Forthcoming events in this series


Mon, 13 Nov 2017

14:15 - 15:15
L3

Lie-Butcher series and rough paths on homogeneous manifolds I+II

HANS MUNTHE-KASS
(Bergen University)
Abstract

Abstract: Butcher’s B-series is a fundamental tool in analysis of numerical integration of differential equations. In the recent years algebraic and geometric understanding of B-series has developed dramatically. The interplay between geometry, algebra and computations reveals new mathematical landscapes with remarkable properties. 

The shuffle Hopf algebra,  which is fundamental in Lyons’s groundbreaking work on rough paths,  is based on Lie algebras without additional properties.  Pre-Lie algebras and the Connes-Kreimer Hopf algebra are providing algebraic descriptions of the geometry of Euclidean spaces. This is the foundation of B-series and was used elegantly in Gubinelli’s theory of Branched Rough Paths. 
Lie-Butcher theory combines Lie series with B-series in a unified algebraic structure based on post-Lie algebras and the MKW Hopf algebra, which is giving algebraic abstractions capturing the fundamental geometrical properties of Lie groups, homogeneous spaces and Klein geometries. 

In these talks we will give an introduction to these new algebraic structures. Building upon the works of Lyons, Gubinelli and Hairer-Kelly, we will present a new theory for rough paths on homogeneous spaces built upon the MKW Hopf algebra.

Joint work with: Charles Curry and Dominique Manchon

 

Mon, 06 Nov 2017

15:45 - 16:45
L3

Karhunen Loeve expansions in regularity structures.

SINA NEJAD
(University of Oxford)
Abstract

We consider L^2-approximations of white noise within the framework of regularity structures. Possible applications include support theorems for SPDEs driven by degenerate noises and numerics. Joint work with Ilya Chevyrev, Peter Friz and Tom Klose. 

Mon, 06 Nov 2017

14:15 - 15:15
L3

Volume distribution of nodal domains of random band-limited functions

IGOR WIGMAN
(Kings College London)
Abstract

This talk is based on a joint work with Dmitry Beliaev.

We study the volume distribution of nodal domains of families of naturally arising Gaussian random field on generic manifolds, namely random band-limited functions. It is found that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and continuity of the cumulative probability function, are established.

Mon, 30 Oct 2017

15:45 - 16:45
L3

Statistics and Rough Paths

ANASTASIA PAPAVASILEIOU
(University of Warwick)
Abstract

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.

Mon, 30 Oct 2017

14:15 - 14:45
L3

Loewner equation driven by complex-valued driving functions

HUY TRAN
(UCLA/TU Berlin)
Abstract

Consider the Loewner equation associated to the upper-half plane. This is an equation originated from an extremal problem in complex analysis. Nowadays, it attracts a lot of attention due to its connection to probability. Normally this equation is driven by a real-valued function. In this talk, we will show that the equation still makes sense when being driven by a complex-valued function. We will relate this situation to the classical situation and also to complex dynamics. 

Mon, 23 Oct 2017

15:45 - 16:45
L3

The signature approach for the supervised learning problem with sequential data input and its application

Hao Ni
(University College London)
Abstract

In the talk, we discuss how to combine the recurrent neural network with the signature feature set to tackle the supervised learning problem where the input is a data stream. We will apply this method to different datasets, including the synthetic datasets( learning the solution to SDEs ) and empirical datasets(action recognition) and demonstrate the effectiveness of this method.

 

Mon, 23 Oct 2017

14:15 - 15:15
L3

On some heavy-tail phenomena occurring in large deviations

FANNY AUGERI
(Weizmann Institute Israel)
Abstract

In this talk, we will revisit the proof of the large deviations principle of Wiener chaoses partially given by Borell, and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \mathbb{R}^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the probability measure $\nu_{\alpha}^n$, where $\nu_{\alpha} =Z_{\alpha}^{-1}e^{-|x|^{\alpha}}dx$, $\alpha \in (0,2]$, for which the large deviations are due to translations. We retrieve, as an application, the large deviations principles known for the so-called Wigner matrices without Gaussian tails of the empirical spectral measure, the largest eigenvalue, and traces of polynomials. We also apply our large deviations result to the last-passage time which yields a large deviations principle when the weight matrix has law $\mu_{\alpha}^{n^2}$, where $\mu_{\alpha}$ is the probability measure on $\mathbb{R}^+$ with density $2Z_{\alpha}^{-1}e^{-x^{\alpha}}$ when $\alpha \in (0,1)$.

 

Mon, 16 Oct 2017

15:45 - 16:45
L3

A signature-based machine learning model for bipolar disorder and borderline personality disorder

IMANOL PEREZ
(University of Oxford)
Abstract

The signature of a path has many properties that make it an excellent feature to be used in machine learning. We exploit this properties to analyse a stream of data that arises from a psychiatric study whose objective is to analyse bipolar and borderline personality disorders. We build a machine learning model based on signatures that tries to answer two clinically relevant questions, based on observations of their reported state over a short period of time: is it possible to predict if a person is healthy, has bipolar disorder or has borderline personality disorder? And given a person or borderline personality disorder, it is possible to predict his or her future mood? Signatures proved to be very effective to tackle these two problems.

