Leeds University

I will explain how the model-theoretic algebraic closure in Zilber’s pseudo-exponential field can be described in terms of the self-sufficient closure. I will sketch a proof and show how the Mordell-Lang conjecture for algebraic tori comes into play. If time permits, I’ll also talk about the characterisation of strongly minimal sets and their geometries. This is joint work (still in progress) with Jonathan Kirby.

Zarankiewicz's problem for hypergraphs asks for upper bounds on the number of edges of a hypergraph that has no complete sub-hypergraphs of a given size. Let M be an o-minimal structure. Basit-Chernikov-Starchenko-Tao-Tran (2021) proved that the following are equivalent:

(1) "linear Zarankiewicz's bounds" hold for hypergraphs whose edge relation is induced by a fixed relation definable in M

(2) M does not define an infinite field.

We prove that the following are equivalent:

(1') linear Zarankiewicz bounds hold for sufficiently "distant" hypergraphs whose edge relation is induced by a fixed relation definable in M

(2') M does not define a full field (that is, one whose domain is the whole universe of M).

This is joint work (in progress) with Aris Papadopoulos.

 I will report on my recent work on dimension, definable groups, and definable fields in vector spaces over algebraically closed [real closed] fields equipped with a non-degenerate alternating bilinear form or a non-degenerate [positive-definite] symmetric bilinear form. After a brief overview of the background, I will discuss a notion of dimension and some other ingredients of the proof of the main result, which states that, in the above context, every definable group is (algebraic-by-abelian)-by-algebraic [(semialgebraic-by-abelian)-by-semialgebraic]. It follows from this result that every definable field is definable in the field of scalars, hence either finite or definably isomorphic to it [finite or algebraically closed or real closed].
 

We consider a non-linear PDE on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition. To conclude we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.

We study the value of a zero-sum stopping game in which the terminal payoff function depends on the underlying process and on an additional randomness (with finitely many states) which is known to one player but unknown to the other. Such asymmetry of information arises naturally in insider trading when one of the counterparties knows an announcement before it is publicly released, e.g., central bank's interest rates decision or company earnings/business plans. In the context of game options this splits the pricing problem into the phase before announcement (asymmetric information) and after announcement (full information); the value of the latter exists and forms the terminal payoff of the asymmetric phase.

The above game does not have a value if both players use pure stopping times as the informed player's actions would reveal too much of his excess knowledge. The informed player manages the trade-off between releasing information and stopping optimally employing randomised stopping times. We reformulate the stopping game as a zero-sum game between a stopper (the uninformed player) and a singular controller (the informed player). We prove existence of the value of the latter game for a large class of underlying strong Markov processes including multi-variate diffusions and Feller processes. The main tools are approximations by smooth singular controls and by discrete-time games.

In this talk I will present three families of differential equations (SDEs, BSDEs and PDEs) and their links to each other. The novel fact is that some of the coefficients are generalised functions living in a fractional Sobolev space of negative order. I will discuss the appropriate notion of solution for each type of equation and show existence and uniqueness results. To do so, I will use tools from analysis like semigroup theory, pointwise products, theory of function spaces, as well as classical tools from probability and stochastic analysis. The link between these equations will play a fundamental role, in particular the results on the PDE are used to give a meaning and solve both the forward and the backward stochastic differential equations.  

The rapidly increasing number of cores in high-performance computing systems causes a multitude of challenges for developers of numerical methods. New parallel algorithms are required to unlock future growth in computing power for applications and energy efficiency and algorithm-based fault tolerance are becoming increasingly important. So far, most approaches to parallelise the numerical solution of partial differential equations focussed on spatial solvers, leaving time as a bottleneck. Recently, however, time stepping methods that offer some degree of concurrency, so-called parallel-in-time integration methods, have started to receive more attention.

I will introduce two different numerical algorithms, Parareal (by Lions et al., 2001) and PFASST (by Emmett and Minion, 2012), that allow to exploit concurrency along the time dimension in parallel computer simulations solving partial differential equations. Performance results for both methods on different architectures and for different equations will be presented. The PFASST algorithm is based on merging ideas from Parareal, spectral deferred corrections (SDC, an iterative approach to derive high-order time stepping methods by Dutt et al. 2000) and nonlinear multi-grid. Performance results for PFASST on close to half a million cores will illustrate the potential of the approach. Algorithmic modifications like IPFASST will be introduced that can further reduce solution times. Also, recent results showing how parallel-in-time integration can provide algorithm-based tolerance against hardware faults will be shown.

The inverse acoustic obstacle scattering problem, in its most general

form, seeks to determine the nature of an unknown scatterer from knowl-

edge of its far eld or radiation pattern. The problem which is the main

concern here is:

If the scattering cross section, i.e the absolute value of the radiation

pattern, of an unknown scatterer is known determine its shape.

In this talk we explore the problem from a number of points of view.

These include questions of uniqueness, methods of solution including it-

erative methods, the Minkowski problem and level set methods. We con-

clude by looking at the problem of acoustically invisible gateways and its

connections with cloaking

TBA