Past Fridays@4

Speaker: Katherine Staden
Introduced by: Frances Kirwan
Title: Inducibility in graphs
Abstract: What is the maximum number of induced copies of a fixed graph H inside any graph on n vertices? Here, induced means that both edges and non-edges have to be correct. This basic question turns out to be surprisingly difficult, and it is not even known for all 4-vertex graphs H. I will survey the area and discuss some key results, ideas and techniques -- combinatorial, analytical and computer-assisted.

Speaker: Pierre Haas
Introduced by: Alain Goriely
Title: Shape-Shifting Droplets
Abstract: Experiments show that small oil droplets in aqueous surfactant solution flatten, upon slow cooling, into a host of polygonal shapes with straight edges and sharp corners. I will begin by showing how plane (and rather plain) geometry explains the sequence of these polygonal shapes. I will go on to show that geometric considerations of that ilk cannot however explain the three-dimensional polyhedral shapes that the initially spherical droplets evolve through while flattening. I will conclude by showing that the experimental data agree with the predictions of a model based on a partial phase transition of the oil near the droplet edges.

For those who do not have login access to the Mathematical Institute website, please email @email to receive the link to this session.

The pandemic has forced all of us to assess our mental well being and the way in which we care for ourselves. We have learnt that good mental health is not a state but a constant evolution, and that it is natural that changes will take place on a daily and weekly timescale.
In this very timely session, Dr Tim Knowlson, Counselling Psychologist and University of Oxford Peer Support Programme Manager will discuss how we can care for our mental health and how we can develop resilience using current evidence-based research for tackling change and uncertainty that will serve us not only in the current pandemic but also provide us with tips that will serve us long into the future.

In this session we will discuss how interviewing and being interviewed has changed now that interviews are conducted online. We will have a panel comprising Marya Bazzi, Mohit Dalwadi, Sam Cohen, Ian Griffiths and Frances Kirwan who have either experienced being interviewed online and have interviewed online and we will compare experiences with in-person interviews. 

In this new session a speaker tells us about how their area of mathematics can be used in different applications.

In this talk, Yuji Nakatsukasa tells us about how random matrix theory can be used in numerical linear algebra. 

Abstract

Randomized SVD is a topic in numerical linear algebra that draws heavily from random matrix theory. It has become an extremely successful approach for efficiently computing a low-rank approximation of matrices. In particular the paper by Halko, Martinsson, and Tropp (SIREV 2011) contains extensive analysis, and has made it a very popular method. The classical Nystrom method is much faster, but only applicable to positive semidefinite matrices. This work studies a generalization of Nystrom's method applicable to general matrices, and shows that (i) it has near-optimal approximation quality comparable to competing methods, (ii) the computational cost is the near-optimal O(mnlog n+r^3) for a rank-r approximation of dense mxn matrices, and (iii) crucially, it can be implemented in a numerically stable fashion despite the presence of an ill-conditioned pseudoinverse. Numerical experiments illustrate that generalized Nystrom can significantly outperform state-of-the-art methods. In this talk I will highlight the crucial role played by a classical result in random matrix theory, namely the Marchenko-Pastur law, and also briefly mention its other applications in least-squares problems and compressed sensing.

Talking maths on YouTube is a lot of fun. Your audience will contain maths enthusiasts, young people, and the general public. These are people who are interested in what you have to say, and want to learn something new. Maths videos on YouTube can be used to teach maths, or to just show people something interesting. Making videos doesn't have to be technically difficult, but is good practice in explaining difficult concepts in clear and succinct ways. In this session we will discuss how to make your first YouTube video, including questions about content, presentation and video making.

Dr James Grime started making his first maths YouTube videos while working as a postdoc in 2008. James has made maths videos with Cambridge University, the Royal Institution, and MathsWorldUK, and is also a presenter on the popular YouTube channel Numberphile, which now has over 3 million subscribers worldwide.

In this session we discuss techniques to get the most out of your supervision sessions and tips on how to work with different personalities and use your supervisor's skills to your advantage. The session will be run by DPhil students and discussion among students during the session is encouraged.  

Martin Gallauer (North): "Algebraic algebraic geometry"
If a space is described by algebraic equations, its algebraic invariants are endowed with additional structure. I will illustrate this with some simple examples, and speculate on the meaning of the title of my talk.

Zhaohe Dai (South): "Two-dimensional material bubbles"
Two-dimensional (2D) materials are a relatively new class of thin sheets consisting of a single layer of covalently bonded atoms and have shown a host of unique electronic properties. In 2D material electronic devices, however, bubbles often form spontaneously due to the trapping of air or ambient contaminants (such as water molecules and hydrocarbons) at sheet-substrate interfaces. Though they have been considered to be a nuisance, I will discuss that bubbles can be used to characterize 2D materials' bending rigidity after the pressure inside being well controlled. I will then focus on bubbles of relatively large deformations so that the elastic tension could drive the radial slippage of the sheet on its substrate. Finally, I will discuss that the consideration of such slippage is vital to characterize the sheet's stretching stiffness and gives new opportunities to understand the adhesive and frictional interactions between the sheet and various substrates that it contacts.
 

Paolo Aceto

Knot concordance and homology cobordisms of 3-manifolds 

We introduce the notion of knot concordance for knots in the 3-sphere and discuss some key problems regarding the smooth concordance group. After defining homology cobordisms of 3-manifolds we introduce the integral and rational homology cobordism groups and briefly discuss their relationship with the concordance group. We conclude stating a few recent results and open questions on the structure of these groups.

In this session, Laura will explain the process of applying for an EPSRC fellowship. In particular, there will be a discussion on the Future Leaders Fellowships, New Investigator Awards and Standard Grant applications. There will also be a discussion on applying for EPSRC funding more generally. Laura will answer any questions that people have. 

Speaker: Thomas Oliver

Title: Hyperbolic circles and non-trivial zeros

Abstract: L-functions can often be considered as generating series of arithmetic information. Their non-trivial zeros are the subject of many famous conjectures, which offer countless applications to number theory. Using simple geometric observations in the hyperbolic plane, we will study the relationship between the zeros of L-functions and their characterisation amongst more general Dirichlet series.
 

Speaker: Ebrahim Patel

Title: From trains to brains: Adventures in Tropical Mathematics.

Abstract: Tropical mathematics uses the max and plus operator to linearise discrete nonlinear systems; I will present its popular application to solve scheduling problems such as railway timetabling. Adding the min operator generalises the system to allow the modelling of processes on networks. Thus, I propose applications such as disease and rumour spreading as well as neuron firing behaviour.

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.