Past

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.

The introduction of Edwards' curves in 2007 relaunched a
deeper study of the arithmetic of elliptic curves with a
view to cryptographic applications.  In particular, this
research focused on the role of the model of the curve ---
a triple consisting of a curve, base point, and projective
(or affine) embedding. From the computational perspective,
a projective (as opposed to affine) model allows one to
avoid inversions in the base field, while from the
mathematical perspective, it permits one to reduce various
arithmetical operations to linear algebra (passing through
the language of sheaves). We describe the role of the model,
particularly its classification up to linear isomorphism
and its role in the linearization of the operations of addition,
doubling, and scalar multiplication.

In joint work with Olof Sisask, we establish new quantitative bounds for Roth's theorem on arithmetic progressions, showing that a set of integers with no three-term arithmetic progressions must have density O(1/(log N)^{1+c}) for some absolute constant c>0. This is the integer analogue of a result of Bateman and Katz for the model setting of vector spaces over a finite field, and the proof follows a similar structure. 

In this talk, we consider a split connected semisimple group G defined over a global field F. Let A denote the ring of adèles of F and K a maximal compact subgroup of G(A) with the property that the local factors of K are hyperspecial at every non-archimedian place. Our interest is to study a certain subspace of the space of square-integrable functions on the adelic quotient G(F)\G(A). Namely, we want to study functions coming from induced representations from an unramified character of a Borel subgroup and which are K-invariant.

Our goal is to describe how the decomposition of such space can be related with the Plancherel decomposition of a graded affine Hecke algebra (GAHA).

The main ingredients are standard analytic properties of the Dedekind zeta-function as well as known properties of the so-called residue distributions, introduced by Heckman-Opdam in their study of the Plancherel decomposition of a GAHA and a result by M. Reeder on the support of the weight spaces of
the anti-spherical  discrete series representations of affine Hecke algebras. These last ingredients are of a purely local nature.

This talk is based on joint work with V. Heiermann and E. Opdam.

In this talk, we discuss an ongoing calculation of the
four-loop form factors in massless QCD. We begin by discussing our
novel approach to the calculation in detail. Of particular interest
are a new polynomial-time integration by parts reduction algorithm and
a new method to algebraically resolve the IR and UV singularities of
dimensionally-regulated bare perturbative scattering amplitudes.
Although not all integral topologies are linearly reducible for the
more non-trivial color structures, it is nevertheless feasible to
obtain accurate numerical results for the finite parts of the complete
four-loop form factors using publicly available sector decomposition
programs and bases of finite integrals. Finally, we present first
results for the four-loop gluon form factor Feynman diagrams which
contain three closed fermion loops.

Let S be a set of monic degree 2 polynomials over a finite field and let C be the compositional semigroup generated by S. In this talk we establish a necessary and sufficient condition for C to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set S. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree 2 polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions (this is a joint work with A.Ferraguti and R.Schnyder). Time permitting, we will also describe how to use character sum techniques to bound the size of the graph determined by the generating set (this is a joint work with D.R. Heath-Brown).

A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of manifolds with finitely many conformally flat ends and singular points. More precisely, under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a Chern-Gauss-Bonnet type formula with error terms that can be expressed as isoperimetric deficits. This is joint work with Huy Nguyen. 

We characterize cutting arcs on ber surfaces that produce new ber surfaces,
and the changes in monodromy resulting from such cuts. As a corollary, we
characterize band surgeries between bered links and introduce an operation called
generalized Hopf banding. We further characterize generalized crossing changes between
bered links, and the resulting changes in monodromy.

This is joint work with Matt Rathbun, Kai Ishihara and Koya Shimokawa

It is well-known that the stochastic heat equation on R^n has a Hölder continuous function-valued solution in the case n=1, and that in dimensions 2 and above the solution is not function-valued but is forced to take values in some wider space of distributions. So what happens if the space has, in some sense, a dimension in between 1 and 2? We turn to the theory of fractals in order to answer this question. It has been shown (Kigami, 2001) that there exists a class of self-similar sets on which natural Laplacians can be defined, and so an analogue to the stochastic heat equation can be posed. In this talk we cover the following questions: Is the solution to this equation function-valued? If so, is it Hölder continuous? To answer the latter we must first prove an analogue of Kolmogorov's celebrated continuity theorem for the self-similar sets that we are working on. Joint work with Ben Hambly.

