To tie in with mental health awareness week, in this session we'll give a brief overview of the mental health support available through the department and university, followed by a panel discussion on how we can look after our mental health as in an academic setting. We're pleased that several of our department Mental Health First Aiders will be panellists - come along for hints and tips on maintaining good mental health and supporting your colleagues and friends.

For week 4's @email session we welcome the SIAM-IMA student chapter, running their annual Three Minute Thesis competition.

The Three Minute Thesis competition challenges graduate students to present their research in a clear and engaging manner within a strict time limit of three minutes. Each presenter will be allowed to use only one static slide to support their presentation, and the panel of esteemed judges (details TBC) will evaluate the presentations based on criteria such as clarity, pacing, engagement, enthusiasm, and impact. Each presenter will receive a free mug and there is £250 in cash prizes for the winners. If you're a graduate student, sign up here (https://oxfordsiam.com/3mt) by Friday of week 3 to take part! And if not, come along to support your DPhil friends and colleagues, and to learn about the exciting maths being done by our research students.

Title: When data driven reduced order modelling meets full waveform inversion

Abstract:

This talk is concerned with the following inverse problem for the wave equation: Determine the variable wave speed from data gathered by a collection of sensors, which emit probing signals and measure the generated backscattered waves. Inverse backscattering is an interdisciplinary field driven by applications in geophysical exploration, radar imaging, non-destructive evaluation of materials, etc. There are two types of methods:

(1) Qualitative (imaging) methods, which address the simpler problem of locating reflective structures in a known host medium. 

(2) Quantitative methods, also known as velocity estimation. 

Typically, velocity estimation is  formulated as a PDE constrained optimization, where the data are fit in the least squares sense by the wave computed at the search wave speed. The increase in computing power has lead to growing interest in this approach, but there is a fundamental impediment, which manifests especially for high frequency data: The objective function is not convex and has numerous local minima even in the absence of noise.

The main goal of the talk is to introduce a novel approach to velocity estimation, based on a reduced order model (ROM) of the wave operator. The ROM is called data driven because it is obtained from the measurements made at the sensors. The mapping between these measurements and the ROM is nonlinear, and yet the ROM can be computed efficiently using methods from numerical linear algebra. More importantly, the ROM can be used to define a better objective function for velocity estimation, so that gradient based optimization can succeed even for a poor initial guess.

The affinity between statistical physics and machine learning has a long history. Theoretical physics often proceeds in terms of solvable synthetic models; I will describe the related line of work on solvable models of simple feed-forward neural networks. I will then discuss how this approach allows us to analyze uncertainty quantification in neural networks, a topic that gained urgency in the dawn of widely deployed artificial intelligence. I will conclude with what I perceive as important specific open questions in the field.

 

Join us for our first Fridays@4 session of Trinity about different academic routes people take post-PhD, with a particular focus on fellowships and grants. We’ll hear from Jason Lotay about his experiences on both sides of the application process, as well as hear about the experiences of ECRs in the South Wing, North Wing, and Statistics. Towards the end of the hour we’ll have a Q+A session with the whole panel, where you can ask any questions you have around this topic!

Fusion 2-categories were introduced by Douglas and Reutter so as to define a state-sum invariant of 4-manifolds. Categorifying a result of Douglas, Schommer-Pries and Snyder, it was conjectured that, over an algebraically closed field of characteristic zero, every fusion 2-category is a fully dualizable object in an appropriate symmetric monoidal 4-category. I will sketch a proof of this conjecture, which will proceed by studying, and in fact classifying, the Morita equivalence classes of fusion 2-categories. In particular, by appealing to the cobordism hypothesis, we find that every fusion 2-category yields a fully extended framed 4D TQFT. I will explain how these theories are related to the ones constructed using braided fusion 1-categories by Brochier, Jordan, and Snyder.

In 1983 Gromov proved the systolic inequality: if M is a closed, essential n-dimensional Riemannian manifold where every loop of length 2 is null-homotopic, then the volume of M is at least a constant depending only on n.  He also proved a version that depends on the simplicial volume of M, a topological invariant generalizing the hyperbolic volume of a closed hyperbolic manifold.  If the simplicial volume is large, then the lower bound on volume becomes proportional to the simplicial volume divided by the n-th power of its logarithm.  Nabutovsky showed in 2019 that Papasoglu's method of area-minimizing separating sets recovers the systolic inequality and improves its dependence on n.  We introduce simplicial volume to the proof, recovering the statement that the volume is at least proportional to the square root of the simplicial volume.