Fri, 14 Feb 2025
15:00
L4

Distance-from-flat persistent homology transforms

Nina Otter
(Inria Saclay)

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Abstract
The persistent homology transform (PHT) was introduced in the field of Topological Data Analysis about 10 years ago, and has since been proven to be a very powerful descriptor of Euclidean shapes. The PHT consists of scanning a shape from all possible directions and then computing the persistent homology of sublevel set filtrations of the respective height functions; this results in a sufficient and continuous descriptor of Euclidean shapes. 
 
In this talk I will introduce a generalisation of the PHT in which we consider arbitrary parameter spaces and sublevel-set filtrations with respect to any function. In particular, we study transforms, defined on the Grassmannian AG(m,n) of affine subspaces of n-dimensional Euclidean space, which allow to scan a shape by probing it with all possible affine m-dimensional subspaces P, for fixed dimension m, and by then computing persistent homology of sublevel-set filtrations of the function encoding the distance from the flat P. We call such transforms "distance-from-flat PHTs". I will discuss how these transforms generalise known examples, how they are sufficient descriptors of shapes and finally present their computational advantages over the classical persistent homology transform introduced by Turner-Mukherjee-Boyer. 
Mon, 26 May 2025
15:30
L3

Transport of Gaussian measures under the flow of semilinear (S)PDEs: quasi-invariance and singularity.

Dr. Leonardo Tolomeo
(University of Edinburgh)
Abstract

In this talk, we consider the Cauchy problem for a number of semilinear PDEs, subject to initial data distributed according to a family of Gaussian measures.  

 

We first discuss how the flow of Hamiltonian equations transports these Gaussian measures. When the transported measure is absolutely continuous with respect to the initial measure, we say that the initial measure is quasi-invariant. 

 

In the high-dispersion regime, we exploit quasi-invariance to build a (unique) global flow for initial data with negative regularity, in a regime that cannot be replicated by the deterministic (pathwise) theory.  

 

In the 0-dispersion regime, we discuss the limits of this approach, and exhibit a sharp transition from quasi-invariance to singularity, depending on the regularity of the initial measure. 

 

We will also discuss how the same techniques can be used in the context of stochastic PDEs, and how they provide information on the invariant measures for their flow. 

 

This is based on joint works with  J. Coe (University of Edinburgh), J. Forlano (Monash University), and M. Hairer (EPFL).

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Mon, 09 Jun 2025
12:15
L5

$3$-$(\alpha,\delta)$-Sasaki manifolds and strongly positive curvature

Ilka Agricola
(Philipps-Universität Marburg)
Abstract
$3$-$(\alpha,\delta)$-Sasaki manifolds are a natural generalisation of $3$-Sasaki manifolds, which in dimension $7$ are intricately related to $G_2$ geometry. We show how these are closely related to various types of quaternionic Kähler orbifolds via connections with skew-torsion and an interesting canonical submersion. Making use of this relation we discuss curvature operators and show that in dimension 7 many such manifolds have strongly positive curvature, a notion originally introduced by Thorpe. 

 
Modelling cerebrovascular pathology and the spread of amyloid beta in Alzheimer’s disease
 Ahern, A Thompson, T Oliveri, H Lorthois, S Goriely, A Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences volume 481 issue 2311 (02 Apr 2025)

G-Research Oxford Pub Quiz

An open invitation from G-Research to join them for their Oxford Pub Quiz to discover the world of Quantitative Finance through an evening of fun and games. You will need a team of up to 4. Prizes up for grabs: You can register here. 

Wednesday 12th February, 18:00 - 20:00, Oxford (exact location to be confirmed after registration)

1st Place team: £250 Amazon voucher each; 2nd Place team: £100 Amazon voucher each; 3rd Place team: £50 Amazon Voucher each; plus prize rounds throughout the quiz.

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