I will define the notion of "sheaf of categories with a local action of Hochschild cochains" over a stack. (This notion is analogous to D-modules, in the same way as the notion of "sheaf of categories" is analogous to quasi-coherent sheaves.) I will prove that both categories appearing in geometric Langlands carry this structure over the stack of de Rham {\check{G}}-local systems. Using this, I will explain how to glue D-mod(Bun_G) out of *tempered* D-modules associated to smaller Levi subgroups of G.

We give an explicit scheme to reconstruct any C^1 curve from its signature. It is implementable and comes with detailed stability properties. The key of the inversion scheme is the use of a symmetrisation procedure that separates the behaviour of the path at small and large scales. Joint work with Terry Lyons.

M5-branes on 4-manifolds M_4 realized as co-associatives in G_2 give rise to 2d (0,2) superconformal theories. In this talk I will propose a duality between these 2d (0,2) theories and 4d topological theories, which are sigma-models from M_4 into the Nahm moduli space. 

How might you prepare talks for different audiences (specialised seminar, colloquium-style talk, talk to a non-mathematical audience, job interview)?  Join us for advice on this, and on how to connect with your audience and get them to feel involved.

In practice, stochastic decision problems are often based on statistical estimates of probabilities. We all know that statistical error may be significant, but it is often not so clear how to incorporate it into our decision making. In this informal talk, we will look at one approach to this problem, based on the theory of nonlinear expectations. We will consider the large-sample theory of these estimators, and also connections to `robust statistics' in the sense of Huber.

Abstract: We will recall some analogies between structures arising from three-manifold topology and rings of integers in number fields. This can be used to define a Chern-Simons functional on spaces of Galois representations.  Some sample computations and elementary applications will be shown.

We show how to adapt methods originally developed in
model-independent finance / martingale optimal transport to give a
geometric description of optimal stopping times tau of Brownian Motion
subject to the constraint that the distribution of tau is a given
distribution. The methods work for a large class of cost processes.
(At a minimum we need the cost process to be adapted. Continuity
assumptions can be used to guarantee existence of solutions.) We find
that for many of the cost processes one can come up with, the solution
is given by the first hitting time of a barrier in a suitable phase
space. As a by-product we thus recover Anulova's classical solution of
the inverse first passage time problem.

Modelling catalytic kinetics is indispensable for the design of reactors and chemical processes. However, developing accurate and computationally efficient kinetic models remains challenging. Empirical kinetic models incorporate assumptions about rate-limiting steps and may thus not be applicable to operating regimes far from those where they were derived. Detailed microkinetic modelling approaches overcome this issue by accounting for all elementary steps of a reaction mechanism. However, the majority of such kinetic models employ mean-field approximations and are formulated as ordinary differential equations, which neglect spatial correlations. On the other hand, kinetic Monte Carlo (KMC) approaches provide a discrete-space continuous-time stochastic formulation that enables a detailed treatment of spatial correlations in the adlayer (resulting for instance from adsorbate-adsorbate lateral interactions), but at a significant computation expense.1,2

Motivated by these challenges, we discuss the necessity of KMC descriptions that incorporate detailed models of lateral interactions. Focusing on a titration experiment involving the oxidation of pre-adsorbed O by CO gas on Pd(111), we discuss experimental findings that show first order kinetics at low temperature (190 K) and half order kinetics at high temperature (320 K), the latter previously attributed to island formation.3 We perform KMC simulations whereby coverage effects on reaction barriers are captured by cluster expansion Hamiltonians and Brønsted-Evans-Polanyi (BEP) relations.4 By quantifying the effect of adlayer structure versus coverage effects on the observed kinetics, we rationalise the experimentally observed kinetics. We show that coverage effects lead to the half order kinetics at 320 K, rather than O-island formation as previously thought.5,6

