University of Oxford

Orbifolds are a generalisation of manifolds which allow group actions to enter the picture. The most basic examples of orbifolds are quotients of manifolds by (non-free) finite group actions.
I will give an introduction to orbifolds, recalling a number of philosophically different but mathematically equivalent definitions. For starters, I will try to convince you that "a space locally modelled on a quotient of R^n by a finite group" is misleading. I will draw many pictures of orbifolds, make the connection to complexes of groups, and explain the definition of a map of orbifolds. In the process, I hope to demystify the definition of the orbifold fundamental group, the orbifold Euler characteristic and orbifold cohomology.

In this talk, I will discuss the important Grothendieck conjecture which originated Grothendieck-Teichmuller Theory, a bridge between Topology and Number Theory. On the geometric side, there is the study of automorphisms of mapping class groups that satisfy compatibility conditions with respect to subsurface inclusions. On the other side, there is the study of the absolute Galois group of the rationals, one of the most important objects in Number Theory today.
In my talk, I will introduce these objects and discuss the recent progress that has been made in understanding such automorphisms of mapping class groups. No background in Number Theory or Galois Theory is required.

Finiteness properties of groups provide various generalisations of the properties "finitely generated" and "finitely presented." We will define different types of finiteness properties and discuss Bestvina-Brady groups as they provide examples of groups with interesting combinations of finiteness properties.

Given an arbitrary group presentation, often very little can be deduced about the underlying group. It is thus something of a miracle that many properties of one-relator groups can be simply read-off from the defining relator. In this talk, I will discuss some of the classical results in the theory of one-relator groups, as well as the key trick used in many of their proofs. Time-permitting, I'll also discuss more recent work on this subject, including some open problems.

Modelling joint dynamics of liquid vanilla options is crucial for arbitrage-free pricing of illiquid derivatives and managing risks of option trade books. This paper develops a nonparametric model for the European options book respecting underlying financial constraints and while being practically implementable. We derive a state space for prices which are free from static (or model-independent) arbitrage and study the inference problem where a model is learnt from discrete time series data of stock and option prices. We use neural networks as function approximators for the drift and diffusion of the modelled SDE system, and impose constraints on the neural nets such that no-arbitrage conditions are preserved. In particular, we give methods to calibrate neural SDE models which are guaranteed to satisfy a set of linear inequalities. We validate our approach with numerical experiments using data generated from a Heston stochastic local volatility model, and will discuss some initial results using real data.

Based on joint work with Christoph Reisinger and Sheng Wang

The Krull dimension is an ideal-theoretic invariant of an algebra. It has an important meaning in algebraic geometry: the Krull dimension of a commutative algebra is equal to the dimension of the corresponding affine variety/scheme. In my talk I'll explain how this idea can be transformed into a tool for measuring non-commutative rings. I'll illustrate this with important examples and techniques, and describe what is known for Iwasawa algebras of compact $p$-adic Lie groups.

“The Royal Swedish Academy of Sciences has today decided to award the 2021 Nobel Prize in Physics for ground-breaking contributions to our understanding of complex physical systems”

Last Tuesday this announcement got many in our community very excited: never before had the Nobel prize been awarded to a topic so closely related to Network Science. We will try to understand the contributions that have led to this Nobel Prize announcement and their ties with networks science. The presentation will be held by Erik Hörmann, who has been lucky enough to have had the honour and pleasure of studying and working with one of the awardees, Professor Giorgio Parisi, before joining the Mathematical Institute.

 Mathematics for the mind: network dynamical systems for neurodegenerative disease pathology

Travis Thompson

Can mathematics understand neurodegenerative diseases?  The modern medical perspective on neurological diseases has evolved, slowly, since the 20th century but recent breakthroughs in medical imaging have quickly transformed medicine into a quantitative science.  Today, mathematical modeling and scientific computing allow us to go farther than observation alone.  With the help of  computing, experimental and data-informed mathematical models are leading to new clinical insights into how neurodegenerative diseases, such as Alzheimer's disease, may develop in the human brain.  In this talk, I will overview my work in the construction, analysis and solution of data and clinically-driven mathematical models related to AD pathology.  We will see that mathematical modeling and scientific computing are indeed indispensible for cultivating a data-informed understanding of the brain, AD and for developing potential treatments.

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Simplified battery models via homogenisation  

Robert Timms

Lithium-ion batteries (LIBs) are one of the most popular forms of energy storage for many modern devices, with applications ranging from portable electronics to electric vehicles. Improving both the performance and lifetime of LIBs by design changes that increase capacity, reduce losses and delay degradation effects is a key engineering challenge. Mathematical modelling is an invaluable tool for tackling this challenge: accurate and efficient models play a key role in the design, management, and safe operation of batteries. Models of batteries span many length scales, ranging from atomistic models that may be used to predict the rate of diffusion of lithium within the active material particles that make up the electrodes, right through to models that describe the behaviour of the thousands of cells that make up a battery pack in an electric vehicle. Homogenisation can be used to “bridge the gap” between these disparate length scales, and allows us to develop computationally efficient models suitable for optimising cell design.