Past

Thin glass sheets are used in smartphone, battery and semiconductor technology, and may be manufactured by first producing a relatively thick glass slab (known as a preform) and subsequently redrawing it to a required thickness. Theoretically, if the sheet is redrawn through an infinitely long heater zone, a product with the same aspect ratio as the preform may be manufactured. However, in reality the effect of surface tension and the restriction to factories of finite size prevent this. In this talk I will present a mathematical model for a viscous sheet undergoing redraw, and use asymptotic analysis in the thin-sheet, low-Reynolds-number limit to investigate how the product shape is affected by process parameters. 

I will survey of some of the many significant connections between integrable many-body physics and mathematics. I exploit an algebraic structure called a fusion category, familiar from the study of conformal field theory, topological quantum field theory and knot invariants. Rewriting statistical-mechanical models in terms of a fusion category allows the derivation of combinatorial identities for the Tutte polynomial, the analysis of discrete ``holomorphic'' observables in probability, and to defining topological defects in lattice models. I will give a little more detail on topological defects, explaining how they allows exact computations of conformal-field-theory quantities directly on the lattice, as well as a greatly generalised set of duality transformations.
 

We consider the Beris-Edwards model of liquid crystal dynamics. We study a non-dimensionalisation and regime suited for the study of defect patterns, that amounts to a combined high Ericksen and high Reynolds  number regime. 
We identify some of the flow mechanisms responsible for the appearance of localized gradients that increase in time.
This is joint work with Hao Wu (Fudan).
 

Decorate knot cobordisms functorially induce maps on knot Floer homology.
We compute these maps for elementary cobordisms, and hence give a formula for 
the Alexander and Maslov grading shifts. We also show a non-vanishing result in the case of
concordances and present some applications to invertible concordances. 
This is joint work with Marco Marengon.
 

We revisit small-noise expansions in the spirit of Benarous, Baudoin-Ouyang, Deuschel-Friz-Jacquier-Violante for bivariate diffusions driven by fractional Brownian motions with different Hurst exponents. A particular focus is devoted to rough stochastic volatility models which have recently attracted considerable attention.
We derive suitable expansions (small-time, energy, tails) in these fractional stochastic volatility models and infer corresponding expansions for implied volatility. This sheds light (i) on the influence of the Hurst parameter in the time-decay of the smile and (ii) on the asymptotic behaviour of the tail of the smile, including higher orders.

Abstract: (This is joint work with Mark McLean, Stony Brook University N.Y.).

The classical McKay correspondence is a 1-1 correspondence between finite subgroups G of SL(2,C) and simply laced Dynkin diagrams (the ADE classification). These diagrams determine the representation theory of G, and they also describe the intersection theory between the irreducible components of the exceptional divisor of the minimal resolution Y of the simple surface singularity C^2/G. In particular those components generate the homology of Y. In the early 1990s, Miles Reid conjectured a far-reaching generalisation to higher dimensions: given a crepant resolution Y of the singularity C^n/G, where G is a finite subgroup of SL(n,C), the claim is that the conjugacy classes of G are in 1-1 correspondence with generators of the cohomology of Y. This has led to much active research in algebraic geometry in recent years, in particular Batyrev proved the conjecture in 2000 using algebro-geometric techniques (Kontsevich's motivic integration machinery). The goal of my talk is to present work in progress, jointly with Mark McLean, which proves the conjecture using symplectic topology techniques. We construct a certain symplectic cohomology group of Y whose generators are Hamiltonian orbits in Y to which one can naturally associate a conjugacy class in G. We then show that this symplectic cohomology recovers the classical cohomology of Y.

This work is part of a large-scale project which aims to study the symplectic topology of resolutions of singularities also outside of the crepant setup.

SPDEs with Lévy noise can be used to model chemical, physical or biological phenomena which contain uncertainties. When discretising these SPDEs in order to solve them numerically the problem might be of large order. The goal is to save computational time by replacing large scale systems by systems of low order capturing the main information of the full model. In this talk, we therefore discuss balancing related MOR techniques. We summarise already existing results and discuss recent achievements.

