L1

Preparing for Prelims and Part A exams with Dr Vicky Neale

Description: This session will offer guidance for Prelims and Part A students preparing for closed-book, in-person exams this summer, with tips on revision and information about practical arrangements. If you have questions, please send them in advance (by 28 February) via https://vevox.app/#/m/170975861 and we'll try to address as many as possible during the session.

A separate session in Week 6 will be aimed at students doing Part B, Part C and MSc exams.

Great progress has been made recently in exploiting categorical/topological/higher symmetries in quantum field theory. I will explain how the same structure is realised directly in the lattice models of statistical mechanics, generalizing Kramers-Wannier duality to a wide class of models. In particular, I will give an overview of my work with Aasen and Mong on using fusion categories to find and analyse lattice topological defects in two and 1+1 dimensions.  These defects possess a variety of remarkable properties. Not only is the partition function is independent of deformations of their path, but they can branch and fuse in a topologically invariant fashion.  The universal behaviour under Dehn twists gives exact results for scaling dimensions, while gluing a topological defect to a boundary allows universal ratios of the boundary g-factor to be computed exactly on the lattice.  I also will describe how terminating defect lines allows the construction of fractional-spin conserved currents, giving a linear method for Baxterization, I.e. constructing integrable models from a braided tensor category.

 I will present a scenario where the early universe is in a topological phase of gravity.  I will discuss a number of analogies which motivate considering gravity in such a phase. Cosmological puzzles such as the horizon problem provide a phenomenological connection to this phase and can be explained in terms of its topological nature. To obtain phenomenological estimates, a concrete realization of this scenario using Witten's four dimensional topological gravity will be used. In this model, the CMB power spectrum can be estimated by certain conformal anomaly coefficients. A qualitative prediction of this phase is the absence of tensor modes in cosmological fluctuations.

This event will be hybrid and will take place in L1 and on Teams. A link will be available 30 minutes before the session begins.

Pascal Heid
Title: Adaptive iterative linearised Galerkin methods for nonlinear PDEs

Abstract: A wide variety of iterative methods for the solution of nonlinear equations exist. In many cases, such schemes can be interpreted as iterative local linearisation methods, which can be obtained by applying a suitable linear preconditioning operator to the original nonlinear equation. Based on this observation, we will derive an abstract linearisation framework which recovers some prominent iteration schemes. Subsequently, in order to cast this unified iteration procedure into a computational scheme, we will consider the discretisation by means of finite dimensional subspaces. We may then obtain an effective numerical algorithm by an instantaneous interplay of the iterative linearisation and an (optimally convergent) adaptive discretisation method. This will be demonstrated by a numerical experiment for a quasilinear elliptic PDE in divergence form.   

Ilyas Khan
Title: Geometric Analysis: Curvature and Applications

Abstract: Often, one will want to find a geometric structure on some given manifold satisfying certain properties. For example, one might want to find a minimal embedding of one manifold into another, or a metric on a manifold with constant scalar curvature, to name some well known examples of this sort of problem. In general, these problems can be seen as equivalent to solving a system of PDEs: differential relations on coordinate patches that can be assembled compatibly over the whole manifold to give a globally defined geometric equation.

In this talk, we will present the theories of minimal surfaces and mean curvature flow as representative examples of the techniques and philosophy that geometric analysis employs to solve problems in geometry of the aforementioned type. The description of the theory will be accompanied by a number of examples and applications to other fields, including physics, topology, and dynamics. 

This event will be hybrid and will take place in L1 and on Teams. A link will be available 30 minutes before the session begins.

`Conferences and collaboration’ is a Fridays@4 group discussion. The goal is to have an open and honest conversion about the hurdles posed by these things, led by a panel of graduate students and postdocs. Conferences can be both exciting and stressful - they involve meeting new people and learning new mathematics, but can be intimidating new professional experiences. Many of us also will either have never been to one in person, or at least not been to one in the past two years. Optimistically looking towards the world opening up again, we thought it would be a good time to ask questions such as:
-Which talks should I go to?
-How to cope with incomprehensible talks. Is it imposter syndrome or is the speaker just bad?
-Should I/how should I go about introducing myself to more senior people in the field?
-How do you start collaborations? Does it happen at conferences or elsewhere?
-How do you approach workload in collaborations?
-What happens if a collaboration isn’t working out?
-FOMO if you like working by yourself. Over the hour we’ll have a conversation about these hurdles and most importantly, talk about how we can make conferences and collaborations better for everyone early in their careers.

This event will take place on Teams. A link will be available 30 minutes before the session begins.

Sebastien Racaniere is a Staff Research Engineer at DeepMind. His current main interest is in the use of symmetries in Machine Learning. This offers diverse applications, for example in Neuroscience or Theoretical Physics (in particular Lattice Quantum Chromodynamics). Past interests, still in Machine Learning, include Reinforcement Learning (i.e. learning from rewards), generative models (i.e. learn to sample from probability distributions), and optimisation (i.e. how to find 'good' minima of functions)

This event will be hybrid and will take place in L1 and on Teams. A link will be available 30 minutes before the session begins.

This event will take place in L1 and on Teams. A link will be available 30 minutes before the session begins. 

It's hard to believe: I've spent nearly 40 years in STEM. In that time, much changed: we changed from typewriters to PCs, from low performance to high  performance computing, from data-supported research to data-driven research, from traditional languages such as Fortran to a plethora of programming environments. And the rate of change seems to increase constantly. Some things have stayed more or less the same, such as the (lack of) diversity of the STEM community, the level of stress and the struggles we all experience (and the joys!). In this talk, I will reflect on those years, on lessons learned and not learned or unlearned, on things I wish I understood 40 years ago, and on things I still don't understand.

Margot is a professor at Stanford University in the Department of Energy Resources Engineering (ERE) and the Institute of Computational & Mathematical Engineering (ICME). Margot was born and raised in the Netherlands. Her STEM education started in 1982. In 1990 she received a MSc in applied mathematics at Delft University and then left her home country to search for sunnier and hillier places. She moved to Colorado and a year later to California to join the PhD program in Scientific Computing and Computational Mathematics at Stanford. During her PhD, Margot spent several quarters at Oxford University (with very good memories). Before returning to Stanford as faculty member in ERE, Margot spent 5 years as lecturer at the University of Auckland, New Zealand. From 2010-2018, Margot was the director of ICME. During this directorship, she founded the Women in Data Science initiative, which is now a global organization in over 70 countries. From 2015-2020, Margot was also the Senior Associate Dean of Educational Affairs at Stanford's school of Earth, Energy & Environmental Sciences. Currently, Margot still co-directs WiDS and is the Chair of the Board of SIAM. She has since moved back to the mountains (still sunny too) and now lives in Bend, Oregon.