Mon, 16 Oct 2017

14:15 - 15:15
L3

On uniqueness and blowup properties for a class of second order SDES

EYAL NEUMAN
(Imperial College London)
Abstract

 

Abstract. As the first  step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multi-plicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (Xt; Yt) of the equations dXt = Ytdt, dYt = jXtj_dBt, (X0; Y0) = (x0; y0). In particular, we prove that solutions arenonunique if 0 < _ < 1 and (x0; y0) = (0; 0) and unique if 1=2 < _ and (x0; y0) 6= (0; 0). We also show that blowup in _nite time holds if _ > 1 and (x0; y0) 6= (0; 0).

This is a joint work with A. Gomez, J.J. Lee, C. Mueller and M. Salins.

 

Mon, 09 Oct 2017

14:15 - 15:15
L3

Inverting the signature of a path

JIAWEI CHANG
(University of Oxford)
Abstract

Inverting the signature of a path with ideas from linear algebra with implementations.

Mon, 12 Jun 2017

15:45 - 16:45
L3

A weak universality result for the parabolic Anderson model

NICOLAS PERKOWSKI
(HU Berlin)
Abstract

We consider a class of nonlinear population models on a two-dimensional lattice which are influenced by a small random potential, and we show that on large temporal and spatial scales the population density is well described by the continuous parabolic Anderson model, a linear but singular stochastic PDE. The proof is based on a discrete formulation of paracontrolled distributions on unbounded lattices which is of independent interest because it can be applied to prove the convergence of a wide range of lattice models. This is joint work with Jörg Martin.

Mon, 05 Jun 2017

15:45 - 16:45
L3

A coupling approach to the kinetic Langevin equation

ANDREAS EBERLE
(University of Bonn)
Abstract


The (kinetic) Langevin equation is an SDE with degenerate noise that describes the motion of a particle in a force field subject to damping and random collisions. It is also closely related to Hamiltonian Monte Carlo methods. An important open question is, why in certain cases kinetic Langevin diffusions seem to approach equilibrium faster than overdamped Langevin diffusions. So far, convergence to equilibrium for kinetic Langevin diffusions has almost exclusively been studied by analytic techniques. In this talk, I present a new probabilistic approach that is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. The approach yields rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime, and it may help to shed some light on the open question mentioned above.

Mon, 05 Jun 2017

14:15 - 15:15
L3

Derivative formulae and estimates for diffusion processes and semigroups

DAVID ELWORTHY
(Warwick University)
Abstract

 There is a routine for obtaining formulae for derivatives of smooth heat semigroups,and for certain heat semigroups acting on differential forms etc, established some time ago by myself, LeJan, & XueMei Li.  Following a description of this in its general form, I will discuss its applicability in some sub-Riemannian situations and to higher order derivatives.

 

Mon, 22 May 2017

15:45 - 16:45
L3

A Stratonovich-to-Skorohod conversion formula for integrals with respect to Gaussian rough paths

THOMAS CASS
(Imperial College London)
Abstract

Lyons’ theory of rough paths allows us to solve stochastic differential equations driven by a Gaussian processes X of finite p-variation. The rough integral of the solutions against X again exists. We show that the solution also belong to the domain of the divergence operator of the Malliavin derivative, so that the 'Skorohod integral' of the solution with respect to X can also be defined. The latter operation has some properties in common with the Ito integral, and a natural question is to find a closed-form conversion formula between this rough integral and its Malliavin divergence. This is particularly useful in applications, where often one wants to compute the (conditional) expectation of the rough integral. In the case of Brownian motion our formula reduces to the classical Stratonovich-to-Ito conversion formula. There is an interesting difference between the formulae obtained in the cases 2<=p<3 and 3<=p<4, and we consider the reasons for this difference. We elaborate on the connection with previous work in which the integrand is generally assumed to be the gradient of a smooth function of X_{t}; we show that our formula can recover these results as special cases. This is joint work with Nengli Lim.

Mon, 22 May 2017

14:15 - 15:15
L3

Convergence of percolation on uniform quadrangulations

JASON MILLER
(University of Cambridge)
Abstract

Let Q be a uniformly random quadrangulation with simple boundary decorated by a critical (p=3/4) face percolation configuration.  We prove that the chordal percolation exploration path on Q between two marked boundary edges converges in the scaling limit to SLE(6) on the Brownian disk (equivalently, a Liouville quantum gravity surface).  The topology of convergence is the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces.  Our method of proof is robust and, up to certain technical steps, extends to any percolation model on a random planar map which can be explored via peeling.  Joint work with E. Gwynne.

