Past

The coefficients in mathematical models of physical processes are often impossible to determine fully or accurately, and are hence subject to uncertainty. It is of great importance to quantify the uncertainty in the model outputs based on the (uncertain) information that is available on the model inputs. This invariably leads to very high dimensional quadrature problems associated with the computation of statistics of quantities of interest, such as the time it takes a pollutant plume in an uncertain subsurface flow problem to reach the boundary of a safety region or the buckling load of an airplane wing. Higher order methods, such as stochastic Galerkin or polynomial chaos methods, suffer from the curse of dimensionality and when the physical models themselves are complex and computationally costly, they become prohibitively expensive in higher dimensions. Instead, some of the most promising approaches to quantify uncertainties in continuum models are based on Monte Carlo sampling and the “multigrid philosophy”. Multilevel Monte Carlo (MLMC) Methods have been introduced recently and successfully applied to many model problems, producing significant gains. In this talk I want to recall the classical MLMC method and then show how the gains can be improved further (significantly) by using quasi-Monte Carlo (QMC) sampling rules. More importantly the dimension independence and the improved gains can be justified rigorously for an important model problem in subsurface flow. To achieve uniform bounds, independent of the dimension, it is necessary to work in infinite dimensions and to study quadrature in sequence spaces. I will present the elements of this new theory for the case of lognormal random coefficients in a diffusion problem and support the theory with numerical experiments.

Rationally and polynomially convex domains in ${\mathbb C}^n$ are fundamental objects of study in the theory of functions of several complex variables. After defining and illustrating these notions, I will explain joint work with Y.Eliashberg giving a complete characterization of the possible topologies of such domains in complex dimension at least three. The proofs are based on recent progress in symplectic topology, most notably the h-principles for loose Legendrian knots and Lagrangian caps.

In this talk, I will introduce an internal, structural 
characterisation of certain convergence properties (Fréchet-Urysohn, or 
more generally, radiality) and apply this structure to understand when 
Stone spaces have these properties. This work can be generalised to 
certain Zariski topologies and perhaps to larger classes of spaces 
obtained from other structures.

The Dehn function of a group measures the complexity of the group's word problem, being the upper bound on the number of relations from a group presentation required to prove that a word in the generators represents the identity element. The Filling Theorem which was first stated by Gromov connects this to the isoperimetric functions of Riemannian manifolds. In this talk, we will see the classification of hyperbolic groups as those with a linear Dehn function, and give Bowditch's proof that a subquadratic isoperimetric inequality implies a linear one (which gives the only gap in the "isoperimetric spectrum" of exponents of polynomial Dehn functions).

The curve graph of a surface has a vertex for each curve on the surface and an edge for each pair of disjoint curves. Although it deals with very simple objects, it has connections with questions in low-dimensional topology, and some properties that encourage people to study it. Yet it is more complicated than it may look from its definition: in particular, what happens if we start following a 'diverging' path along this graph? It turns out that the curves we hit get so complicated that eventually give rise to a lamination filling up the surface. This can be understood by drawing some train track-like pictures on the surface. During the talk I will keep away from any issue that I considered too technical.

http://hottformath.wikispaces.com

 Recently several conjectures about l2-invariants of
CW-complexes have been disproved. At the heart of the counterexamples
is a method of computing the spectral measure of an element of the
complex group ring. We show that the same method can be used to
compute the finite field analog of the l2-Betti numbers, the homology
gradient. As an application we point out that (i) the homology
gradient over any field of characteristic different than 2 can be an
irrational number, and (ii) there exists a CW-complex whose homology
gradients over different fields have infinitely many different values.
 

One general approach to random number generation is to take a uniformly distributed (0,1) random variable and then invert the cumulative distribution function (CDF) to generate samples from another distribution.  This talk follows this approach, approximating the inverse CDF for the Poisson distribution in a way which is particularly efficient for vector execution on NVIDIA GPUs.

This talk describes a method to compute supersymmetric tree amplitudes and loop integrands in ten-dimensional super Yang-Mills theory. It relies on the constructive interplay between their cubic graph organization and BRST invariance of the underlying pure spinor superspace description. After a general introduction to this kind of superspace, we discuss a canonical set of multiparticle building blocks which represent tree level subdiagrams and are guided by their BRST transformation. These building blocks are shown to yield a compact solution for tree level amplitudes, and the applicability of the BRST approach to loop integrands is exemplified through recent examples at one-loop.