Calabi-Yau spaces provide well-understood examples of supersymmetric vacua in supergravity. The supersymmetry conditions on such spaces can be rephrased as the existence and integrability of a particular geometric structure. When fluxes are allowed, the conditions are more complicated and the analogue of the geometric structure is not well understood.
In this talk, I will review work that defines the analogue of Calabi-Yau geometry for generic D=4, N=2 supergravity backgrounds. The geometry is characterised by a pair of structures in generalised geometry, where supersymmetry is equivalent to integrability of the structures. I will also discuss the extension AdS backgrounds, where deformations of these geometric structures correspond to exactly marginal deformations of the dual field theories.

The so-called Ax-Lindemann theorem asserts that the Zariski closure of a certain subset of a homogeneous variety (typically abelian or Shimura) is itself a homogeneous variety. This theorem has recently been proven in full generality by Klingler-Ullmo-Yafaev and Gao. This statement leads to a variety of questions about topological and Zariski closures of certain sets in  homogeneous varieties which can be approached by both ergodic and o-minimal techniques.  In a series of recent papers with E. Ullmo, we formulate conjectures and prove a certain number of results  of this type.  In this talk I will present these conjectures and results and explain the ideas of proofs
 

Some relish the idea of working with users of research and having an impact on the outside world - some view it as a ridiculous government agenda which interferes with academic freedom.  We’ll give an overview of  the political and practical aspects of impact and identify things you might want to consider when deciding whether, and how, to get involved.

Niall Bootland (Scalable Two-Phase Flow Solvers)

Sourav Mondal (Electrohydrodynamics in microchannel)

Abstract: Flow of liquid due to an electric potential gradient is possible when the channel walls bear a surface charge and liquid contains free charges (electrolyte). Inclusion of electrokinetic effects in microchannel flows has an added advantage over Poiseuille flow - depending upon the electrolyte concentration, the Debye layer thickness is different, which allows for tuning of flow profiles and the associated mass transport. The developed mathematical model helps in probing the mass transfer effects through a porous walled microchannel induced by electrokinetic forces.

The challenge is to develop an automated process that transforms an initial desired design of turbine rotor and blades in to a close approximation having eigenfrequencies that avoid the operating frequency (and its first harmonic) of the turbine.

I will revisit the theory of Hodge-Tate local systems in the light of the p-adic Simpson correspondence. This is a joint work with Michel Gros.

The study of moving contact lines is challenging for various reasons: Physically no sliding motion is allowed with a standard no-slip boundary condition over a solid substrate. Mathematically one has to deal with a free-boundary problem which contains certain singularities at the contact line. Instabilities can lead to topological transition in configurations space - their rigorous mathematical understanding is highly non-trivial. In this talk some state-of-the-art modeling and numerical techniques for such challenges will be presented. These will be applied to flows over solid and liquid substrates, where we perform detailed comparisons with experiments.

 The matrix logarithm, when applied to symmetric positive definite matrices, is known to satisfy a notable concavity property in the positive semidefinite (Loewner) order. This concavity property is a cornerstone result in the study of operator convex functions and has important applications in matrix concentration inequalities and quantum information theory.
In this talk I will show that certain rational approximations of the matrix logarithm remarkably preserve this concavity property and moreover, are amenable to semidefinite programming. Such approximations allow us to use off-the-shelf semidefinite programming solvers for convex optimization problems involving the matrix logarithm. These approximations are also useful in the scalar case and provide a much faster alternative to existing methods based on successive approximation for problems involving the exponential/relative entropy cone. I will conclude by showing some applications to problems arising in quantum information theory.

This is joint work with James Saunderson (Monash University) and Pablo Parrilo (MIT)

A Kähler group is a group which can be realised as fundamental group of a compact Kähler manifold. I shall begin by explaining why such groups are not arbitrary and then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler. Work of Bridson, Howie, Miller and Short reduces this to the case of subgroups which are not of type $\mathcal{F}_r$ for some $r$. We will give a new construction producing Kähler groups with exotic finiteness properties by mapping products of closed Riemann surfaces onto an elliptic curve. We will then explain how this construction can be generalised to higher dimensions. This talk is independent of last weeks talk on Kähler groups and all relevant notions will be explained.