commodity storage problems but in general the problem of optimally controlling a mean reverting processes with transaction costs.

• Chong Luo - Microscopic models for planar bistable liquid crystal device
• Laura Gallimore - Modelling Cell Motility
• Yi Ming Lai - Stochastic Oscillators in Biology

We also state some connections to some open problems.

Hollow vortices are vortices whose interior is at rest. They posses vortex sheets on their boundaries and can be viewed as a desingularization of point vortices. We give a brief history of point vortices. We then obtain exact solutions for hollow vortices in linear and nonlinear strain and examine the properties of streets of hollow vortices. The former can be viewed as a canonical example of a hollow vortex in an arbitrary flow, and its stability properties depend. In the latter case, we reexamine the hollow vortex street of Baker, Saffman and Sheffield and examine its stability to arbitrary disturbances, and then investigate the double hollow vortex street. Implications and extensions of this work are discussed.

The geometric Langlands correspondence relates rank n integrable connections 
on a complex Riemann surface $X$ to regular holonomic D-modules on 
$Bun_n(X)$, the moduli stack of rank n vector bundles on $X$.  If we replace 
$X$ by a smooth irreducible curve over a finite field of characteristic p 
then there is a version of the geometric Langlands correspondence involving 
$l$-adic perverse sheaves for $l\neq p$.  In this lecture we consider the 
case $l=p$, proposing a $p$-adic version of the geometric Langlands 
correspondence relating convergent $F$-isocrystals on $X$ to arithmetic 
$D$-modules on $Bun_n(X)$.

The geometric Langlands correspondence relates rank n integrable connections on a complex Riemann surface $X$ to regular holonomic D-modules on  $Bun_n(X)$, the moduli stack of rank n vector bundles on $X$.  If we replace $X$ by a smooth irreducible curve over a finite field of characteristic p then there is a version of the geometric Langlands correspondence involving $l$-adic perverse sheaves for $l\neq p$.  In this lecture we consider the case $l=p$, proposing a $p$-adic version of the geometric Langlands correspondence relating convergent $F$-isocrystals on $X$ to arithmetic $D$-modules on $Bun_n(X)$.