Past

Variational phase-field models of fracture have been at the center of a multidisciplinary effort involving a large community of mathematicians, mechanicians, engineers, and computational scientists over the last 25 years or so.

I will start with a modern interpretation of Griffith's classical criterion as a variational principle for a free discontinuity energy and will recall some of the milestones in its analysis. Then, I will introduce the phase-field approximation per se and describe its numerical implementation. I illustrate how phase-field models have led to major breakthroughs in the predictive simulation of fracture in complex situations.

I then will turn my attention to current issues, with a specific emphasis on crack nucleation in nominally brittle materials. I will recall the fundamental incompatibility between Griffith’s theory and nucleation criteria based on a stress yield surface: the strength vs. toughness paradox. I will then present several attempts at addressing this issue within the realm of phase-fracture and discuss their respective strengths and weaknesses. 

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 In this talk, we show some recent results related to the study of mechanical instabilities in slender structures. First, we propose a model of metamaterial sheets inspired by the pellicle of Euglenids, unicellular organisms capable of swimming due to their ability of changing their shape. These structures are composed of interlocking elastic rods which can freely slide along their edges. We characterize the kinematics and the mechanics of these structures using the special Cosserat theory of rods and by assuming axisymmetric deformations of the tubular assembly. We also characterize the mechanics of a single elastic beam constrained to smoothly slide along a rigid support, where the distance between the rod midline and the constraint is fixed and finite. In the presence of a straight support, the rod can deform into shapes exhibiting helices and perversions, namely transition zones connecting together two helices with opposite chirality.

Finally, we develop a mathematical model of damaged axons based on the theory of continuum mechanics and nonlinear elasticity. In several pathological conditions, such as coronavirus infections, multiple sclerosis, Alzheimer's and Parkinson's diseases, the physiological shape of axons is altered and a periodic sequence of bulges appears. The axon is described as a cylinder composed of an inner passive part, called axoplasm, and an outer active cortex, composed mainly of F-actin and able to contract thanks to myosin-II motors. Through a linear stability analysis, we show that, as the shear modulus of the axoplasm diminishes due to the disruption of the cytoskeleton, the active contraction of the cortex makes the cylindrical configuration unstable to axisymmetric perturbations, leading to a beading pattern.

We will consider the viscous Burgers driven by a localised one-dimensional control. The problem is considered in a bounded domain and is supplemented with the Dirichlet boundary condition. We will prove that any solution of the equation in question can be exponentially stabilised. Combining this result with an earlier result on local exact controllability we will show global exact controllability by a localised control. This is a joint work with A. Shirikyan.

Several natural measures of complexity can be attached to an
existentially definable ("diophantine") subset of a field. One of these
is the minimal number of existential quantifiers required to define it,
while others are of a more geometric nature. I shall define these
measures and discuss interesting interactions and behaviours, some of
which depend on properties of the field (e.g. imperfection and
ampleness). We shall see for instance that the set of n-tuples of field
elements consisting of n squares is usually definable with a single
quantifier, but not always. I will also discuss connections with
Hilbert's 10th Problem and a number of open questions.
This is joint work with Nicolas Daans and Arno Fehm.

Group theoretic Dehn filling, motivated by Dehn filling in the theory of 3- manifolds, is a process of constructing quotients of a given group. This technique is usually applied to groups with certain negative curvature feature, for example word-hyperbolic groups, to construct exotic and useful examples of groups. In this talk, I will start by recalling the notion of word-hyperbolic groups, and then show that how group theoretic Dehn filling can be used to answer the Burnside Problem and questions about mapping class groups of surfaces.

The AdS/CFT correspondence provides a rich setup to study the properties of gauge theories and the dual theories of gravity, in particular their thermodynamic properties. On RxS3, the maximally supersymmetric Yang-Mills theory with gauge group U(N) exhibits a phase transition that resembles the confinement-deconfinement transition of QCD. For infinite N, this transition is characterized by Hagedorn behavior. We show how the corresponding Hagedorn temperature can be calculated at any value of the ’t Hooft coupling via integrability. For large but finite N, we show how the Hagedorn behavior is replaced by Lee-Yang behavior.

This will be a zoom seminar with communal viewing in L4

I will talk about some recent joint work with H. Bui and J. Keating where we study the Ratios Conjecture for the family of quadratic L-functions over function fields. I will also discuss the closely related problem of obtaining upper bounds for negative moments of L-functions, which allows us to obtain partial results towards the Ratios Conjecture in the case of one over one, two over two and three over three L-functions. 

