Past

Recent results (starting with Scheffer and Shnirelman and continuing with De Lellis and Szekelhyhidi ) underline the importance of considering solutions of the incompressible Euler equations as limits of solutions of more physical examples like Navier-Stokes or Boltzmann.
I intend to discuss several examples illustrating this issue.

In Riemannian geometry there are several notions of rank

defined for non-positively curved manifolds and with natural extensions

for groups acting on non-positively curved spaces.

The talk shall explain how various notions of rank behave for

mapping class groups of surfaces. This is joint work with J. Behrstock.

We will give a quick overview of the semigroup perspective on splitting schemes for S(P)DEs which present a robust, "easy to implement" numerical method for calculating the expected value of a certain payoff of a stochastic process driven by a S(P)DE. Having a high numerical order of convergence enables us to replace the Monte Carlo integration technique by alternative, faster techniques. The numerical order of splitting schemes for S(P)DEs is bounded by 2. The technique of combining several splittings using linear combinations which kills some additional terms in the error expansion and thus raises the order of the numerical method is called the extrapolation. In the presentation we will focus on a special extrapolation of the Lie-Trotter splitting: the symmetrically weighted sequential splitting, and its subsequent extrapolations. Using the semigroup technique their convergence will be investigated. At the end several applications to the S(P)DEs will be given.

Based on ideas from rough path analysis and operator splitting, the Kusuoka-Lyons-Victoir scheme provides a family of higher order methods for the weak approximation of stochastic differential equations. Out of this family, the Ninomiya-Victoir method is especially simple to implement and to adjust to various different models. We give some examples of models used in financial engineering and comment on the performance of the Ninomiya-Victoir scheme and some modifications when applied to these models.

String theory on a torus requires the introduction of dual coordinates

conjugate to string winding number. This leads to physics and novel geometry in a doubled space. This will be

compared to generalized geometry, which doubles the tangent space but not the manifold.

For a d-torus,   string theory can be formulated in terms of an infinite

tower of fields depending on both the d torus coordinates and the d dual

coordinates. This talk focuses on a finite subsector  consisting of a metric

and B-field (both d x d matrices) and a dilaton all depending on the 2d

doubled torus coordinates.

The double field theory is constructed and found to have a novel symmetry

that reduces to diffeomorphisms and anti-symmetric tensor gauge

transformations in certain circumstances. It also has manifest T-duality

symmetry which provides a generalisation of the usual Buscher rules to

backgrounds without isometries. The theory has a real dependence on the full

doubled geometry:  the dual dimensions are not auxiliary. It is concluded

that the doubled geometry is physical and dynamical.

Inverse methods are frequently used in geosciences to estimate model parameters from indirect measurements. A common inverse problem encountered when modelling the flow of large ice masses such as the Greenland and the Antarctic ice sheets is the determination of basal conditions from surface data. I will present an overview over some of the inverse methods currently used to tackle this problem and in particular discuss the use of Bayesian inverse methods in this context. Examples of the use of adjoint methods for large-scale optimisation problems that arise, for example, in flow modelling of West-Antarctica will be given.

Absence of arbitrage is a highly desirable feature in mathematical models of financial markets. In its pure form (whether as NFLVR or as the existence of a variant of an equivalent martingale measure R), it is qualitative and therefore robust towards equivalent changes of the underlying reference probability (the "real-world" measure P). But what happens if we look at more quantitative versions of absence of arbitrage, where we impose for instance some integrability on the density dR/dP? To which extent is such a property robust towards changes of P? We discuss these uestions and present some recent results.

The talk is based on joint work with Tahir Choulli (University of Alberta, Edmonton).

In this lecture I will exploit a model of asset prices where speculators overconfidence is a source of heterogeneous beliefs and arbitrage is limited. In the model, asset buyers are the most positive investors, but prices exceed their optimistic valuation because the owner of an asset has the option of reselling it in the future to an even more optimistic buyer. The value of this resale option can be identified as a bubble. I will focus on assets with a fixed terminal date, as is often the case with credit instruments. I will show that the size of a bubble satisfies a Partial Differential Equation that is similar to the equation satisfied by an American option and use the PDE to evaluate the impact of parameters such as interest rates or a “Tobin tax” on the size of the bubble and on trading volume.

Feature Selection is a ubiquitous problem in across data mining,

bioinformatics, and pattern recognition, known variously as variable

selection, dimensionality reduction, and others. Methods based on

information theory have tremendously popular over the past decade, with

dozens of 'novel' algorithms, and hundreds of applications published in

domains across the spectrum of science/engineering. In this work, we

asked the question 'what are the implicit underlying statistical

assumptions of feature selection methods based on mutual information?'

