Past Fridays@4

Elena Gal
Categorification, Quantum groups and TQFTs

Quantum groups are mathematical objects that encode (via their "category of representations”) certain symmetries which have been found in the last several dozens of years to be connected to several areas of mathematics and physics. One famous application uses representation theory of quantum groups to construct invariants of 3-dimensional manifolds. To extend this theory to higher dimensions we need to “categorify" quantum groups - in essence to find a richer structure of symmetries. I will explain how one can approach such problem.

Carolina Urzua-Torres
Why you should not do boundary element methods, so I can have all the fun.

Boundary integral equations offer an attractive alternative to solve a wide range of physical phenomena, like scattering problems in unbounded domains. In this talk I will give a simple introduction to boundary integral equations arising from PDEs, and their discretization via Galerkin BEM. I will discuss some nice mathematical features of BEM, together with their computational pros and cons. I will illustrate these points with some applications and recent research developments.
 

Dr Rachel Philip will discuss her experiences working at the interface between academic mathematics and industry. Oxford University Innovation will discuss how they can help academics when interacting with industry. 

Dominic Vella will talk about writing grants, Anna Seigal will talk about writing research fellow applications and Jason Lotay will talk about his experience and tips for applying for faculty positions. 

Speaker: Daniel Woodhouse (North)
Title: Generalizing Leighton's Graph Covering Theorem
Abstract: Before he ran off and became a multimillionaire, exploiting his knowledge of network optimisation, the computer scientist F. Thomas Leighton proved an innocuous looking result about finite graphs. The result states that any pair of finite graphs with isomorphic universal covers have isomorphic finite covers. I will explain what all this means, and why this should be of tremendous interest to group theorists and topologists.

Speaker: Benjamin Fehrman (South)
Title: Large deviations for particle processes and stochastic PDE
Abstract: In this talk, we will introduce the theory of large deviations through a simple example based on flipping a coin.  We will then define the zero range particle process, and show that its diffusive scaling limit solves a nonlinear diffusion equation.  The large deviations of the particle process about its scaling limit formally coincide with the large deviations of a certain ill-posed, singular stochastic PDE.  We will explain in what sense this relationship has been made mathematically precise.

Speaker: Joseph Keir (North)
Title: Dispersion (or not) in nonlinear wave equations
Abstract: Wave equations are ubiquitous in physics, playing central roles in fields as diverse as fluid dynamics, electromagnetism and general relativity. In many cases of these wave equations are nonlinear, and consequently can exhibit dramatically different behaviour when their solutions become large. Interestingly, they can also exhibit differences when given arbitrarily small initial data: in some cases, the nonlinearities drive solutions to grow larger and even to blow up in a finite time, while in other cases solutions disperse just like the linear case. The precise conditions on the nonlinearity which discriminate between these two cases are unknown, but in this talk I will present a conjecture regarding where this border lies, along with some conditions which are sufficient to guarantee dispersion.

Speaker: Priya Subramanian (South)
Title: What happens when an applied mathematician uses algebraic geometry?
Abstract: A regular situation that an applied mathematician faces is to obtain the equilibria of a set of differential equations that govern a system of interest. A number of techniques can help at this point to simplify the equations, which reduce the problem to that of finding equilibria of coupled polynomial equations. I want to talk about how homotopy methods developed in computational algebraic geometry can solve for all solutions of coupled polynomial equations non-iteratively using an example pattern forming system. Finally, I will end with some thoughts on what other 'nails' we might use this new shiny hammer on.

Aden Forrow
Optimal transport and cell differentiation

Abstract
Optimal transport is a rich theory for comparing distributions, with both deep mathematics and application ranging from 18th century fortification planning to computer graphics. I will tie its mathematical story to a biological one, on the differentiation of cells from pluripotency to specialized functional types. First the mathematics can support the biology: optimal transport is an apt tool for linking experimental samples across a developmental time course. Then the biology can inspire new mathematics: based on the branching structure expected in differentiation pathways, we can find a regularization method that dramatically improves the statistical performance of optimal transport.

Paul Ziegler
Geometry and Arithmetic

Abstract
For a family of polynomials in several variables with integral coefficients, the Weil conjectures give a surprising relationship between the geometry of the complex-valued roots of these polynomials and the number of roots of these polynomials "modulo p". I will give an introduction to this circle of results and try to explain how they are used in modern research.
 