Subsequently, we discuss our ongoing work in the development of approximations that capture such coverage effects but are much more computationally efficient than KMC, making it possible to use such models in reactor design. We focus on a model for NO oxidation incorporating first nearest neighbour lateral interactions and construct a sequence of approximations of progressively higher accuracy, starting from the mean-field treatment and continuing with a sequence of Bethe-Peierls models with increasing cluster sizes. By comparing the turnover frequencies of these models with those obtained from KMC simulation, we show that the mean-field predictions deviate by several orders of magnitude from the KMC results, whereas the Bethe-Peierls models exhibit progressively higher accuracy as the size of the explicitly treated cluster increases. While more computationally intensive than mean-field, these models still enable significant computational savings compared to a KMC simulation, thereby paving the road for employing them in multiscale modelling frameworks.

References

1    M. Stamatakis and D. G. Vlachos, ACS Catal. 2 (12), 2648 (2012).

2    M. Stamatakis, J Phys-Condens Mat 27 (1), 013001 (2015).

3    I. Nakai, H. Kondoh, T. Shimada, A. Resta, J. N. Andersen, and T. Ohta, J. Chem. Phys. 124 (22), 224712 (2006).

4    J. Nielsen, M. d’Avezac, J. Hetherington, and M. Stamatakis, J. Chem. Phys. 139 (22), 224706 (2013).

5    M. Stamatakis and S. Piccinin, ACS Catal. 6 (3), 2105 (2016).

6    S. Piccinin and M. Stamatakis, ACS Catal. 4, 2143 (2014).

Rational Krylov methods are applicable to a wide range of scientific computing problems, and ​the rational Arnoldi algorithm is a commonly used procedure for computing an ​orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this​ ​algorithm is the solution of a large linear system of equations at each iteration. We explore the​ ​option of solving several linear systems simultaneously, thus constructing the rational Krylov​ ​basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly​ ​conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the​ ​new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that ​allows to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm.
​ ​
The computational benefits are illustrated using several numerical examples from different application areas.
​ ​
This ​talk is based on joint work with Mario Berljafa  available as an Eprint at http://eprints.ma.man.ac.uk/2503/
 

I'll discuss the problem of controlling energy concentration in YM flow over a four-manifold. Based on a study of the rotationally symmetric case, it was conjectured in 1997 that bubbling can only occur at infinite time. My thesis contained some strong elementary results on this problem, which I've now solved in full generality by a more involved method.

A Kähler group is a group which can be realised as the fundamental group of a close Kähler manifold. We will prove that for a Kähler group $G$ we have that $G$ is residually free if and only if $G$ is a full subdirect product of a free abelian group and finitely many closed hyperbolic surface groups. We will then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler: We explain how to construct subgroups of direct products of surface groups which have even first Betti number but are not Kähler. All relevant notions will be explained in the talk.

The Algebraic Eraser is a cryptosystem (more precisely, a class of key
agreement schemes) introduced by Anshel, Anshel, Goldfeld and Lemieux
about 10 years ago. There is a concrete instantiation of the Algebraic
Eraser called the Colored Burau Key Agreement Protocol (CBKAP), which
uses a blend of techniques from permutation groups, matrix groups and
braid groups. SecureRF, the company owning the trademark to the
Algebraic Eraser, is marketing this system for lightweight
environments such as RFID tags and other Internet of Things
applications; they have proposed making this scheme the basis for an
ISO RFID standard.

This talk gives an introduction to the Algebraic Eraser, a brief
history of the attacks on this scheme using ideas from group-theoretic
cryptography, and describes the countermeasures that have been
proposed. I would not recommend the scheme for the proposed
applications: the talk ends with a brief sketch of a recent convincing
cryptanalysis of this scheme due to Ben-Zvi, Blackburn and Tsaban
(which appeared at CRYPTO this summer), and significant attacks
on the protocol in the proposed ISO standard due to Blackburn and
Robshaw (which appeared at ACNS earlier this year).