University of Oxford

In this talk, we consider the setting: a random realization of an evolving dynamical system, and explain how, using notions common in the theory of rough paths, such as the signature, and shuffle product, one can provide a new united approach to the fundamental problem of predicting the conditional distribution of the near future given the past. We will explain how the problem can be reduced to a linear regression and least squaresanalysis. The approach is clean and systematic and provides a clear gradation of finite dimensional approximations. The approach is also non-parametric and very general but still presents itself in computationally tractable and flexible restricted forms for concrete problems. Popular techniques in time series analysis such as GARCH can be seen to be restricted special cases of our approach but it is not clear they are always the best or most informative choices. Some numerical examples will be shown in order to compare our approach and standard time series models.

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.

It is well known that the trapezoid rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with powerful algorithms all across scientific computing, including double exponential and Gauss quadrature, computation of inverse Laplace transforms, special functions, computational complex analysis, the computation of functions of matrices and operators, rational approximation, and the solution of partial differential equations.

This talk represents joint work with Andre Weideman of the University of Stellenbosch.

I will outline Bergeron-Wise’s proof that the Virtual Haken Conjecture follows from Wise’s Conjecture on virtual specialness of non-positively curved cube complexes. If time permits, I will sketch some highlights from the proof of Wise’s Conjecture due to Agol and based on the Weak Separation Theorem of Agol-Groves-Manning.

Self-similarity is a fundamental idea in many areas of mathematics. In this talk I will explain how it has entered group theory and the links between self-similar groups and other areas of research. There will also be pretty pictures.

First of all, I will give an overview of what the phenomenon of homological stability is and why it's useful, with plenty of examples. I will then introduce configuration spaces -- of various different kinds -- and give an overview of what is known about their homological stability properties. A "configuration" here can be more than just a finite collection of points in a background space: in particular, the points may be equipped with a certain non-local structure (an "orientation"), or one can consider unlinked embedded copies of a fixed manifold instead of just points. If by some miracle time permits, I may also say something about homological stability with local coefficients, in general and in particular for configuration spaces.

The theory of equations over groups goes back to the very beginning of group theory and is linked to many deep problems in mathematics, such as the Diophantine problem over rationals. In this talk, we shall survey some of the key results on equations over groups, give an outline of the Makanin-Razborov process (an algorithm for solving equations over free groups) and its connections to other results in group theory and low-dimensional topology.

We will discuss (very) recent work by Hensel; Przytycki and Webb, who describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

The eponymous result is due to Bridson and Vogtmann, and was proven in their paper "Automorphisms of Automorphism Groups of Free Groups" (Journal of Algebra 229). While I'll remind you all the basic definitions, it would be very helpful to be already somewhat familiar with the outer space.

I'll discuss work of Wise and Ollivier-Wise that gives cubulations of certain small cancellation and random groups, which in turn shows that they do not have property (T).