Trophic coherence, a measure of a graph’s hierarchical organisation, has been shown to be linked to a graph’s structural and dynamical aspects such as cyclicity, stability and normality. Trophic levels of vertices can reveal their functional properties, partition and rank the vertices accordingly. Trophic levels and hence trophic coherence can only be defined on graphs with basal vertices, i.e. vertices with zero in-degree. Consequently, trophic analysis of graphs had been restricted until now. In this talk I will introduce a novel  framework which can be defined on any simple graph. Within this general framework, I'll illustrate several new metrics: hierarchical levels, a generalisation of the notion of trophic levels, influence centrality, a measure of a vertex’s ability to influence dynamics, and democracy coefficient, a measure of overall feedback in the system. I will then discuss what new insights are illuminated on the topological and dynamical aspects of graphs. Finally, I will show how the hierarchical structure of a network relates to the incidence rate in a SIS epidemic model and the economic insights we can gain through it.

Article link: https://www.nature.com/articles/s41598-021-93161-4

A complex plane curve singularity gives rise to two objects: (1) a moduli space that representation theorists call an affine Springer fiber, and (2) a topological link up to isotopy. Roughly a decade ago, Oblomkov–Rasmussen–Shende conjectured a striking identity relating the homology of the affine Springer fiber to the so-called HOMFLYPT homology of the link. In unpublished writing, Shende speculated that it would follow from advances in nonabelian Hodge theory: the study of transcendental diffeomorphisms relating “Hitchin” and “Betti” moduli spaces. We make this dream precise by expressing HOMFLYPT homology in terms of the homology of a “Betti”-type space, which, we conjecture, deformation-retracts onto the affine Springer fiber. In doing so, we recast the whole story in terms of an arbitrary semisimple group. We give evidence for the nonabelian Hodge conjecture at the numerical level, using a mysterious formula that involves rational Cherednik algebras and the degrees of unipotent principal-series representations.

The $n$-queens problem asks how many ways there are to place $n$ queens on an $n \times n$ chessboard so that no two queens can attack one another, and the toroidal $n$-queens problem asks the same question where the board is considered on the surface of a torus. Let $Q(n)$ denote the number of $n$-queens configurations on the classical board and $T(n)$ the number of toroidal $n$-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that $T(n)>0$ if and only if $n \equiv 1,5 \mod 6$. Much more recently Luria showed that $T(n)\leq ((1+o(1))ne^{-3})^n$ and conjectured equality when $n \equiv 1,5 \mod 6$. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) $n \equiv 1,5 \mod 6$. We also show that $Q(n)\geq((1+o(1))ne^{-3})^n$ for all $n \in \mathbb{N}$ which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both $Q(n)$ and $T(n)$. 

In this talk we'll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a $4$-partite $4$-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost' configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.

This is joint work with Peter Keevash.

Michael Negus

Modelling high-speed droplet impact onto an elastic membrane

The impact of a high-speed droplet onto an elastic membrane is a highly nonlinear process and poses a formidable modelling challenge due to both the multi-scale nature of the flow and the fluid-structure interaction between the droplet and the membrane. We present two modelling approaches for droplet impact onto elastic membranes: matched asymptotics and direct numerical simulations (DNS). Inviscid Wagner theory is used in the former to derive analytical expressions which approximate the behaviour of the droplet during the early stages of impact, while the DNS builds on the open-source volume-of-fluid code Basilisk. We demonstrate the strong influence that the thickness, tension and stiffness of the membrane have on the dynamics of the droplet and the membrane. We also quantitatively show that the speed the droplet spreads across the substrate is notably decreased when the membrane is more compliant, which is consistent with experimental findings that splashing can be inhibited by impacting onto a soft substrate. We conclude by showing how these methods are complementary, as a combination of both can lead to a thorough understanding of the droplet impact across timescales.

Helen Saville

Lubrication model of a valve-controlled, gravity-driven bioreactor

Hospitals sometimes experience shortages of donor blood platelet supplies, motivating research into in vitro production of platelets. We model a novel platelet bioreactor described in Shepherd et al. [1]. The bioreactor consists of an upper channel, a lower channel, and a cell-seeded porous collagen scaffold situated between the two. Flow is driven by gravity, and controlled by valves on the four inlets and outlets. The bioreactor is long relative to its width, a feature which we exploit to derive a lubrication reduction of Navier-Stokes flow coupled to Darcy. Models for two cases are considered: small amplitude valve oscillations, and order one amplitude valve oscillations. The former model is a systematic reduction; the latter incorporates a phenomenological approximation for the cross-sectional flow profile. As the shear stress experienced by cells influences platelet production, we use our model to quantify the effect of valve dynamics on shear stress.

