Past

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.

Graphs and digraphs behave quite differently, and many classical results for graphs are often trivially false when extended to general digraphs. Therefore it is usually necessary to restrict to a smaller family of digraphs to obtain meaningful results. One such very natural family is Eulerian digraphs, in which the in-degree equals out-degree at every vertex.

In this talk, we discuss several natural parameters for Eulerian digraphs and study their connections. In particular, we show that for any Eulerian digraph G with n vertices and m arcs, the minimum feedback arc set (the smallest set of arcs whose removal makes G acyclic) has size at least $m^2/2n^2+m/2n$, and this bound is tight. Using this result, we show how to find subgraphs of high minimum degrees, and also long cycles in Eulerian digraphs. These results were motivated by a conjecture of Bollob\'as and Scott.

Joint work with Ma, Shapira, Sudakov and Yuster

Stability plays an important role in engineering, for it limits the load carrying capacity of all kinds of structures. Many failure mechanisms in advanced engineering materials are stability-related, such as localized deformation zones occurring in fiber-reinforced composites and cellular materials, used in aerospace and packaging applications. Moreover, modern biomedical applications, such as vascular stents, orthodontic wire etc., are based on shape memory alloys (SMA’s) that exploit the displacive phase transformations in these solids, which are macroscopic manifestations of lattice-level instabilities.

The presentation starts with the introduction of the concepts of stability and bifurcation for conservative elastic systems with a particular emphasis on solids with periodic microstructures. The concept of Bloch wave analysis is introduced, which allows one to find the lowest load instability mode of an infinite, perfect structure, based solely on unit cell considerations. The relation between instability at the microscopic level and macroscopic properties of the solid is studied for several types of applications involving different scales: composites (fiber-reinforced), cellular solids (hexagonal honeycomb) and finally SMA's, where temperature- or stress-induced instabilities at the atomic level have macroscopic manifestations visible to the naked eye.

This talk will attempt to say something about the p-adic zeta function, a p-adic analytic object which encodes information about Galois cohomology of Tate twists in its special values. We first explain the construction of the p-adic zeta function, via p-adic Fourier theory. Then, after saying something about Coleman integration, we will explain the interpretation of special values of the p-adic zeta function as limiting values of p-adic polylogarithms, in analogy with the Archimedean case. Finally, we will explore the consequences for the de Rham and etale fundamental groupoids of the projective line minus three points.

In this talk, I will focus on an infinite class of 3d N=4 gauge theories

which can be constructed from a certain set of ordered pairs of integer

partitions. These theories can be elegantly realised on brane intervals in

string theory.  I will give an elementary review on such brane constructions

and introduce to the audience a symmetry, known as mirror symmetry, which

exchanges two different phases (namely the Higgs and Coulomb phases) of such

theories.  Using mirror symmetry as a tool, I will discuss a certain

geometrical aspect of the vacuum moduli spaces of such theories in the

Coulomb phase. It turns out that there are certain infinite subclasses of

such spaces which are special and rather simple to study; they are complete intersections. I will mention some details and many interesting features of these spaces.

 Consider a fully-connected social network of people, companies,
or countries, modeled as an undirected complete graph with real numbers on
its edges. Positive edges link friends; negative edges link enemies.
I'll discuss two simple models of how the edge weights of such networks
might evolve over time, as they seek a balanced state in which "the enemy of
my enemy is my friend." The mathematical techniques involve elementary
ideas from linear algebra, random graphs, statistical physics, and
differential equations. Some motivating examples from international
relations and social psychology will also be discussed. This is joint work
with Seth Marvel, Jon Kleinberg, and Bobby Kleinberg. 

Data assimilation in highly nonlinear and high dimensional systems is a hard

problem. We do have efficient data-assimilation methods for high-dimensional

weakly nonlinear systems, exploited in e.g. numerical weather forecasting.

And we have good methods for low-dimensional (

In this talk, we elaborate on the notions of no-free-lunch that have proved essential in the theory of financial mathematics---most notably, arbitrage of the first kind. Focus will be given in most recent developments. The precise connections with existence of deflators, numeraires and pricing measures are explained, as well as the consequences that these notions have in the existence of bubbles and the valuation of illiquid assets in the market.

ARKeX is a geophysical exploration company that conducts airborne gravity gradiometer surveys for the oil industry. By measuring the variations in the gravity field it is possible to infer valuable information about the sub-surface geology and help find prospective areas.

A new type of gravity gradiometer instrument is being developed to have higher resolution than the current technology. The basic operating principles are fairly simple - essentially measuring the relative displacement of two proof masses in response to a change in the gravity field. The challenge is to be able to see typical signals from geological features in the presence of large amounts of motional noise due to the aircraft. Fortunately, by making a gradient measurement, a lot of this noise is cancelled by the instrument itself. However, due to engineering tolerances, the instrument is not perfect and residual interference remains in the measurement.

Accelerometers and gyroscopes record the motional disturbances and can be used to mathematically model how the noise appears in the instrument and remove it during a software processing stage. To achieve this, we have employed methods taken from the field of system identification to produce models having typically 12 inputs and a single output. Generally, the models contain linear transfer functions that are optimised during a training stage where controlled accelerations are applied to the instrument in the absence of any anomalous gravity signal. After training, the models can be used to predict and remove the noise from data sets that contain signals of interest.

High levels of accuracy are required in the noise correction schemes to achieve the levels of data quality required for airborne exploration. We are therefore investigating ways to improve on our existing methods, or find alternative techniques. In particular, we believe non-linear and non-stationary models show benefits for this situation.

Nematic liquid crystals (NLCs) are materials that flow like liquids, but have some crystalline features. Their molecules are typically long and thin, and tend to align locally, which imparts some elastic character to the NLC. Moreover at interfaces between the NLC and some other material (such as a rigid silicon substrate, or air) the molecules tend to have a preferred direction (so-called "surface anchoring"). This preferred behaviour at interfaces, coupled with the internal "elasticity", can give rise to complex instabilities in spreading free surface films. This talk will discuss modelling approaches to describe such flows. The models presented are capable of capturing many of the key features observed experimentally, including arrested spreading (with or without instability). Both 2D and 3D spreading scenarios will be considered, and simple ways to model nontrivial surface anchoring patterns, and "defects" within the flows will also be discussed.