In this talk I present work in progress with Wolfgang König and Nicolas Perkowski on the parabolic Anderson model (PAM) with white noise potential in 2d. We show the behavior of the total mass as the time tends to infinity. By using partial Girsanov transform and singular heat kernel estimates we can obtain the mass-asymptotics by using the eigenvalue asymptotics that have been showed in another work in progress with Khalil Chouk. 

We prove an a priori bound for solutions of the dynamic $\Phi^4_3$ equation.

This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions.

We treat the  large and small scale behaviour of solutions with completely different arguments.For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. We stress immediately that our proof is fully self-contained, but we give a detailed explanation of how our arguments relate to Hairer's. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure.

The fact that our bounds do not depend on space-time boundary conditions makes them useful for the analysis of large scale properties of solutions. They can for example be used in a compactness argument to construct solutions on the full space and their invariant measures

Abstract: We use groupoids to describe a geometric framework which can host a generalisation of Hairer's regularity structures to manifolds. In this setup, Hairer's re-expansionmap (usually denoted \Gamma) is a (direct) connection on a gauge groupoid and can therefore be viewed as a groupoid counterpart of a (local) gauge field. This definitions enables us to make the link between re-expansion maps (direct connections), principal connections and path connections, to understand the flatness of the direct connection in terms of that of the manifold and, finally, to easily build a polynomial regularity structure which we compare to the one given by Driver, Diehl and Dahlquist. (Join work with Sara Azzali, Alessandra Frabetti and Sylvie Paycha).

Quantum Yang-Mills theory is an important part of the Standard model built by physicists to describe elementary particles and their interactions. One approach to the mathematical substance of this theory consists in constructing a probability measure on an infinite-dimensional space of connections on a principal bundle over space-time. However, in the physically realistic 4-dimensional situation, the construction of this measure is still an open mathematical problem. The subject of this talk will be the physically less realistic 2-dimensional situation, in which the construction of the measure is possible, and fairly well understood.

In probabilistic terms, the 2-dimensional Yang-Mills measure is the distribution of a stochastic process with values in a compact Lie group (for example the unitary group U(N)) indexed by the set of continuous closed curves with finite length on a compact surface (for example a disk, a sphere or a torus) on which one can measure areas. It can be seen as a Brownian motion (or a Brownian bridge) on the chosen compact Lie group indexed by closed curves, the role of time being played in a sense by area.

In this talk, I will describe the physical context in which the Yang-Mills measure is constructed, and describe it without assuming any prior familiarity with the subject. I will then present a set of results obtained in the last few years by Antoine Dahlqvist, Bruce Driver, Franck Gabriel, Brian Hall, Todd Kemp, James Norris and myself concerning the limit as N tends to infinity of the Yang-Mills measure constructed with the unitary group U(N). 

 

We address the problem of pricing and hedging general exotic derivatives. We study this problem in the scenario when one has access to limited price data of other exotic derivatives. In this presentation I explore a nonparametric approach to pricing exotic payoffs using market prices of other exotic derivatives using signatures.

Abstract: Consider a gaussian field f on R^2 and a level l. One can define a random coloring of the plane by coloring a point x in black if f(x)>-l and in white otherwise. The topology of this coloring is interesting in many respects. One can study the "small scale" topology by counting connected components with fixed topology, or study the "large scale" topology by considering black crossings of large rectangles. I will present results involving these quantities.

In this talk, we consider solving nonlinear systems of equations and the unconstrained minimization problem using Newton’s method methods from the Halley class. The methods in this class have in general local and third order rate of convergence while Newton’s method has quadratic convergence. In the unconstrained optimization case, the Halley methods will require the second and third derivative. Third-order methods will, in most cases, use fewer iterations than a second-order method to reach the same accuracy. However, the number of arithmetic operations per iteration is higher for third-order methods than for a second-order method. We will demonstrate that for a large class of problems, the ratio of the number of arithmetic operations of Halley’s method and Newton’s method is constant per iteration (independent of the number of unknowns).

We say that the sparsity pattern of the third derivative (or tensor) is induced by the sparsity pattern of the Hessian matrix. We will discuss some datastructures for matrices where the indices of nonzero elements of the tensor can be computed. Historical notes will be merged into the talk.

The 23rd Annual Meeting of the SIAM UKIE Section will take place on Friday 11th January 2019 at the Mathematical Institute at the University of Oxford.

The meeting will feature five invited speakers covering a broad range of industrial and applied mathematics: 

- Lisa Fauci, Tulane University, Incoming SIAM President
- Des Higham, Strathclyde University 
- Carola-Bibiane Schoenlieb (IMA sponsored speaker), University of Cambridge 
- Kirk Soodhalter, Trinity College Dublin 
- Konstantinos Zygalakis, University of Edinburgh 

There will also be a poster session, open to PhD students and postdocs. Travel support will be available for PhD students with an accepted poster presentation, and Best Poster prizes will be awarded. 

All talks will take place in room L3 in the Andrew Wiles Building (Mathematical Institute, University of Oxford). 

Programme 
09:30 - 10:00 Registration, tea/coffee 
10:00 - 10:15 Welcome 
10:15 - 11:00 Des Higham: Our Friends are Cooler than Us 
11:00 - 11:45 Lisa Fauci: Complex dynamics of fibers in flow at the microscale 
11:45 - 12:15 Poster Blitz 
12:15 - 13:30 Lunch and Poster session 
13:30 - 14:00 SIAM UKIE Business Meeting, open to all 
14:00 - 14:45 Kirk Soodhalter: Augmented Arnoldi-Tikhonov Methods for Ill-posed Problems 
14:45 - 15:30 Konstantinos Zygalakis: Explicit stabilised Runge-Kutta methods and their application to Bayesian inverse problems 
15:30 - 16:00 Tea/coffee 
16:00 - 16:45 Carola-Bibiane Schoenlieb (IMA sponsored speaker): Variational models and partial differential equations for mathematical imaging 
16:45 - 17:00 Poster prize announcement

The well known (and notoriously hard) P vs NP problem asks whether every Boolean function with polynomial-size proofs is also computable in
polynomial time.

The standard approach to the P vs NP problem is via circuit complexity. For progressively richer classes of Boolean circuits (networks of AND, OR and NOT
logic gates), one wishes to show super-polynomial lower bounds on the sizes of circuits (as a function of the size of the input) computing some Boolean
function known to be in NP, such as the Satisfiability problem.

However, there is a more logic-oriented approach initiated by Cook and Reckhow, going through proof complexity rather than circuit complexity. For
progressively richer proof systems, one wishes to show super-polynomial lower bounds on the sizes of proofs (as a function of the size of the tautology) of
some sequence of propositional tautologies.

I will give a brief overview on known results along these two directions, and on their limitations. Somewhat surprisingly, similar techniques have been found
to be useful for these seemingly different approaches. I will say something about known connections between the approaches, and pose the question of
whether there are deeper connections.

Finally, I will discuss how the perspective of proof complexity can be used to formalize the difficulty of proving lower bounds on the sizes of computations
(or of proofs).