University of Warwick

A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about the numerical error. In this paper we propose a novel statistical model for this numerical error set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging.

I will talk about some basic facts about slope stable sheaves and the Bogomolov inequality.  New techniques from stability conditions will imply new stronger bounds on Chern characters of stable sheaves on some special varieties, including  Fano varieties, quintic threefolds and etc. I will discuss the progress in this direction and some related open problems.

I will talk about R^n valued random processes driven by a "noise", which is generated by a deterministic dynamical system, randomness coming from the choice of the initial condition.

Such processes were considered by D.Kelly and I.Melbourne.I will present our joint work with I.Chevyrev, P.Friz, I.Melbourne and H.Zhang, where we consider the noise with long term memory. We prove convergence to solution of a stochastic differential equation which is, depending on the noise, driven by either a Brownian motion (optimizing the assumptions of Kelly-Melbourne) or a Lévy process.Our work is made possible by recent progress in rough path theory for càdlàg paths in p-variation topology.

The classical Lorentz gas model introduced by Lorentz in 1905, studied further by Sinai in the 1960s, provides a rich source of examples of chaotic dynamical systems with strong stochastic properties (despite being entirely deterministic).  Central limit theorems and convergence to Brownian motion are well understood, both with standard n^{1/2} and nonstandard (n log n)^{1/2} diffusion rates.

In joint work with Paulo Varandas, we discuss examples with diffusion rate n^{1/a}, 1<a<2, and prove convergence to an a-stable Levy process.  This includes to the best of our knowledge the first natural examples where the M_2 Skorokhod topology is the appropriate one.



 

In the 80s, Hatcher and Thurston introduced the pants graph as a tool to prove that the mapping class group of a closed, orientable surface is finitely presented. The pants graph remains relevant for the study of the mapping class group, sitting between the marking graph and the curve graph. More precisely, there is a sequence of natural coarse lipschitz maps taking the marking graph via the pants graph to the curve graph.

A second motivation for studying the pants graph comes from Teichmüller theory. Brock showed that the pants graph can be interpreted as a combinatorial model for Teichmüller space with the Weil-Petersson metric.

In this talk I will introduce the pants graph, discuss some of its properties and state a few open questions.

In 2012 Eskin, Fisher and Whyte proved there was a locally finite vertex transitive graph which was not quasi-isometric to any connected locally finite Cayley Graph. This motivates the study of vertex transitive graphs from a geometric group theory point of view. We will discus how concepts and problems from group theory generalise to this setting. Constructing one framework in which problems can be framed so that techniques from group theory can be applied. This is work in progress with Agelos Georgakopoulos.

In a given ambient Riemannian manifold with boundary, geometric objects of particular interest are those properly embedded submanifolds that are critical points of the volume functional, when allowed variations are only those that preserve (but not necessarily fix) the ambient boundary. This variational condition translates into a quite nice geometric condition, namely, minimality and orthogonal intersection with the ambient boundary. Even when the ambient manifold is simply a ball in the Euclidean space, the theory of these objects is very rich and interesting. We would like to discuss several aspects of the theory, including our own contributions to the subject on topics such as: classification results, index estimates and compactness (joint works with different groups of collaborators - I. Nunes, A. Carlotto, B. Sharp, R. Buzano -  will be appropriately mentioned). 

After a brief introduction to the synthetic notions of Ricci curvature lower bounds in terms of optimal transportation, due to Lott-Sturm-Villani, I will discuss some applications to smooth Riemannian manifolds. These include: rigidity and stability of Levy- Gromov inequality, an almost euclidean isoperimetric inequality motivated by the celebrated Perelman’s Pseudo-Locality Theorem for Ricci flow. Joint work with F. Cavalletti.

We propose a new splitting algorithm to solve a class of quasilinear PDEs with convex and quadratic growth gradients. 

By splitting the original equation into a linear parabolic equation and a Hamilton-Jacobi equation, we are able to solve both equations explicitly. 

In particular, we solve the associated Hamilton-Jacobi equation by the Hopf-Lax formula, 

and interpret the splitting algorithm as a stochastic Hopf-Lax approximation of the quasilinear PDE.  

We show that the numerical solution will converge to the viscosity solution of the equation.  

The upper bound of the convergence rate is proved based on Krylov's shaking coefficients technique, 

while the lower bound is proved based on Barles-Jakobsen's optimal switching approximation technique. 

Based on joint work with Shuo Huang and Thaleia Zariphopoulou.

Having made sense of differential equations driven by rough paths, we now have a new set of models available but when it comes to calibrating them to data, the tools are still underdeveloped. I will present some results and discuss some challenges related to building these tools.