Past

It is thought that the hairy legs of water walking arthropods are able to remain clean and dry because the flexibility of the hairs spontaneously moves drops off the hairs. We present a mathematical model of this bending-induced motion, or bendotaxis, and study how it performs for wetting and non-wetting drops. Crucially, we show that both wetting and non-wetting droplets move in the same direction (using physical arguments and numerical solutions). This suggests that a surface covered in elastic filaments (such as the hairy leg of insects) may be able to universally self-clean. To quantify the efficiency of this effect, we explore the conditions under which drops leave the structure by ‘spreading’ rather than translating and also how long it takes to do so.

In a recent preprint, Basterra, Bobkova, Ponto, Tillmann and Yeakel
defined operads with homological stability (OHS) and showed that after
group-completion, algebras over an OHS group-complete to infinite loop
spaces. This can in particular be used to put a new infinite loop space
structure on stable moduli spaces of high-dimensional manifolds in the
sense of Galatius and Randal-Williams, which are known to be infinite
loop spaces by a different method.

To complicate matters further, I shall introduce a mild strengthening of
the OHS condition and construct yet another infinite loop space
structure on these stable moduli spaces. This structure turns out to be
equivalent to that constructed by Basterra et al. It is believed that
the infinite loop space structure due to Galatius--Randal-Williams is
also equivalent to these two structures.

l-adic cohomology was built to provide an etale cohomology with coefficients in a field of characteristic 0. This, via the Grothendieck trace formula, gives  a cohomological interpretation of L-functions - a fundamental tool in Deligne's theory of weights developed in Weil II. Instead of l-adic coefficients one can consider coefficients in ultra products of finite fields. I will state the fundamental theorem of Weil II for curves in this setting and explain briefly what are the difficulties to overcome to adjust Deligne's proof. I will then discuss how this ultra product variant of Weil II allows to extend to arbitrary coefficients  previous results of Gabber and Hui, Tamagawa and myself for constant $\mathbb{Z}_\ell$-coefficients.  For instance,  it implies that, in an $E$-rational compatible system of smooth $\overline{\mathbb{Q}}_\ell$-sheaves all what is true for $\overline{\mathbb{Q}}_\ell$-coefficients (semi simplicity, irreducibility, invariant dimensions etc) is true for $\overline{\mathbb{F}}_\ell$-coefficients provided $\ell$ is large enough or that the $\overline{\mathbb{Z}}_\ell$-models are unique with torsion-free cohomology provided $\ell$ is large enough.

Many species of insects adhere to vertical and inverted surfaces using footpads that secrete thin films of a mediating fluid. The fluid bridges the gap between the foot and the target surface. The precise role of this liquid is still subject to debate, but it is thought that the contribution of surface tension to the adhesive force may be significant. It is also known that the footpad is soft, suggesting that capillary forces might deform its surface. Inspired by these physical ingredients, we study a model problem in which a thin, deformable membrane under tension is adhered to a flat, rigid surface by a liquid droplet. We find that there can be multiple possible equilibrium states, with the number depending on the applied tension and aspect ratio of the system. The presence of elastic deformation  ignificantly enhances the adhesion force compared to a rigid footpad. A mathematical model shows that the equilibria of the system can be controlled via two key parameters depending on the imposed separation of the foot and target surface, and the tension applied to the membrane. We confirm this finding experimentally and show that the system may transition rapidly between two states as the two parameters are varied. This suggests that different strategies may be used to adhere strongly and then detach quickly.

In this talk, I consider  the problem of
hedging European and Bermudan option with a given probability. This 
question is
more generally linked to portfolio optimisation problems under weak
stochastic target constraints.
I will recall, in a Markovian framework, the characterisation of the 
solution by
non-linear PDEs. I will then discuss various numerical algorithms
to compute in practice the quantile hedging price.

This presentation is based on joint works with B. Bouchard (Université 
Paris Dauphine), G. Bouveret (University of Oxford) and ongoing work 
with C. Benezet (Université Paris Diderot).