A panel discussion on non-academic careers for mathematicians with PhDs, featuring Katia Babbar (AI Wealth Technologies & QuantBright), Jara Imbers (Risk Management Solutions) and Tom Hawes (Smith Institute).
 

What is the point of giving a talk?  What is the point of going to a talk?  In this presentation, which is intended to have a lot of audience participation, I would like to explore how one should prepare talks for different audiences and different occasions, and what one should try to get out of going to a talk.

Valérie Voorsluijs
Deterministic limit of intracellular calcium spikes
Abstract: In non-excitable cells, global calcium spikes emerge from the collective dynamics of clusters of calcium channels that are coupled by diffusion. Current modeling approaches have opposed stochastic descriptions of these systems to purely deterministic models, while both paradoxically appear compatible with experimental data. Combining fully stochastic simulations and mean-field analyses, we demonstrate that these two approaches can be reconciled. Our fully stochastic model generates spike sequences that can be seen as noise-perturbed oscillations of deterministic origin while displaying statistical properties in agreement with experimental data. These underlying deterministic oscillations arise from a phenomenological spike nucleation mechanism.

Matthias Nagel
Knots in dimensions three and four
Abstract: Knot theory studies the various embeddings of a circle into three-dimensional space. I will describe an equivalence relation on knots, called "concordance", which takes the fourth dimension into account. The study of concordance is intimately related with many problems at the heart of the topology of four-manifolds, such as the difference between the smooth and the topological category, and I will discuss results that illuminate this relation.

Aura Raulo (Ecological and Evolutionary Dynamics) and Marie-Claire Koschowitz (Vertebrate Palaeobiology) discuss their work and its mathematical challenges.

Aura Raulo

" Aura Raulo is a graduate student in Zoology Department working on transmission of symbiotic bacteria in the social networks of their animal hosts"
Title: Heaps in networks - How we share our microbiota through kisses
Abstract: Humans, like all vertebrates have a microbiome, a diverse community of symbiotic bacteria that live in and on us and are crucial for our functioning. These bacteria help us digest food, regulate our mood and function as a key part of our immune system. Intriguingly, while they are part of us, they are, unlike our other cells, in constant flux between us, challenging the traditional definition of a biological individual. Many of these bacteria need intimate social contact to be transmitted from human to human, making social network analysis tools handy in explaining their community dynamics.What then is a recipe for a ``good microbiome”? Theories and evidence implies that the most healthy and immunologically robust microbiome composition is both diverse, semi-stable and somewhat synchronized among closely interacting individuals, but little is known about what kind of transmission landscapes determine these bacterial cocktails. In my talk, I will present humanmicrobiome as a network trait: a metacommunity of cells shaped by an equilibrium of isolation and contact among their hosts. I propose that we do notnecessarily need to think of levels of life (e.g. cells, individuals, populations) as being neatly nested inside of each other. Rather, aggregations of cooperating cells (both bacteria and human cells) can be considered as mere tighter clusters in their interaction network, dynamically creating de novo defined units of life. I will present a few game theoretical evolutionary dilemmas following from this perspective and highlight outstanding questions in mapping how network position of the host translates into community composition of bacteria in flux.

Marie Koschowitz
“Marie Koschowitz is a PhD student in the Department of Zoology and the Department of Earth Sciences, working on comparative physiology and large scale evolutionary patterns in reptiles such as crocodiles, birds and dinosaurs."
Title: Putting the maths into dinosaurs – A zoologist's perspective
Abstract: Contemporary palaeontology is a subject area that often deals with sparse data.Therefore, palaeontologists became rather inventive in pursuit of getting the most out of what is available. If we find a dinosaur’s skull that shows prominent, but puzzling, bony ridges without any apparent function, how can we make meaningful interpretations of its purpose in the living animal that was? If we are confronted with a variety of partially preserved bones from animals looking anatomically similar, but not quite alike, how can we infer relationships in the absence of genetic data?Some methods that resolve these questions, such as finite element analysis, were borrowed from engineering. Others, like comparative phylogenetics or MCMC generalised mixed effects models, are even more directly based on mathematical computations. All of these approaches help us to calculate things like a raptors bite-force and understand the ins and outsof their skulls anatomy, or why pterosaurs and plesiosaurs aren’t exactly dinosaurs. This talk aims to presents a selection of current approaches to applied mathematics which have been inspired by interdisciplinary research – and to foster awareness of all the ways how mathematicians can get involved in “dinosaur research”, if they feel inclined to do so.


