Mon, 02 Mar 2015

14:15 - 15:15
Oxford-Man Institute

tba

Michael Kozdron
(University of Regina)
Abstract

tba

Mon, 16 Feb 2015

15:45 - 16:45
Oxford-Man Institute

tba

Dmitry Chellak
Abstract

tba

Mon, 16 Feb 2015

14:15 - 15:15
Oxford-Man Institute

Learning with Cross-Kernel Matrices and Ideal PCA

Franz Kiraly
(University College London)
Abstract

 We describe how cross-kernel matrices, that is, kernel matrices between the data and a custom chosen set of `feature spanning points' can be used for learning. The main potential of cross-kernel matrices is that (a) they provide Nyström-type speed-ups for kernel learning without relying on subsampling, thus avoiding potential problems with sampling degeneracy, while preserving the usual approximation guarantees and the attractive linear scaling of standard Nyström methods and (b) the use of non-square matrices for kernel learning provides a non-linear generalization of the singular value decomposition and singular features. We present a novel algorithm, Ideal PCA (IPCA), which is a cross-kernel matrix variant of PCA, showcasing both advantages: we demonstrate on real and synthetic data that IPCA allows to (a) obtain kernel PCA-like features faster and (b) to extract novel features of empirical advantage in non-linear manifold learning and classification.

Mon, 09 Feb 2015

15:45 - 16:45
Oxford-Man Institute

tba

tba
Abstract

tba

Mon, 09 Feb 2015

14:15 - 15:15
Oxford-Man Institute

The Renormalization Group as a tool of Rigorous Probability Theory

Ajay Chandra
(Warwick University)
Abstract

The Renormalization Group (RG) was pioneered by the physicist Kenneth Wilson in the early 70's and since then it has become a fundamental tool in physics. RG remains the most general philosophy for understanding how many models in statistical mechanics behave near their critical point but implementing RG analysis in a mathematically rigorous way remains quite challenging.

I will describe how analysis of RG flows translate into statements about continuum limits, universality, and cross-over phenomena - as a concrete example I will speak about some joint work with Abdelmalek Abdesselam and Gianluca Guadagni.

Mon, 02 Feb 2015

15:45 - 16:45
Oxford-Man Institute

Spectral volume and surface measures via the Dixmier trace for local symmetric Dirichlet spaces with Weyl type eigenvalue asymptotics

Naotaka Kajino
Abstract

Spectral volume and surface measures via the Dixmier trace for local symmetric Dirichlet spaces with Weyl type eigenvalue asymptotics

 

The purpose of this talk is to present the author's recent results of on an

operator theoretic way of looking atWeyl type Laplacian eigenvalue asymptotics

for local symmetric Dirichlet spaces.

For the Laplacian on a d-dimensional Riemannian manifoldM, Connes' trace

theorem implies that the linear functional  coincides with

(a constant multiple of) the integral with respect to the Riemannian volume

measure of M, which could be considered as an operator theoretic paraphrase

of Weyl's Laplacian eigenvalue asymptotics. Here  denotes a Dixmier trace,

which is a trace functional de_ned on a certain ideal of compact operators on

a Hilbert space and is meaningful e.g. for compact non-negative self-adjoint

operators whose n-th largest eigenvalue is comparable to 1/n.

The first main result of this talk is an extension of this fact in the framework

of a general regular symmetric Dirichlet space satisfying Weyl type asymptotics

for the trace of its associated heat semigroup, which was proved for Laplacians

on p.-c.f. self-simiar sets by Kigami and Lapidus in 2001 under a rather strong

assumption.

Moreover, as the second main result of this talk it is also shown that, given a

local regular symmetric Dirichlet space with a sub-Gaussian heat kernel upper

bound and a (sufficiently regular) closed subset S, a “spectral surface measure"

on S can be obtained through a similar linear functional involving the Lapla-

cian with Dirichlet boundary condition on S. In principle, corresponds to the

second order term for the eigenvalue asymptotics of this Dirichlet Laplacian, and

when the second order term is explicitly known it is possible to identify  For

example, in the case of the usual Laplacian on Rd and a Lipschitz hypersurface S,is a constant multiple of the usual surface measure on S.

Mon, 02 Feb 2015

14:15 - 15:15
Oxford-Man Institute

Maximal couplings and geometry

Sayan Banerjee
(Warwick University)
Abstract

Maximal couplings are couplings of Markov processes where the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian couplings are coupling strategies where neither process is allowed to look into the future of the other before making the next transition. These are easier to describe and play a fundamental role in many branches of probability and analysis. Hsu and Sturm proved that the reflection coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada proved that to have a MMC for Brownian motions on a Riemannian manifold, the manifold should have a reflection structure, and thus proved the first result connecting this purely probabilistic phenomenon (MMC) to the geometry of the underlying space.

Mon, 19 Jan 2015

15:45 - 16:45
Oxford-Man Institute

A stochastic free boundary problem

Martin Keller-Ressel
(Dresden University of Technology)
Abstract

Motivated by stochastic models for order books in stock exchanges we consider stochastic partial differential equations with a free boundary condition. Such equations can be considered generalizations of the classic (deterministic) Stefan problem of heat condition in a two-phase medium. 

Extending results by Kim, Zheng & Sowers we allow for non-linear boundary interaction, general Robin-type boundary conditions and fairly general drift and diffusion coefficients. Existence of maximal local and global solutions is established by transforming the equation to a fixed-boundary problem and solving a stochastic evolution equation in suitable interpolation spaces. Based on joint work with Marvin Mueller.

Mon, 19 Jan 2015

14:15 - 15:15
Oxford-Man Institute

'Optimal Switching in Finite Horizon under State Constraints’

Idris Kharoubbi
(Université Paris Dauphine)
Abstract

'We study an optimal switching problem with a state constraint: the controller is only allowed to choose strategies that keep the controlled diffusion in a closed domain. We prove that the value function associated to the weak formulation of this problem is the limit of the value function associated to an unconstrained switching problem with penalized coefficients, as the penalization parameter goes to infinity. This convergence allows to set a dynamic programming principle for the constrained switching problem. We then prove that the value function is a constrained viscosity solution to a system of variational inequalities (SVI for short). We finally prove that the value function is the maximal solution to this SVI. All our results are obtained without any regularity assumption on the constraint domain.’

Tue, 24 Feb 2015
12:30
Oxford-Man Institute

Measuring and predicting human behaviour using online data

Tobias Preis
(University of Warwick)
Abstract

In this talk, I will outline some recent highlights of our research, addressing two questions. Firstly, can big data resources provide insights into crises in financial markets? By analysing Google query volumes for search terms related to finance and views of Wikipedia articles, we find patterns which may be interpreted as early warning signs of stock market moves. Secondly, can we provide insight into international differences in economic wellbeing by comparing patterns of interaction with the Internet? To answer this question, we introduce a future-orientation index to quantify the degree to which Internet users seek more information about years in the future than years in the past. We analyse Google logs and find a striking correlation between the country's GDP and the predisposition of its inhabitants to look forward. Our results illustrate the potential that combining extensive behavioural data sets offers for a better understanding of large scale human economic behaviour.

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