In highly concentrated surfactant solutions the surfactant molecules self-assemble into long flexible "wormy" structures. Properties of these wormlike micellar solutions make them ideal for use in oil recovery and in body care products (shampoo). These properties depend strongly on temperature and concentration conditions.   In solution the "worms" entangle, forming a network, but also continuously break and reform, thus earning the name ‘living polymers’. In flow these fluids exhibit spatial inhomogeneities,  shear-banding, and dynamic elastic recoil. In this talk a rheological equation of state that is capable of describing these fluids is described   The resultant governing  macroscale equations consist of a coupled nonlinear partial differential equation system.  Model predictions are presented, contrasted with experimental results, and contrasted with predictions of other existing models.  Generalizations of the model to allow the capturing of  behaviors under changing concentration or temperature conditions, namely power law and stretched exponential relaxation as opposed to exponential relaxation, will be discussed and  particularly a mesoscale stochastic simulation network model will be presented.  

We will discuss Raynaud's classical theory on Néron models of Jacobians of curves, and mention some tropical aspects of the theory that help us understand modular curves from a modern non-Archimedean viewpoint. There will be an annoyingly large number of examples illustrating the key principles throughout. 

The celebrated strong cosmic censorship conjecture in general relativity in particular suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present a recent result proving that the Cauchy horizon is stable in the sense that spacetime arising from data close to that of Kerr has a continuous metric up to the Cauchy horizon. We discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.

In this talk we will present some recent developments in the theory of ambit fields with a particular focuson limit theorems.
Ambit fields is a tempo-spatial class of models, which has been originally introduced by Barndorff-Nielsen and Schmiegel in the context of turbulence,
but found applications also in biology and finance. Its purely temporal analogue, Levy semi-stationary processes, has a continuous moving average structure
with an additional multiplicative random input (volatility or intermittency). We will briefly describe the main challenges of ambit stochastics, which
include questions from stochastic analysis, statistics and numerics. We will then focus on certain type of high frequency functionals typically called power variations.
We show some surprising non-standard limit theorems, which strongly depend on the driving Levy process. The talk is based on joint work with O.E. Barndorff-Nielsen, A. Basse-O'Connor,
J.M. Corcuera and R. Lachieze-Rey. 

I will describe a generalization of toric symplectic geometry to a new class of Poisson manifolds which are
symplectic away from a collection of hypersurfaces forming a normal crossing configuration.  Using a "tropical
moment map",  I will describe the classification of such manifolds in terms of decorated log affine polytopes,
in analogy with the Delzant classification of toric symplectic manifolds. 

Motivated by stochastic models for order books in stock exchanges we consider stochastic partial differential equations with a free boundary condition. Such equations can be considered generalizations of the classic (deterministic) Stefan problem of heat condition in a two-phase medium. 

Extending results by Kim, Zheng & Sowers we allow for non-linear boundary interaction, general Robin-type boundary conditions and fairly general drift and diffusion coefficients. Existence of maximal local and global solutions is established by transforming the equation to a fixed-boundary problem and solving a stochastic evolution equation in suitable interpolation spaces. Based on joint work with Marvin Mueller.

@email 

One of the main obstacles to forecasting sea level rise over the coming centuries is the problem of predicting changes in the flow of ice sheets, and in particular their fast-flowing outlet glaciers. While numerical models of ice sheet flow exist, they are often hampered by a lack of input data, particularly concerning the bedrock topography beneath the ice. Measurements of this topography are relatively scarce, expensive to obtain, and often error-prone. In contrast, observations of surface elevations and velocities are widespread and accurate.

In an ideal world, we could combine surface observations with our understanding of ice flow to invert for the bed topography. However, this problem is ill-posed, and solutions are both unstable and non-unique. Conventionally, this problem is circumvented by the use of regularization terms in the inversion, but these are often arbitrary and the numerical methods are still somewhat unstable.

One philosophically appealing option is to apply a fully Bayesian framework to the problem. Although some success has been had in this area, the resulting distributions are extremely difficult to work with, both from an interpretive standpoint and a numerical one. In particular, certain forms of prior information, such as constraints on the bedrock slope and roughness, are extremely difficult to represent in this framework.

A more profitable avenue for exploration is a semi-Bayesian approach, whereby a classical inverse method is regularized using terms derived from a Bayesian model of the problem. This allows for the inclusion of quite sophisticated forms of prior information, while retaining the tractability of the classical inverse problem. In particular, we can account for the severely non-Gaussian error distribution of many of our measurements, which was previously impossible.

1. Minimising Regret in Portfolio Optimisation (Simões)

When looking for an "optimal" portfolio the traditional approach is to either try to minimise risk or maximise profit. While this approach is probably correct for someone investing their own wealth, usually traders and fund managers have other concerns. They are often assessed taking into account others' performance, and so their decisions are molded by that. We will present a model for this decision making process and try to find our own "optimal" portfolio.

2. Systemic risk in financial networks (Murevics)

Abstract: In this paper I present a framework for studying systemic risk and financial contagion in interbank networks. The current financial health of institutions is expressed through an abstract measure of robustness, and the evolution of robustness in time is described through a system of stochastic differential equations. Using this model I then study how the structure of the interbank lending network affects the spread of financial contagion through different contagion channels and compare the results for different network structures. Finally I outline the future directions for developing this model.