Past

I shall talk about Subgroup Membership Problem for amalgamated products of finite rank free groups. I'm going to show how one can solve different versions of this problem in amalgams of free groups and give an estimate of the complexity of some algorithms involved.  This talk is based on a joint paper with A. J. Duncan.

In my talk I shall give a small survey on some algorithmic properties of amalgamated products of finite rank 
free groups. In particular, I'm going to concentrate on Membership Problem for this groups. Apart from being algorithmically interesting, amalgams of free groups admit a lot of interpretations. I shall show how to 
characterize this construction from the point of view of geometry and linguistic.  

I will explain some recent joint work with Georgios Dimitroglou Rizell in which we use moduli spaces of holomorphic discs with boundary on a monotone Lagrangian torus in ${\mathbb C}^n$ to prove that all such tori are smoothly isotopic when $n$ is odd and at least 5

The quantification and management of risk in financial markets
is at the center of modern financial mathematics. But until recently, risk
assessment models did not consider the effects of inter-connectedness of
financial agents and the way risk diversification impacts the stability of
markets. I will give an introduction to these problems and discuss the
implications of some mathematical models for dealing with them. 

Exact Lagrangian immersions are governed by an h-principle, whilst exact Lagrangian

embeddings are well-known to be constrained by strong rigidity theorems coming from

holomorphic curve theory. We consider exact Lagrangian immersions in Euclidean space with a

prescribed number of double points, and find that the borderline between flexibility and

rigidity is more delicate than had been imagined. The main result obtains constraints on such

immersions with exactly one double point which go beyond the usual setting of Morse or Floer

theory. This is joint work with Tobias Ekholm, and in part with Ekholm, Eliashberg and Murphy.

We relate the expected signature to the Fourier transform of n-point functions, first studied by O. Schramm, and subsequently
by J. Cardy and Simmon, D. Belyaev and J. Viklund. We also prove that the signatures determine the paths in the complement of a Chordal SLE null set. In the end, we will also discuss an idea on how to extend the uniqueness of signatures result by Hambly and Lyons (2006) to paths with finite 1<p<2variations.

Modelling reactive flows, diffusion, transport and mechanical interactions in media consisting of multiple phases, e.g. of a fluid and a solid phase in a porous medium, is giving rise to many open problems for multi-scale analysis and simulation. In this lecture, the following processes are studied:

diffusion, transport, and reaction of substances in the fluid and the solid phase,

mechanical interactions of the fluid and solid phase,

change of the mechanical properties of the solid phase by chemical reactions,

volume changes (“growth”) of the solid phase.

These processes occur for instance in soil and in porous materials, but also in biological membranes, tissues and in bones. The model equations consist of systems of nonlinear partial differential equations, with initial-boundary conditions and transmission conditions on fixed or free boundaries, mainly in complex domains. The coupling of processes on different scales is posing challenges to the mathematical analysis as well as to computing. In order to reduce the complexity, effective macroscopic equations have to be derived, including the relevant information from the micro scale.

In case of processes in tissues, a homogenization limit leads to an effective, mechanical system, containing a pressure gradient, which satisfies a generalized, time-dependent Darcy law, a Biot-law, where the chemical substances satisfy diffusion-transport-reaction equations and are influencing the mechanical parameters.

The interaction of the fluid and the material transported in a vessel with its flexible wall, incorporating material and changing its structure and mechanical behavior, is a process important e.g. in the vascular system (plague-formation) or in porous media.

The lecture is based on recent results obtained in cooperation with A. Mikelic, M. Neuss-Radu, F. Weller and Y. Yang.

Abstract: The aim of this lecture is to give a general introduction to the theory of interacting particle methods and an overview of its applications to numerical finance. We survey the main techniques and results on interacting particle systems and explain how they can be applied to deal with a variety of financial numerical problems such as: pricing complex path dependent European options, computing sensitivities, American option pricing or solving numerically partially observed control problems.

In this talk, we will introduce how to apply Green's function method to get  pointwise estimates for solutions of the Cauchy problem of nonlinear evolution equations with dissipative  structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and exhibit the generalized Huygen's principle. Then, for other nonlinear dissipative evolution equations, we will  introduce some recent results and give brief explanations. Our approach is based on the detailed analysis of the Green's function of the linearized system and micro-local analysis, such as frequency decomposition and so on.

