Past

Stallings' theorem states that a finitely generated group splits over a finite subgroup if and only if it has more than one end. As a consequence of this, group splittings over finite subgroups are invariant under quasi-isometry. I will discuss a generalisation of Stallings' theorem which shows that under suitable hypotheses, group splittings over classes of infinite groups, namely coarse $PD_n$ groups, are also invariant under quasi-isometry.

Not considering classified work, the first person to have asked and solved the problem of secure communication over insecure communication channels was Ralph Merkle, in a project for a Computer securitjohn y course at UC Berkeley in 1974. In this work, he gave a protocol that allow two legitimate parties to establish a secret key with an effort of the order of N, but such that an eavesdropper can not discover the secret key with non-vanishing probability if he is not willing to spend an effort of at least the order of N^2.
In this talk, we will consider key exchange protocols in the presence of a quantum eavesdropper. Unfortunately, it is easy to see that in this case, breaking Merkle’s original protocol only requires an effort of the order of N, similar to the one of the legitimate parties. We will show how to restore the security by presenting two sequences of protocols with the following properties:
- In the first sequence, the legitimate parties have access to a quantum computer, and the eavesdropper's effort is arbitrarily close to N^2.
- In the second sequence, the protocols are classical, but the eavesdropper’s effort is arbitrarily close to N^{3/2}.
We will show the key exchange protocols, the quantum attacks with the proof of their optimality. We will focus mostly on the techniques from quantum algorithms and complexity theory used to devise quantum algorithms and to prove lower bounds. The underlying tools are the quantum walk formalism, and the quantum adversary lower bound method, respectively. Finally, we will introduce a new method to prove average-case quantum query complexity lower bounds.

The problem of computing the Galois group of an irreducible, rational polynomial has been studied for many years. I will discuss the methods developed over the years to approach this problem, and give some examples of them in practice. These methods mainly involve constructing and factorising resolvent polynomials, and thereby determining better upper bounds for the conjugacy class of the Galois group within the symmetric group, i.e. describe its action on the roots of the polynomial explicitly. I will describe how using approximations to the zeros of the polynomial allows us to construct resolvents, and in particular, how using p-adic approximations can be advantageous over numerical approximations, and how this can yield a direct and systematic method of determining the Galois group.

The Hall algebra can be constructed using the Waldhausen S-construction. We will give a systematic recipe for this and show how it extends naturally to give a bi-algebraic structure. As a result we obtain a more transparent proof of Green's theorem about the bi-algebra structure on the Hall algebra.

For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show that for fixed $k\geq2$ and $n$ odd and sufficiently large,
$$
R_k(C_n)=2^{k-1}(n-1)+1.
$$
This resolves a conjecture of Bondy and Erdős for large $n$. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation.  This allows us to prove a stability-type generalisation of the above and establish a correspondence between extremal $k$-colourings for this problem and perfect matchings in the $k$-dimensional hypercube $Q_k$.

It has long been expected, and is now proved in many important cases, 
that quantum algebras are more rigid than their classical limits. That is, they 
have much smaller automorphism groups. This begs the question of whether this 
broken symmetry can be recovered.

I will outline an approach to this question using the ideas of noncommutative 
projective geometry, from which we see that the correct object to study is a 
groupoid, rather than a group, and maps in this groupoid are the replacement 
for automorphisms. I will illustrate this with the example of quantum 
projective space.

This is joint work with Nicholas Cooney (Clermont-Ferrand).

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.

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.