The main result I will present is a unifying probabilistic framework for

information theoretic feature selection, bringing almost two decades of

research on heuristic methods under a single theoretical interpretation.

Exponential time integrators are a powerful tool for numerical solution

of time dependent problems. The actions of the matrix functions on vectors,

necessary for exponential integrators, can be efficiently computed by

different elegant numerical techniques, such as Krylov subspaces.

Unfortunately, in some situations the additional work required by

exponential integrators per time step is not paid off because the time step

can not be increased too much due to the accuracy restrictions.

To get around this problem, we propose the so-called time-stepping-free

approach. This approach works for linear ordinary differential equation (ODE)

systems where the time dependent part forms a small-dimensional subspace.

In this case the time dependence can be projected out by block Krylov

methods onto the small, projected ODE system. Thus, there is just one

block Krylov subspace involved and there are no time steps. We refer to

this method as EBK, exponential block Krylov method. The accuracy of EBK

is determined by the Krylov subspace error and the solution accuracy in the

projected ODE system. EBK works for well for linear systems, its extension

to nonlinear problems is an open problem and we discuss possible ways for

such an extension.

Min-Max equations, also called Isaacs equations, arise from many applications, eg in game theory or mathematical finance. For their numerical solution, they are often discretised by finite difference

methods, and, in a second step, one is then faced with a non-linear discrete system. We discuss how upper and lower bounds for the solution to the discretised min-max equation can easily be computed.

In this seminar I will present two results regarding the uniqueness (and further properties) for the two-dimensional continuity equation

and the ordinary differential equation in the case when the vector field is bounded, divergence free and satisfies additional conditions on its distributional curl. Such settings appear in a very natural way in various situations, for instance when considering two-dimensional incompressible fluids. I will in particular describe the following two cases:\\

(1) The vector field is time-independent and its curl is a (locally finite) measure (without any sign condition).\\

(2) The vector field is time-dependent and its curl belongs to L^1.\\

Based on joint works with: Giovanni Alberti (Universita' di Pisa), Stefano Bianchini (SISSA Trieste), Francois Bouchut (CNRS &

Universite' Paris-Est-Marne-la-Vallee) and Camillo De Lellis (Universitaet Zuerich).

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 $Bun_n(X)$.

We present recent numerical techniques for the treatment of integral formulations of Helmholtz boundary value problems in the case of high frequencies. The combination of $H^2$-matrices with further developments of the adaptive cross approximation allows to solve such problems with logarithmic-linear complexity independent of the frequency. An advantage of this new approach over existing techniques such as fast multipole methods is its stability over the whole range of frequencies, whereas other methods are efficient either for low or high frequencies.

We find and describe four futures markets where the bid-ask spread is bid down to the fixed price tick size practically all the time, and which match coun- terparties using a pro-rata rule. These four markets’ offered depths at the quotes on average exceed mean market order size by two orders of magnitude, and their order cancellation rates (the probability of any given offered lot being cancelled) are significantly over 96 per cent. We develop a simple theoretical model to explain these facts, where strategic complementarities in the choice of limit order size cause traders to risk overtrading by submitting over-sized limit orders, most of which they expect to cancel.

Joint work with Jonathan Field.

We shall dedicate the first half of the talk to introduce

classical Higgs bundles and describe the fibres of the corresponding

Hitchin fibration in terms of spectral data. Then, we shall define

principal Higgs bundles and look at some examples. Finally, we

consider the particular case of $SL(2,R)$, $U(p,p)$ and $Sp(2p,2p)$ Higgs

bundles and study their spectral data. Time permitting, we shall look

at different applications of our new methods.

In this talk, we make quantitative comparisons between two widely-used liquid crystal modelling approaches - the continuum Landau-de Gennes theory and mesoscopic mean-field theories, such as the Maier-Saupe and Onsager theories. We use maximum principle arguments for elliptic partial differential equations to compute explicit bounds for the norm of static equilibria within the Landau-de Gennes framework. These bounds yield an explicit prescription of the temperature regime within which the LdG and the mean-field predictions are consistent, for both spatially homogeneous and inhomogeneous systems. We find that the Landau-de Gennes theory can make physically unrealistic predictions in the low-temperature regime. In my joint work with John Ball, we formulate a new theory that interpolates between mean-field and continuum approaches and remedies the deficiencies of the Landau-de Gennes theory in the low-temperature regime. In particular, we define a new thermotropic potential that blows up whenever the mean-field constraints are violated. The main novelty of this work is the incorporation of spatial inhomogeneities (outside the scope of mean-field theory) along with retention of mean-field level information.