Mon, 15 May 2017

15:45 - 16:45
L3

Well-posedness by noise for scalar conservation laws

BENJAMIN GESS
(Max Plank Institute and Unviersidad Bielefeld)
Abstract

In certain cases of (linear) partial differential equations random perturbations have been observed to cause regularizing effects, in some cases even producing the uniqueness of solutions. In view of the long-standing open problems of uniqueness of solutions for certain PDE arising in fluid dynamics such results are of particular interest. In this talk we will extend some known results concerning the well-posedness by noise for linear transport equations to the nonlinear case.

Mon, 15 May 2017

14:15 - 15:15
L3

Renormalisation of SPDE's

ILYA CHEVYREV
(University of Oxford)
Abstract

Recent work in regularity structures has provided a robust solution theory for a wide class of singular SPDEs. While much progress has been made on understanding the analytic and algebraic aspects of renormalisation of the driving signal, the action of the renormalisation group on the equation still needed to be performed by hand. In this talk, we aim to give a systematic description of the renormalisation procedure directly on the level of the PDE, which allows for explicit computation of the form of the renormalised equation. Joint work with Yvain Bruned, Ajay Chandra, and Martin Hairer.

 

Mon, 08 May 2017

15:45 - 16:45
L3

Weak Solutions of a Stochastic Landau-Lifshitz-Gilbert Equation Driven by Pure Jump Noise

ZDZISLAW BRZEZNIAK
(York University)
Abstract

In this work we study a stochastic three-dimensional Landau-Lifschitz-Gilbert equation perturbed by pure jump noise in the Marcus canonical form. We show existence of weak martingale solutions taking values in a two-dimensional sphere $\mathbb{S}^3$ and discuss certain regularity results. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. This is a joint work with Utpal Manna (Triva

Mon, 08 May 2017

14:15 - 15:15
L3

Characterising path-independence of Girsanov transform for stochastic differential equations

JIANG-LUN WU
(Swansea University)
Abstract

This talk will address a new link from stochastic differential equations (SDEs) to nonlinear parabolic PDEs. Starting from the necessary and sufficient condition of the path-independence of the density of Girsanov transform for SDEs, we derive characterisation by nonlinear parabolic equations of Burgers-KPZ type. Extensions to the case of SDEs on differential manifolds and the case od SDEs with jumps as well as to that of (infinite dimensional) SDEs on separable Hilbert spaces will be discussed. A perspective to stochastically deformed dynamical systems will be briefly considered.

Mon, 24 Apr 2017

15:45 - 16:45
L3

An analytic BPHZ theorem for regularity structures

AJAY CHANDRA
(University of Warwick)
Abstract

I will described how ideas from constructive quantum field theory can be adapted to produce a systematic approach for analytic renormalization in the theory of regularity structures.

Mon, 24 Apr 2017

14:15 - 15:15
L3

The harmonic measure on the boundary of Hastings-Levitov clusters

VITTORIA SILVESTRI
(University of Cambridge)
Abstract

The Hastings-Levitov models describe the growth of random sets (or clusters) in the complex plane as the result of iterated composition of random conformal maps. The correlations between these maps are determined by the harmonic measure density profile on the boundary of the clusters. In this talk I will focus on the simplest case, that of i.i.d. conformal maps, and obtain a description of the local fluctuations of the harmonic measure density around its deterministic limit, showing that these are Gaussian. This is joint work with James Norris.

Mon, 06 Mar 2017

15:45 - 16:45
L3

Percolation of random nodal lines

DAMIEN GAYET
(Universite Grenoble-Alpes)
Abstract

If we fix a rectangle in the affine real space and if we choose at random a real polynomial with given degree d, the probability P(d) that a component of its vanishing locus crosses the rectangle in its length is clearly positive. But is P(d) uniformly bounded from below when d increases? I will explain a positive answer to a very close question involving real analytic functions. This is a joint work with Vincent Beffara.

 

Mon, 06 Mar 2017

14:15 - 15:15
L3

Mathematical connection between Statistical Mechanics and Conformal Field Theory: an Ising model perspective

CLEMENT HONGLER
(EPFL (Ecole Polytechnique Federale de Lausanne))
Abstract

The Ising model is one of the most classical statistical mechanics model, which has seen spectacular mathematical and physical developments for almost a century. The description of its scaling limit at the phase transition is at the center of a fascinating (conjectured) connection between statistical mechanics and field theories. I will discuss how recent mathematical progress allows one to make the connection between the two-dimensional Ising model and Conformal Field Theory rigorous. If time allows, I will discuss the insight this gives one into related models and field theories.

Based off joint works with S. Benoist, D. Chelkak, H. Duminil-Copin, R. Gheissari, K. Izyurov, F. Johansson-Viklund, K. Kytölä, S. Park and S. Smirnov