1: Shepherd, J.H., Howard, D., Waller, A.K., Foster, H.R., Mueller, A., Moreau, T., Evans, A.L., Arumugam, M., Chalon, G.B., Vriend, E. and Davidenko, N., Biomaterials, 182, pp.135-144. (2018)

We will report on recent progress regarding the near-critical behavior of certain statistical physics models in dimension 3. Our results deal with the second-order phase transition associated to two percolation problems involving the Gaussian free field in 3D. In one case, they determine a unique ``fixed point'' corresponding to the transition, which is proved to obey one of several scaling relations. Such laws are classically conjectured to hold by physicists on the grounds of a corresponding scaling ansatz. 

The aim of this talk is to understand the qualitative properties that emerge from a PDE model inspired from neurosciences, in order to understand what are the key processes that lead to mathematical complex patterns for the solutions of this equation. 

It is a widely accepted philosophy in additive number theory that convex sets ought not to exhibit much additive structure. We could measure this by estimating the sizes of their sumsets. In this talk, we will hopefully move from the philosophical to the concrete, by giving ways to see that convex sets and functions have poor additive structure. We will also discuss some recent developments in the area.

In a joint work with Peter Ozsvath we have developed algebraic invariants for knots using a family of bordered knot algebras. The goal of this lecture is to review these constructions and discuss some of the latest developments.

Cohomological Hall algebras and vertex algebras are two structures whose origins are (at least in part) from physics. I will explain what these objects are, how the latter was related to moduli stacks by Joyce, and a theorem relating these two structures. The main tool is torus localisation, a method for "turning geometry into combinatorics", or rather a new formulation of it which works in the singular setting.

We consider the problem of parameter estimation for a McKean stochastic differential equation, and the associated system of weakly interacting particles. The problem is motivated by many applications in areas such as neuroscience, social sciences (opinion dynamics, cooperative behaviours), financial mathematics, statistical physics. We will first survey some model properties related to propagation of chaos and ergodicity and then move on to discuss the problem of parameter estimation both in offline and on-line settings. In the on-line case, we propose an online estimator, which evolves according to a continuous-time stochastic gradient descent algorithm on the asymptotic log-likelihood of the interacting particle system. The talk will present our convergence results and then show some numerical results for two examples, a linear mean field model and a stochastic opinion dynamics model. This is joint work with Louis Sharrock, Panos Parpas and Greg Pavliotis. Preprint: https://arxiv.org/abs/2106.13751

The analytic structure of scattering amplitudes exhibit striking
properties that are not at all evident from the first principles of
Quantum Field Theory. These are often rich and powerful enough to be
considered as their defining features, and this makes the problem of
finding a set of universal rules a compelling one. I will review the
recently mounting evidence for the relevance of tropical Grassmannians
in this respect, including implications on symbol alphabets and
adjacency conditions

In this talk I will review the “holomorphic modular bootstrap,” i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters. By making use of the representation theory of PSL(2, Zn), we describe a method to classify allowed central charges and weights (c, hi) for theories with any number of characters d. This allows us to avoid various bottlenecks encountered previously in the literature, and leads to a classification of consistent characters up to d = 5 whose modular differential equations are uniquely fixed in terms of (c, hi). In the process, we identify the full set of constraints on the allowed values of the Wronskian index for fixed d ≤ 5.

This session will take place live in L1 only and not online on Teams. 

Are you interested in sharing your love of Maths with the next generation of mathematicians, but you don’t know where to start? In this session we will discuss some basic principles and top tips for creating a workshop for students aged 14–16, and get you started on developing your own. There will also be the opportunity to work on this further afterwards and potentially deliver your session as part of the Oxfordshire Maths Masterclasses (for local school students) in Hilary Term. Bring along your favourite bit of maths and a willingness to have a go.

In this talk, I'll present inequalities bounding the number of critical cells in a filtered cell complex on the one hand, and the entries of the Betti tables of the multi-parameter persistence modules of such filtrations on the other hand. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for the number of critical cells. Furthermore, we prove a sharp upper bound for the minimal number of critical cells, expressed again in terms of the entries of Betti tables. This is joint work with Andrea Guidolin (KTH, Stockholm). The full paper is posted online as arxiv:2108.11427.

The Specht modules are of fundamental importance to the representation theory of the symmetric group, and their 0th cohomology is understood through entirely combinatorial methods due to Gordon James. Over fields of odd characteristic, Hemmer proposed a similar combinatorial approach to calculating their 1st degree cohomology, or extensions by the trivial module. This combinatorial approach motivates the definition of universal $p$-ary designs, which we shall classify. We then explore the consequences of this classification to problem of determining extensions of Specht modules. In particular, we classify all extensions of Specht modules indexed by two-part partitions by the trivial module and shall see some far-reaching conditions on when the first cohomology of a Specht module is trivial.