Past

I will explore the geometrical structure of Higgs branches of quantum field theories with 8 supercharges in 3, 4, 5 and 6 dimensions. They are hyperkahler singularities, and as such they can be described by a Hasse diagram built from a family of elementary transitions. This corresponds physically to the partial Higgs mechanism. Using brane systems and recently introduced notions of magnetic quivers and quiver subtraction, we formalise the rules to obtain the Hasse diagrams.

Viruses encapsulate their genetic material into protein containers that act akin to molecular Trojan horses, protecting viral genomes between rounds of infection and facilitating their release into the host cell environment. In the majority of viruses, including major human pathogens, these containers have icosahedral symmetry. Mathematical techniques from group, graph and tiling theory can therefore be used to better understand how viruses form, evolve and infect their hosts, and point the way to novel antiviral solutions.

In this talk, I will present an overarching theory of virus architecture, that contains the seminal Caspar Klug theory as a special case and solves long-standing open problems in structural virology. Combining insights into virus structure with a range of different mathematical modelling techniques, such as Gillespie algorithms, I will show how viral life cycles can be better understood through the lens of viral geometry. In particular, I will discuss a recent model for genome release from the viral capsid. I will also demonstrate the instrumental role of the Hamiltonian path concept in the discovery of a virus assembly mechanism that occurs in many human pathogens, such as Picornaviruses – a family that includes the common cold virus– and Hepatitis B and C virus. I will use multi-scale models of a viral infection and implicit fitness landscapes in order to demonstrate that therapeutic interventions directed against this mechanism have advantages over conventional forms of anti-viral therapy. The talk will finish with a discussion of how the new mathematical and mechanistic insights can be exploited in bio-nanotechnology for applications in vaccination and gene therapy.

What should you expect in intercollegiate classes?  What can you do to get the most out of them?  In this session, experienced class tutors will share their thoughts, and a current student will offer tips and advice based on their experience.  

All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. (Students who attended the Part C/OMMS induction event will find significant overlap between the advice offered there and this session!)

In this talk I shall describe recent work inspired by problems in cell biology, namely how the dynamics of small G-proteins underlies polarity formation. Their dynamics is such that their active membrane bound form diffuses more slowly. Hence you might expect Turing patterns. Yet how do cells form backs and fronts or single isolated patches. In understanding these questions we shall show that the key is to identify the parameter region where Turing bifurcations are sub-critical. What emerges is a unified 2-parameter bifurcation diagram containing pinned fronts, localised spots, localised patterns. This diagram appears in many canonical models such as Schnakenberg and Brusselator, as well as biologically more realistic systems. A link is also found between theories of semi-string interaction asymptotics and so-called homoclinic snaking. I will close with some remarks about relevance to root hair formation and to the importance of subcriticality in biology. 

Machine learning for scientific applications faces the challenge of limited data. To reduce overfitting, we propose a framework to leverage as much as possible a priori-known physics for a problem. Our approach embeds a deep neural network in a partial differential equation (PDE) system, where the pre-specified terms in the PDE capture the known physics and the neural network will learn to describe the unknown physics. The neural network is estimated from experimental and/or high-fidelity numerical datasets. We call this approach a “deep learning PDE model” (DPM). Once trained, the DPM can be used to make out-of-sample predictions for new physical coefficients, geometries, and boundary conditions. We implement our approach on a classic problem of the Navier-Stokes equation for turbulent flows. The numerical solution of the Navier-Stokes equation (with turbulence) is computationally expensive and requires a supercomputer. We show that our approach can estimate a (computationally fast) DPM for the filtered velocity of the Navier-Stokes equations. 

I will discuss work in progress with Sary Drappeau and Maksym Radziwill on low-lying zeros of Dirichlet L-functions. By way of motivation I will discuss some results on the spacings of zeros of the Riemann zeta function, and the conjectures of Katz and Sarnak relating the distribution of low-lying zeros of L-functions to eigenvalues of random matrices. I will then describe some ideas behind the proof of our theorem.
 

Springer theory is an important branch of geometric representation theory. It is a beautiful interplay between combinatorics, geometry and representation theory.
It started with Springer correspondence, which yields geometric construction of irreducible representations of symmetric groups, and Ginzburg's construction of universal enveloping algebra U(sl_n).

Here I will present a view of two-row Springer theory of type A (thus looking at nilpotent elements with two Jordan blocks) from a scope of a symplectic topologist (hence the title), that yields connections between symplectic-topological invariants and link invariants (Floer homology and Khovanov homology) and connections to representation theory (Fukaya category and parabolic category O), thus summarising results by Abouzaid,
Seidel, Smith and Mak on the subject.

There is a very simple algorithm for the inference of posteriors for probability models on trees. This algorithm, known as "Belief Propagation" is widely used in coding theory, in machine learning, in evolutionary inference, among many other areas. The talk will be devoted to the analysis of Belief Propagation in some of the simplest probability models. We will highlight the interplay between Belief Propagation, linear estimators (statistics), the Kesten-Stigum bound (probability) and Replica Symmetry Breaking (statistical physics). We will show how the analysis of Belief Propagation allowed proof phase transitions for phylogenetic reconstruction in evolutionary biology and developed optimal algorithms for inference of block models. Finally, we will discuss the computational complexity of this 'simple' algorithm.

