Past Forthcoming Seminars

9 July 2020
16:00
Horatio Boedihardjo
Abstract

The concept of path signatures has been widely used in several areas of pure mathematics including in applications to data science. However, we remain unable to answer even the most basic questions about it. For instance, how to fully characterise the set of (untruncated) signatures of bounded variation paths? Can certain norms on signatures be related to the length of a path, like in Fourier isometry? In this talk, we will review some known results, explain the open problems and discuss their difficulties.

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2 July 2020
16:00
to
17:30
Nigel Higson

Further Information: 

Part of UK virtual operator algebra seminar: https://sites.google.com/view/uk-operator-algebras-seminar/home

Abstract

Abstract: John Roe was a much admired figure in topology and noncommutative geometry, and the creator of the C*-algebraic approach to coarse geometry. John died in 2018 at the age of 58. My aim in the first part of the lecture will be to explain in very general terms the major themes in John’s work, and illustrate them by presenting one of his best-known results, the partitioned manifold index theorem. After the break I shall describe a later result, about relative eta invariants, that has inspired an area of research that is still very active.


Assumed Knowledge: First part: basic familiarity with C*-algebras, plus a little topology. Second part: basic familiarity with K-theory for C*-algebras.

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  • Functional Analysis Seminar
30 June 2020
15:30
Gaultier Lambert
Abstract

In the first part of the talk, I will review the basic ideas behind Stein’s method for normal approximation and present a general result which we obtained in arXiv:1706.10251 (joint work with Michel Ledoux and Christian Webb). This result states that for a Gibbs measure, an eigenfunction of the corresponding infinitesimal generator is approximately Gaussian in a sense which will be made precise. In the second part, I will report on several applications in random matrix theory. This includes a proof of Johansson’s central limit theorem for linear statistics of beta-ensembles on \R, as well as an application to circular beta-ensembles in the high temperature regime (based on arXiv:1909.01142, joint work with Adrien Hardy).

  • Random Matrix Theory Seminars
25 June 2020
17:00
Atul Sharma
Abstract

It is a well-known fact that conformal structures on Riemann surfaces are in 1:1 correspondence with complex structures, but have you ever wondered whether this is just a fluke in 2 dimensions? In this talk, I will explain the concept of Penrose's "non-linear graviton", a fancy name for the twistor space of a hyperkahler manifold and one of the major historical achievements of Oxford maths. The twistor correspondence associates points of the hyperkahler manifold with certain holomorphic rational curves embedded in twistor space. We will see how information of the hyperkahler metric can be encoded purely in the complex structure on twistor space, giving a partial but welcome generalization of the 2-dimensional "fluke". Then I will outline a recently found Dolbeault-framework for the metric's reconstruction from local representatives of this complex structure. This provides an explicit integral formula for Kahler forms and consequently for the hyperkahler metric in terms of holomorphic data on twistor space. Finally, time permitting, I will discuss some interesting applications to (some or all of) PDEs, hyperkahler quotients, and the physics of "quantum gravity".
 

  • Junior Geometry and Topology Seminar
25 June 2020
16:00
to
18:00
Abstract

We present a method for obtaining approximate solutions to the problem of optimal execution, based on a signature method. The framework is general, only requiring that the price process is a geometric rough path and the price impact function is a continuous function of the trading speed. Following an approximation of the optimisation problem, we are able to calculate an optimal solution for the trading speed in the space of linear functions on a truncation of the signature of the price process. We provide strong numerical evidence illustrating the accuracy and flexibility of the approach. Our numerical investigation both examines cases where exact solutions are known, demonstrating that the method accurately approximates these solutions, and models where exact solutions are not known. In the latter case, we obtain favourable comparisons with standard execution strategies.

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23 June 2020
12:00
Matteo Parisi
Abstract

In this talk I will present some recent explorations of cluster-algebraic patterns in the building blocks of scattering amplitudes in N = 4 super Yang-Mills theory. In particular, I will first briefly introduce the main characters on stage, i.e. Leading Singularities, Landau singularities, the amplituhedron and cluster algebras. I will then present my main conjecture, "LL-adjacency", which makes all the above characters play together: given a maximal cut of a loop amplitude, Landau singularities and poles of each Yangian invariant appearing in any representation of the corresponding Leading Singularities can be found together in a cluster.  I will explain how the conjecture has been tested for all one-loop amplitudes up to 9 points using cluster algebraic and amplituhedron-based methods.  Finally, I will discuss implications for computing loop amplitudes and their singularity structure, and open research directions.

This is based on the joint work with Ömer Gürdoğan (arXiv: 2005.07154).

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22 June 2020
16:00
Abstract

Most of the basic results on martingale problems extend to the setting in which the generator depends on a control.  The “control” could represent a random environment, or the generator could specify a classical stochastic control problem.  The equivalence between the martingale problem and forward equation (obtained by taking expectations of the martingales) provides the tools for extending linear programming methods introduced by Manne in the context of controlled finite Markov chains to general Markov stochastic control problems.  The controlled martingale problem can also be applied to the study of constrained Markov processes (e.g., reflecting diffusions), the boundary process being treated as a control.  The talk includes joint work with Richard Stockbridge and with Cristina Costantini. 

  • Stochastic Analysis & Mathematical Finance Seminars
22 June 2020
15:45
Yair Minsky
Abstract

There is a well-known correspondence between Weil-Petersson geodesic loops in the moduli space of a surface S and hyperbolic 3-manifolds fibering over the circle with fibre S. Much is unknown, however, about the detailed relationship between geometric features of the loops and those of the 3-manifolds.

In work with Leininger-Souto-Taylor we study the relation between WP length and 3-manifold volume, when the length (suitably normalized) is bounded and the fiber topology is unbounded. We obtain a WP analogue of a theorem proved by Farb-Leininger-Margalit for the Teichmuller metric. In work with Modami, we fix the fiber topology and study connections between the thick-thin decomposition of a geodesic loop and that of the corresponding 3-manifold. While these decompositions are often in direct correspondence, we exhibit examples where the correspondence breaks down. This leaves the full conjectural picture somewhat mysterious, and raises many questions. 

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