The Ross Recovery Theorem gives sufficient conditions under which the

market’s beliefs

can be recovered from risk-neutral probabilities. His approach places

mild restrictions on the form of the preferences of

the representative investor. We present an alternative approach which

has no restrictions beyond preferring more to less,

Instead, we restrict the form and risk-neutral dynamics of John Long’s

numeraire portfolio. We also replace Ross’ finite state Markov chain

with a diffusion with bounded state space. Finally, we present some

preliminary results for diffusions on unbounded state space.

In particular, our version of Ross recovery allows market beliefs to be

recovered from risk neutral probabilities in the classical Cox

Ingersoll Ross model for the short interest rate.

The original transport problem is to optimally move a pile of soil to an excavation.

Mathematically, given two measures of equal mass, we look for an optimal bijection that takes

one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed

this problem by considering a measure whose marginals agree with given two measures instead of

a bijection. This generalization linearizes the problem. Hence, allows for an easy existence

result and enables one to identify its convex dual.

In robust hedging problems, we are also given two measures. Namely, the initial and the final

distributions of a stock process. We then construct an optimal connection. In general, however,

the cost functional depends on the whole path of this connection and not simply on the final value.

Hence, one needs to consider processes instead of simply the maps S. The probability distribution

of this process has prescribed marginals at final and initial times. Thus, it is in direct analogy

with the Kantorovich measure. But, financial considerations restrict the process to be a martingale

Interestingly, the dual also has a financial interpretation as a robust hedging (super-replication)

problem.

In this talk, we prove an analogue of Kantorovich duality: the minimal super-replication cost in

the robust setting is given as the supremum of the expectations of the contingent claim over all

martingale measures with a given marginal at the maturity.

This is joint work with Yan Dolinsky of Hebrew University.

Please note the change of venue!

Suppose there is a system where certain objects move through a network. The objects are detected only when they pass through a sparse set of points in the network. For example, the objects could be vehicles moving along a road network, and observed by a radar or other sensor as they pass through (or originate or terminate at) certain key points in the network, but which cannot be observed continuously and tracked as they travel from one point to another. Alternatively they could be data packets in a computer network. The detections only record the time at which an object passes by, and contain no information about identity that would trivially allow the movement of an individual object from one point to another to be deduced. It is desired to determine the statistics of the movement of the objects through the network. I.e. if an object passes through point A at a certain time it is desired to determine the probability density that the same object will pass through a point B at a certain later time.

The system might perhaps be represented by a graph, with a node at each point where detections are made. The detections at each node can be represented by a signal as a function of time, where the signal is a superposition of delta functions (one per detection). The statistics of the movement of objects between nodes must be deduced from the correlations between the signals at each node. The problem is complicated by the possibility that a given object might move between two nodes along several alternative routes (perhaps via other nodes or perhaps not), or might travel along the same route but with several alternative speeds.

What prior knowledge about the network, or constraints on the signals, are needed to make this problem solvable? Is it necessary to know the connections between the nodes or the pdfs for the transition time between nodes a priori, or can this be deduced? What conditions are needed on the information content of the signals? (I.e. if detections are very sparse on the time scale for passage through the network then the transition probabilities can be built up by considering each cascade of detections independently, while if detections are dense then it will presumably be necessary to assume that objects do not move through the network independently, but instead tend to form convoys that are apparent as a pattern of detections that persist for some distance on average). What limits are there on the noise in the signal or amount of unwanted signal, i.e. false detections, or objects which randomly fail to be detected at a particular node, or objects which are detected at one node but which do not pass through any other nodes? Is any special action needed to enforce causality, i.e. positive time delays for transitions between nodes?

One of the most subtle aspects of the correspondence between automorphic and Galois representations is the weight part of Serre conjectures, namely describing the weights of modular forms corresponding to mod p congruence class of Galois representations. We propose a direct geometric approach via studying the mod p cohomology groups of certain integral models of modular or Shimura curves, involving Deligne-Lusztig curves with the action of GL(2) over finite fields. This is a joint work with James Newton.

In this talk we will discuss the mathematical modelling of the performance of Lithium-ion batteries. A mathematical model based on a macro-homogeneous approach developed by John Neuman will be presented. The uniqueness and existence of solution of the corresponding problem will be also discussed.

Very-large scale data analytics are an alleged golden goose for efforts in parallel and distributed computing, and yet contemporary statistics remain somewhat of a dark art for the uninitiated. In this presentation, we are going to take a mathematical and algorithmic look beyond the veil of Big Data by studying the structure of the algorithms and data, and by analyzing the fit to existing and proposed computer systems and programming models. Towards highly scalable kernels, we will also discuss some of the promises and challenges of approximation algorithms using randomization, sampling, and decoupled processing, touching some contemporary topics in parallel numerics.