During this seminar, we will present a new mathematical model describing the transport of nitric oxide (NO) in a realistic geometrical representation of the lungs. Nitric oxide (NO) is naturally produced in the bronchial region of the lungs. It is a physiological molecule that has antimicrobial properties and allows the relaxation of muscles. It is well known that the measurement of the molar fraction of NO in the exhaled air, the so-called FeNO, allows a monitoring of asthmatic patients, since the production of this molecule in the lungs is increased in case of inflammation. However, recent clinical studies have shown that the amount of NO in the exhaled air can also be affected by « non-inflammatory » processes, such as the action of a bronchodilator or a respiratory physiotherapy session for a patient with cystic fibrosis. Using our new model, we will highlight the complex interplay between different transport phenomena in the lungs. More specifically, we will show why changes taking place in the deepest part of the lungs are expected to impact the FeNO. This gives a new light on the clinical studies mentioned below, allowing to confer a new role to the NO for the management of various pulmonary pathologies.

Deep learning has revolutionized image, text, and speech recognition. Motivated by this success, there is growing interest in developing deep learning methods for financial applications. We will present some of our recent results in this area, including deep learning models of high-frequency data. In the second part of the talk, we prove a law of large numbers for single-layer neural networks trained with stochastic gradient descent. We show that, depending upon the normalization of the parameters, the law of large numbers either satisfies a deterministic partial differential equation or a random ordinary differential equation. Using similar analysis, a law of large numbers can also be established for reinforcement learning (e.g., Q-learning) with neural networks. The limit equations in each of these cases are discussed (e.g., whether a unique stationary point and global convergence can be proven).  

A fundamental problem in the context of Einstein's equations of general relativity is to understand precisely the dynamical evolution of small perturbations of stationary black hole solutions. It is expected that there is a discrete set of characteristic frequencies that play a dominant role at late time intervals and carry information about the nature of the black hole, much like the normal frequencies of a vibrating string. These frequencies are called quasi-normal frequencies or resonances and they are closely related to scattering resonances in the study of Schrödinger-type equations. I will discuss a new method of defining and studying resonances for linear wave equations on asymptotically flat black holes, developed from joint work with Claude Warnick.

Every Cayley graph of a finitely generated group has some basic properties: they are locally finite, connected, and vertex-transitive. These are not sufficient conditions, there are some well known examples of graphs that have all these properties but are non-Cayley. These examples do however "look like" Cayley graphs, which leads to the natural question of if there exist any vertex-transitive graphs that are completely unlike any Cayley graph. I plan to give some of the history of this question, as well as the construction of the example that finally answered it.

The triangular inequality is central in Mathematics. What would happen if we reverse it? We only obtain trivial spaces. However, if we mix it with an order structure, we obtain interesting spaces. We'll present linear antimetrics, prove a "masking theorem", and then look at a corollary which tells us about the "twin paradox" in special relativity; time is antimetric!

In this talk I will give an overview of Seba's billiard as a popular model in the field of Quantum Chaos. Consider a rectangular billiard with a Dirac mass placed in its interior. Whereas this mass has essentially no effect on the classical dynamics, it does have an effect on the quantum dynamics, because quantum wave packets experience diffraction at the point obstacle. Numerical investigations of this model by Petr Seba suggested that the spectrum and the eigenfunctions of the Seba billiard resemble the spectra and eigenfunctions of billiards which are classically chaotic.

I will give an introduction to this model and discuss recent results on quantum ergodicity, superscars and the validity of Berry's random wave conjecture. This talk is based on joint work with Par Kurlberg and Zeev Rudnick.

Given a smooth variety X with a normal crossings divisor D (or more generally a smooth log variety) we consider the ring of logarithmic differential operators: the subring of differential operators on X generated by vector fields tangent to D. Modules over this ring are called logarithmic D-modules and generalize the classical theory of regular meromorphic connections. They arise naturally when considering compactifications.

We will discuss which parts of the theory of D-modules generalize to the logarithmic setting and how to overcome new challenges arising from the logarithmic structure. In particular, we will define holonomicity for log D-modules and state a conjectural extension of the famous Riemann-Hilbert correspondence. This talk will be very example-focused and will not require any previous knowledge of D-modules or logarithmic geometry. This is joint work with Mattia Talpo.
 

Partial differential equations governing unknown or evolving geometries are ubiquitous in applications and challenging to discretize. A great deal of numerical work in this area has focused on extrinsic geometric flows, where the evolving geometry is a curve or surface embedded in Euclidean space. Much less attention has been paid to the discretization of intrinsic geometric flows, where the evolving geometry is described by a Riemannian metric. This talk will present finite element discretizations of such flows.
 