Spline curves represent a simple and efficient tool for data interpolation in Euclidean space. During the past decades, however, more and more applications have emerged that require interpolation in (often high-dimensional) nonlinear spaces such as Riemannian manifolds. An example is the generation of motion sequences in computer graphics, where the animated figure represents a curve in a Riemannian space of shapes. Two particularly useful spline interpolation methods derive from a variational principle: linear splines minimize the average squared velocity and cubic splines minimize the average squared acceleration among all interpolating curves. Those variational principles and their discrete analogues can be used to define continuous and discretized spline curves on (possibly infinite-dimensional) Riemannian manifolds. However, it turns out that well-posedness of cubic splines is much more intricate on nonlinear and high-dimensional spaces and requires quite strong conditions on the underlying manifold. We will analyse and discuss linear and cubic splines as well as their discrete counterparts on Riemannian manifolds and show a few applications.

In the study of variational models for non-linear elasticity in the context of proving regularity we are led to the challenging so-called Ball-Evan's problem of approximating a Sobolev homeomorphism with diffeomorphisms in its Sobolev space. In some cases however we are not able to guarantee that the limit of a minimizing sequence is a homeomorphism and so the closure of Sobolev homeomorphisms comes into the game. For $p\geq 2$ they are exactly Sobolev monotone maps and for $1\leq p<2$ the monotone maps are intricately related to these limits. In our paper we prove that monotone maps can be approximated by diffeomorphisms in their Sobolev (or Orlicz-Sobolev) space including the case $p=1$ not proven by Iwaniec and Onninen.
 

Warped cones are infinite metric spaces that are associated with actions by homeomorphisms on metric spaces. In this talk I will try to explain why the coarse geometry of warped cones can be seen as an invariant of the action and what it can tell us about the acting group.

Broken cryptographic algorithms and hardness assumptions are a constant
threat to real-world protocols. Prominent examples are
hash functions for which collisions become known, or number-theoretic
assumptions which are threatened by advances in quantum computing.
Especially when it comes to key exchange protocols, the switch to
quantum-resistant primitives has begun and aims to protect today’s
secrets against future developments, moving from common Diffie–Hellman
based solutions to Learning-With-Errors-based approaches. Remarkably,
the authentication step in such protocols is usually still carried out
with quantum-vulnerable signature schemes. The intuition here is that
the adversary would need to break this protocol primitive today, without
having quantum power yet. The question we address here is if this
intuition is justified, and if so, if we can show this rigorously. We
particularly consider the authenticated variant of the recently
introduced post-quantum secure key exchange protocol NewHope (Alkim et
al., USENIX Security 2016), as well as by TLS 1.3, which is currently
being developed by the Internet Engineering Task Force.

I was speak on the way Newton carries out his calculus in the Principia in the framework of classical geometry rather than with fluxions, his deficiencies, and the relation of this work to inverse-square laws.

Understanding the distribution of subgraph counts has long been a central question in the study of random graphs. In this talk, we consider the distribution of Sn, the number of K4 subgraphs, in the Erdös Rényi random graph G(n, p). When the edge probability p \in (0, 1) is constant, a classical central limit theorem for Sn states that (Sn−µn)/σn converges in distribution. We establish a stronger form of convergence, namely the corresponding local limit theorem, which is joint work with O. Riordan.
 

(Joint with Gareth Boxall) In this talk I will introduce some properties of distal theories. I will remark that distality is preserved neither under reducts nor expansions of the language. I will then go on to discuss a recent result that the Shelah expansion of a theory is distal if and only if the theory itself is distal. 

In mirror symmetry, the prepotential on the Kahler side has an expansion, the constant term of which is a rational multiple of zeta(3)/(2 pi i)^3 after an integral symplectic transformation. In this talk I will explain the connection between this constant term and the period of a mixed Hodge-Tate structure constructed from the limit MHS at large complex structure limit on the complex side. From Ayoub’s works on nearby cycle functor, there exists an object of Voevodsky’s category of mixed motives such that the mixed Hodge-Tate structure is expected to be a direct summand of the third cohomology of its Hodge realisation. I will present the connections between this constant term and conjecture about how mixed Tate motives sit inside Voevodsky’s category, which will also provide a motivic interpretation to the occurrence of zeta(3) in prepotential. 

I will describe a novel algorithm for computing the Walsh Hadamard Transform (WHT) which consists entirely of Haar wavelet transforms. The algorithm shares precisely the same serial complexity as the popular divide-and-conquer algorithm for the WHT. There is also a natural way to parallelize the algorithm which appears to have a number of attractive features.

A class C of graphs has polynomial expansion if there exists a polynomial p such that for every graph G from C and for every integer r, each minor of G obtained by contracting disjoint subgraphs of radius at most r is p(r)-degenerate. Classes with polynomial expansion exhibit interesting structural, combinatorial, and algorithmic properties. In the talk, I will survey these properties and propose further research directions.

