Past

In this paper we prove a weak necessary and sufficient maximum principle for Markov regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.

“Flash sintering” is a process reported by R Raj and co-workers in which very rapid densification of a ceramic powder compact is achieved by the passage of an electrical current through the specimen. Full density can be achieved in a few seconds (sintering normally takes several hours) and at furnace temperatures several hundred Kelvin below the temperatures required with conventional sintering. The name of the process comes from a runaway power spike that is observed at the point of sintering. Although it is acknowledged by Raj that Joule heating plays a role in the process, he and his co-authors claim that this is of minor importance and that entirely new physical effects must also be involved. However, the existence and possible relevance of these other effects of the electric field/current remains controversial. The aim of this workshop is to introduce the subject and to stimulate discussion of how mathematics could shed light on some the factors that are difficult to measure and understand experimentally.

A mini-lecture series consisting of four 1 hour lectures.

The Schelling segregation model has been extensively studied, by researchers in fields as diverse as economics, physics and computer science. While the explicit concern when the model was first introduced back in 1969, was to model the kind for racial segregation observed in large American cities, the model is sufficiently abstract to apply to almost situation in which agents or nodes arrange themselves geographically according to a preference not to be of a minority type within their own neighbourhhood. Kirman and Vinkovik have established, for example, that Schelling's model is a finite difference version of a differential equation describing interparticle forces (and applied in the modelling of cluster formation). Despite the large literature relating to the model, however, it has largely resisted rigorous analysis -- it has not been possible to prove the segregation behaviour easily observed when running simulations. For the first time we have now been able to rigorously analyse the model, and have also established some rather surprising threshold behaviour.

This talk will require no specialist background knowledge.

I'll discuss questions about the structure of long sums of

Dirichlet characters --- that is, sums of length comparable to the modulus.

For example: How often do character sums get large? Where do character sums

get large? What do character sums "look like" when then get large? This will

include some combination of theorems and experimental data.

A program for Geometric Unity is presented to argue that the seemingly baroque features of the standard model of particle physics are in fact inexorable and geometrically natural when generalizations of the Yang-Mills and Dirac theories are unified with one of general relativity.

We investigate the effect of mass transfer on the evolution of a thin two-dimensional partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, the effects of inter alia gravity, surface tension gradients, vapour transport and heat transport are neglected in favour of mathematical tractability. Our matched asymptotic analysis reveals that the leading-order outer formulation and contact-line law that is selected in the small-slip limit depends delicately on both the sign and size of the mass transfer flux. We analyse the resulting evolution of the drop and report good agreement with numerical simulations.

"Among the first successes of the h-cobordism theorem was the classification of simply connected closed 5-manifolds. Dimension five is sufficiently large to be able to implement the tools of surgery theory, yet low enough to allow an explicit classification of the manifolds. These traits make dimension five interesting in terms of existence results of geometric structures, like Riemannian metrics of positive Ricci/nonnegative sectional/positive sectional curvature, Einstein metrics, contact structures, Sasakian structures, among others. The talk will be a limited survey of the five-dimensional symbiosis between topology and geometry"

In imaging science, efficient acquisition of images by few samples with the possibility to precisely recover the complete image is a topic of significant interest. The area of compressed sensing, which in particular advocates random sampling strategies, has had already a tremendous impact on both theory and applications. The necessary requirement for such techniques to be applicable is the sparsity of the original data within some transform domain. Recovery is then achieved by, for instance, $\ell_1$ minimization. Various applications however do not allow complete freedom in the choice of the samples. Take Magnet Resonance Imaging (MRI) for example, which only provides access to Fourier samples. For this particular application, empirical results still showed superior performance of compressed sensing techniques.

\\

\\

In this talk, we focus on sparse sampling strategies under the constraint that only Fourier samples can be accessed. Since images -- and in particular images from MRI -- are governed by anisotropic features and shearlets do provide optimally sparse approximations of those, compactly supported shearlet systems will be our choice for the reconstruction procedure. Our sampling strategy then exploits a careful variable density sampling of the Fourier samples with $\ell_1$-analysis based reconstruction using shearlets. Our main result provides success guarantees and shows that this sampling and reconstruction strategy is optimal.