Combinatorial anabelian geometry is a modern branch of anabelian geometry which deals with those aspects of anabelian geometry which manifest themselves over algebraically closed fields of characteristic zero. The origin of combinatorial anabelian geometry is in S. Mochizuki’s pioneering papers from 2007, in which he reinterpreted and generalised some key components of his earlier famous proof of the Grothendieck conjecture. S. Mochizuki  discovered that one can separate arguments which work over algebraically closed fields from arithmetic arguments, and study the former by using combinatorial methods. This led to a very nontrivial development of the theory of combinatorial anabelian geometry by S. Mochizuki and Y. Hoshi and other mathematicians. In this talk, after introducing the theory of combinatorial anabelian geometry I will discuss  applications of combinatorial anabelian geometry to the study of the absolute Galois group of number fields and of p-adic local fields and to the study of the Grothendieck-Teichmueller group. In particular, I will talk about the recent construction of a splitting of the natural inclusion of the absolute Galois group of p-adic numbers to the (largest) p-adic Grothendieck–Teichmueller group and a splitting of the natural embedding of the absolute Galois group of rationals into the commensurator of the absolute Galois group of the maximal abelian extension of rationals in the Grothendieck–Teichmueller group.
 

Decision problems – those in which the goal is to return an answer of “YES" or “NO" – are at the heart of the theory of computational complexity, and remain the most studied problems in the theoretical algorithms community. However, in many real-world applications it is not enough just to decide whether the problem under consideration admits a solution: we often want to find all solutions, or at least count (either exactly or approximately) their  total number. It is clear that finding or counting all solutions is at least as computationally difficult as deciding whether there exists a single solution, and  indeed in many cases it is strictly harder (assuming P is not equal NP) even to count approximately the number of solutions than it is to decide whether there exists at least one.

In this talk I will discuss a restricted family of problems, in which we are interested in solutions of a given size: for example, solutions could be copies of a specific k-vertex graph H in a large host graph G, or more generally k-vertex subgraphs of G that have some specified property (e.g. k-vertex subgraphs that are connected). In this setting, although exact counting is strictly harder than decision (assuming standard assumptions in parameterised complexity), the methods typically used to separate approximate counting from decision break down. Indeed, I will demonstrate a method that, subject to certain additional assumptions, allows us to transform an efficient decision algorithm for a problem of this form into an approximate counting algorithm with essentially the same running time.

This is joint work with John Lapinskas (Bristol) and Holger Dell (ITU Copenhagen).

The two courses I took from Wilkinson as a graduate student at Stanford influenced me greatly.  Along with some reminiscences of those days, this talk will touch upon backward error analysis, Gaussian elimination, and Evariste Galois.  It was originally presented at the Wilkinson 100th Birthday conference in Manchester earlier this year.

Many soft materials, such as elastomers and hydrogels, are made of long chain molecules crosslinked to form a three-dimensional network. Their mechanical properties depend on network parameters such as chain density, chain length distribution and the functionality of the crosslinks. Understanding the relationships between the topology of polymer networks and their mechanical properties has been a long-standing challenge in polymer physics.

In this work, we focus on so-called “near-ideal” networks, which are produced by the cross-coupling of star-like macromolecules with well-defined chain length. We developed a computational approach based on random discrete networks, according to which the polymer network is represented by an assembly of non-linear springs connected at junction points representing crosslinks. The positions of the crosslink points are determined from the conditions of mechanical equilibrium. Scaling relations for the elastic modulus and maximum extensibility of the network were obtained. Our scaling relations contradict some predictions of classical estimates of rubber elasticity and have implications for the interpretation of experimental data for near-ideal polymer networks.

Reference: G. Alame, L. Brassart. Relative contributions of chain density and topology to the elasticity of two-dimensional polymer networks. Soft Matter 15, 5703 (2019).

Magnetic skyrmions are topological solitons which occur in a large class
of ferromagnetic materials and which are currently attracting much
attention in the condensed matter community because of  their possible
use  in future magnetic information storage technology.  The talk is
about an integrable model for magnetic skyrmions, introduced in a recent
paper (arxiv 1812.07268) and generalised in (arxiv 1905.06285). The
model can be solved by interpreting it as a gauged nonlinear sigma
model. In the talk will explain the model and the geometry behind its
integrability, and discuss some of the solutions and their physical
interpretation.

In 1941, Chabauty gave a way to compute the set of rational points on specific curves. In 2004, Minhyong Kim showed how to extend Chabauty's method to a bigger class of curves using anabelian methods. In the talk, I will explain Chabauty's method and give an outline of how Kim extended those methods.

Description:In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions in terms of their ability to access the boundary (Feller's test for explosions) and to enter the interior from the boundary. Feller's technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille-Yosida theory. In the present article we study exit and entrance from infinity for jump diffusions driven by a stable process.Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller's work but exit and entrance from infinite boundaries has long remained open. We show that the these processes have features not observes in the diffusion setting. We derive necessary and sufficient conditions on σ so that (i) non-exploding solutions exist and (ii) the corresponding transition semigroup extends to an entrance point at `infinity'. Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti-Kiu representation and new Wiener-Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz-Bogdan--Żak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt-Nagasawa duality and Getoor's characterisation of transience and recurrence.