Medical imaging is a key diagnostic tool, and is paramount for disease detection and for patient monitoring during ongoing care. Often, to reduce the amount of radiation that a patient is subjected to, there is a strong incentive to consider image reconstruction from incomplete sets of measurements, and so the imaging process is formulated as a compressed sensing problem.

In this talk, we will focus on compressed sensing for digital tomosynthesis (DTS), in which three-dimensional images are reconstructed from a set of two-dimensional X-ray projections. We first discuss a reconstruction approach for static bodies, with a particular interest in the choice of basis for the image representation. We will then focus on the need for accurate image reconstructions when the body of interest is not stationary, but is undergoing simple motion, discussing two different approaches for tackling this dynamic problem.

This will be a discussion of the paper https://arxiv.org/abs/1604.02618

A great deal of effort has gone into trying to model social influence --- including the spread of behavior, norms, and ideas --- on networks. Most models of social influence tend to assume that individuals react to changes in the states of their neighbors without any time delay, but this is often not true in social contexts, where (for various reasons) different agents can have different response times. To examine such situations, we introduce the idea of a timer into threshold models of social influence. The presence of timers on nodes delays the adoption --- i.e., change of state --- of each agent, which in turn delays the adoptions of its neighbors. With a homogeneous-distributed timer, in which all nodes exhibit the same amount of delay, adoption delays are also homogeneous, so the adoption order of nodes remains the same. However, heterogeneously-distributed timers can change the adoption order of nodes and hence the "adoption paths" through which state changes spread in a network. Using a threshold model of social contagions, we illustrate that heterogeneous timers can either accelerate or decelerate the spread of adoptions compared to an analogous situation with homogeneous timers, and we investigate the relationship of such acceleration or deceleration with respect to timer distribution and network structure. We derive an analytical approximation for the temporal evolution of the fraction of adopters by modifying a pair approximation of the Watts threshold model, and we find good agreement with numerical computations. We also examine our new timer model on networks constructed from empirical data.

Link to arxiv paper: https://arxiv.org/abs/1706.04252

I will review some classical results on geometric scattering
theory for linear hyperbolic evolution equations
on globally hyperbolic spacetimes and its relation to particle and charge
creation in QFT. I will then show that some index formulae for the
scattering matrix can be interpreted as a special case of the  Lorentzian
analog of the Atyiah-Patodi-Singer index theorem. I will also discuss a
local version of this theorem and its relation to anomalies in QFT.
(Joint work with C. Baer)

Many industrial optimisation problems involve the challenging task of efficiently searching for optimal decisions from a huge set of possible combinations. The optimal solution is the one that best optimises a set of objectives or goals, such as maximising productivity while minimising costs. If we have a nice mathematical equation for how each objective depends on the decisions we make, then we can usually employ standard mathematical approaches, such as calculus, to find the optimal solution. But what do we do when we have no idea how our decisions affect the objectives, and thus no equations? What if all we have is a small set of experiments, where we have tried to measure the effect of some decisions? How do we make use of this limited information to try to find the best decisions?

This talk will present a common industrial optimisation problem, known as expensive black box optimisation, through a case study from the manufacturing sector. For problems like this, calculus can’t help, and trial and error is not an option! We will introduce some methods and tools for tackling expensive black-box optimisation. Finally, we will discuss new methodologies for assessing the strengths and weaknesses of optimisation methods, to ensure the right method is selected for the right problem.

I will talk about the diffeomorphism classification of 4-manifolds up to 
connected sums with the complex projective plane, and how the resulting 
equivalence class of a manifold can be detected by algebraic topological 
invariants of the manifold.  I may also discuss related results when one 
takes connected sums with another favourite 4-manifold, S^2 x S^2, instead.

Recurrent major mood episodes and subsyndromal mood instability cause substantial disability in patients with bipolar disorder. Early identification of mood episodes enabling timely mood stabilisation is an important clinical goal. The signature method is derived from stochastic analysis (rough paths theory) and has the ability to capture important properties of complex ordered time series data. To explore whether the onset of episodes of mania and depression can be identified using self-reported mood data.

In this joint work with Amandine Aftalion we study the minimisers of an energy functional in two-dimensions describing a rotating two-component condensate. This involves in particular separating a line-energy term and a vortex term which have different orders of magnitude, and requires new estimates for functionals of the Cahn-Hilliard (or Modica-Mortola) type.