 

What actually happens when you submit an article to a journal? How does refereeing work in practice? How can you keep editors happy as an author or referee? How does one become a referee or editor? What does 'publication' mean with the internet and arXiv?

In this panel we'll discuss what happens between finishing writing a mathematical paper and its final (?) publication, looking at the various roles that people play and how they work best.

Featuring Helen Byrne, Rama Cont and Jonathan Pila.

 

How do you create a self-sustaining, flourishing academic community in a developing country? What kind of challenges need to be overcome to ensure that quality education becomes available? What can we do to help make it happen? In this talk, we will describe our experience visiting the University of Yangon in Myanmar. During the visit, we delivered a course to the academic staff, and discussed future collaborations between Oxford and Yangon, as well as further directions for Mathematical education in Myanmar, all the while marvelling at the wonders of the Burmese culture.

Teaching ethics to the mathematicians who need it most
For the last 20 years it has become increasingly obvious, and increasingly pressing, that mathematicians should be taught some ethical awareness so as to realise the impact of their work. This extends even to those more highly trained, like graduate students and postdocs. But which mathematicians should we be teaching this to, what should we be teaching them, and how should we do it? In this talk I’ll explore the idea that all mathematicians will, at some stage, be faced with ethical challenges stemming from their work, and yet few are ever told beforehand.
 

Thomas Prince The double life of the number 24.

The number 24 appears in a somewhat surprising result in the study of polyhedra with integer lattice points. In a different setting, the number 24 is the Euler characteristic of a K3 surface: a four (real) dimensional object which plays a central role in algebraic geometry. We will hint at why both instances of 24 are in fact the same, and suggest that integral affine geometry can be used to interpolate between the realm of integral polytopes and the world of complex algebraic geometry.

Mohit Dalwadi A multiscale mathematical model of bacterial nutrient uptake

In mathematical models that include nutrient delivery to bacteria, it is prohibitively expensive to include many small bacterial regions acting as volumetric nutrient sinks. To combat this problem, such models often impose an effective uptake instead. However, it is not immediately clear how to relate properties on the bacterial scale with this effective result. For example, one may intuitively expect the effective uptake to scale with bacterial volume for weak first-order uptake, and with bacterial surface area for strong first-order uptake. I will present a general model for bacterial nutrient uptake, and upscale the system using homogenization theory to determine how the effective uptake depends on the microscale bacterial properties. This will show us when the intuitive volume and surface area scalings are each valid, as well as the correct form of the effective uptake when neither of these scalings is appropriate.
 

Robert Timms

Title: Multiscale modelling of lithium-ion batteries

Lithium-ion batteries are one of the most widely used technologies for energy storage, with applications ranging from portable electronics to electric vehicles. Due to their popularity, there is a continued interest in the development of mathematical models of lithium-ion batteries. These models encompass various levels of complexity, which may be suitable to aid with design, or for real-time monitoring of performance. After a brief introduction to lithium-ion batteries, I will discuss some of the modelling efforts undertaken here at Oxford and within the wider battery modelling community.
 

Jan Vonk

Title: Singular moduli for real quadratic fields

At the 1900 ICM, David Hilbert posed a series of problems, of which the 12th remains completely open today. I will discuss how to solve this problem in the simplest open case, by considering certain exotic (so called p-adic) metrics on the set of numbers, and using its concomitant theories of analysis and geometry.
 

Are you teaching intercollegiate classes or tutorials this term? Would you like to explore inclusive teaching strategies that could help all students make the most of your sessions? In this interactive workshop, we'll explore strategies that have been found effective. This will be a self-contained session, but will also be a good introduction to the "Developing Learning and Teaching" course offered by MPLS for graduate students and early career researchers. The session will be led by Vicky Neale (Mathematics) and Delia O'Rourke (Oxford Learning Institute).