\\

\\

This is joint work with Wang-Q Lim (Technische Universit\"at Berlin).

Inspired by some recents developments in the theory of small-strain elastoplasticity, we

both revisit and generalize the formulation of the quasistatic evolutionary problem in

perfect plasticity for heterogeneous materials recently given by Francfort and Giacomini.

We show that their definition of the plastic dissipation measure is equivalent to an

abstract one, where it is defined as the supremum of the dualities between the deviatoric

parts of admissible stress fields and the plastic strains. By means of this abstract

definition, a viscoplastic approximation and variational techniques from the theory of

rate-independent processes give the existence of an evolution statisfying an energy-

dissipation balance and consequently Hill's maximum plastic work principle for an

abstract and very large class of yield conditions.

 A classical question in the model theory of fields is to find out which fields are model complete in the language of rings. It turns out that all well-known examples of model complete fields are quite rigid when it comes to henselianity. We discuss some first results which indicate that in residue characteristic zero, definable henselian valuations prevent model completeness.

In this talk I will describe an attempt to construct a conformal field theory with target space a symmetric product of $R^D$ (referred to by physicists as orbifold sigma model). The construction uses branched covers of $S^2$ to lift the well studied formulation of a sigma model on $S^2$, in terms of vertex operator algebras, to higher genus surfaces. I will motivate and explain this construction.

I will give a gentle introduction to uniformly finite homology. The highlight application will be showing existence of aperiodic tilings of the hyperbolic plane.

The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra

was introduced as a generalised phase space for a quantum system in the

so-called topos approach to quantum theory. Here, it will be shown that

the spectral presheaf has many features of a spectrum of a

noncommutative operator algebra (and that it can be defined for other

classes of algebras as well). The main idea is that the spectrum of a

nonabelian algebra may not be a set, but a presheaf or sheaf over the

base category of abelian subalgebras. In general, the spectral presheaf

has no points, i.e., no global sections. I will show that there is a

contravariant functor from unital C*-algebras to their spectral

presheaves, and that a C*-algebra is determined up to Jordan

*-isomorphisms by its spectral presheaf in many cases. Moreover, time

evolution of a quantum system can be described in terms of flows on the

spectral presheaf, and commutators show up in a natural way. I will

indicate how combining the Jordan and Lie algebra structures may lead to

a full reconstruction of nonabelian C*- or von Neumann algebra from its

spectral presheaf.

The ``k-commodity flow problem'' is: we are given k pairs of vertices of a graph, and we ask whether there are k flows in the graph, where the ith flow is between the ith pair of vertices, and has total value one, and for each edge, the sum of the absolute values of the flows along it is at most one. We may also require the flows to be 1/2-integral, or indeed 1/p-integral for some fixed p.

If the problem is feasible (that is, the desired flows exist) then it is still feasible after contracting any edge, so let us say a flow problem is ``critical'' if it is infeasible, but becomes feasible when we contract any edge. In many special cases, all critical instances have only two vertices, but if we ask for integral flows (that is, p = 1, essentially the edge-disjoint paths problem), then there arbitrarily large critical instances, even with k = 2. But it turns out that p = 1 is the only bad case; if p>1 then all critical instances have bounded size (depending on k, but independent of p), and the same is true if there is no integrality requirement at all.

The proof gives rise to a very simple algorithm for the k edge-disjoint paths problem in 4-edge-connected graphs.

Cutting-edge experiments in quantum communications are reaching regimes

where relativistic effects can no longer be neglected. For example, there

are advanced plans to use satellites to implement teleportation and quantum

cryptographic protocols. Relativistic effects can be expected at these

regimes: the Global Positioning System (GPS), which is a system of

satellites that is used for time dissemination and navigation, requires

relativistic corrections to determine time and positions accurately.

Therefore, it is timely to understand what are the effects of gravity and

motion on entanglement and other quantum properties exploited in quantum

information.

In this talk I will show that entanglement can be created or degraded by

gravity and non-uniform motion. While relativistic effects can degrade the

efficiency of teleportation between moving observers, the effects can also

be exploited in quantum information. I will show that the relativistic

motion of a quantum system can be used to perform quantum gates. Our

results, which will inform future space-based experiments, can be

demonstrated in table-top experiments using superconducting circuits.