In this talk, I am going to report on some on-going research at the interface between Rough Paths Theory and Schramm-Loewner evolutions (SLE). In this project, we try to adapt techniques from Rough Differential Equations to the study of the Loewner Differential Equation. The main ideas concern the restart of the backward Loewner differential equation from the singularity in the upper half plane. I am going to describe some general tools that we developed in the last months that lead to a better understanding of the dynamics in the closed upper half plane under the backward Loewner flow.
Joint work with Prof. Dmitry Belyaev and Prof. Terry Lyons

Associated to a finite graph without loops is the Kac-Moody Lie algebra for the Cartan matrix whose off diagonal entries are (minus) the adjacency matrix for the graph.  Two famous conjectures of Kac, proved by Hausel, Letellier and Villegas, hint that there may be some larger cohomologically graded algebra associated to the graph (even if there are loops), providing "higher" Kac moody Lie algebras, or at least their positive halves.  Using work with Sven Meinhardt, I will give a geometric construction of the (full) Kac-Moody algebra for a general finite graph, using cohomological DT theory.  Along the way we'll see a proof of the positivity conjecture for the modified Kac polynomials of Bozec, Schiffmann and Vasserot counting various types of representations of quivers.

The existence of planetary and stellar magnetic fields is attributed to the dynamo instability, the mechanism by which a background turbulent flow spontaneously generates a magnetic field by the constructive refolding of magnetic field lines. Many efforts have been made by several experimental groups to reproduce the dynamo instability in the laboratory using liquid metals. However, so far, unconstrained dynamos driven by turbulent flows have not been achieved in the intrinsically low magnetic Prandtl number $P_m$ (i.e. $Pm = Rm/Re << 1$) laboratory experiments. In this seminar I will demonstrate that the critical magnetic Reynolds number $Rm_c$ for turbulent non-helical dynamos in the low $P_m$ limit can be significantly reduced if the flow is submitted to global rotation. Even for moderate rotation rates the required energy injection rate can be reduced by a factor more than 1000. Our finding thus points into a new paradigm for the design of new liquid metal dynamo experiments.

Image use continues to increase in both biomedical sciences and clinical practice. State of the art acquisition techniques allow characterisation from subcellular to whole organ scale, providing quantitative information of structure and function. In the heart, for example, images acquired from a single modality (cardiac MRI) can characterise micro- and macrostructure, describe mechanical function and measure blood flow. In the lungs, new contrast agents can be used to visualise the flow of gas in free breathing subjects. This provides rich new sources of information as well as new challenges to extract data in a way that is useful to clinicians as well as computer modellers.
I will describe efforts in my group to use the latest advances in machine learning to analyse images, and explain how we are applying these to the development of accurate computer models of the heart.
 

We discuss pathwise pricing-hedging dualities in continuous time and on a frictionless market consisting of finitely many risky assets with continuous price trajectories.

The costs to Vodafone of calls terminating on other networks – especially fixed networks – are largely determined by the termination charges levied by other telecoms operators.  We interconnect to several other telecoms operators, who charge differently; within one interconnect operator, costs vary depending on which of their switching centres we deliver calls to, and what the terminating phone number is.  So, while these termination costs depend partly on factors that we cannot control (such as the number called, the call duration and the time of day), they are also influenced by some factors that we can control.  In particular, we can route calls within our network before handing them over from our network to the other telecoms operator; where this “handover” occurs has an impact on termination cost.  
Vodafone would like to develop a repeatable capability to determine call delivery cost efficiency and identify where network routing changes can be made to improve matters, and determine traffic growth forecasts.

In this talk I will discuss the problem of finding Einstein metrics in the homogeneous and cohomogeneity one setting. 
In particular, I will describe a recent result concerning existence of solutions to the Dirichlet problem for cohomogeneity one Einstein metrics.

We present a framework for the design, analysis and application of computational multiscale methods for slow-fast high-dimensional stochastic processes. We call these processes "microscopic'', and assume existence of an approximate "macroscopic'' model that captures the slow behaviour of a selected set of macroscopic state variables. The methodology combines short bursts of microscopic simulation with extrapolation at the macroscopic level. The methodology requires the careful study of a few key algorithmic ingredients. First, we need to properly initialise the microscopic system, based on a given macroscopic state and (possibly) a prior microscopic state that contains additional information about the system. Second, we need to control the variance of the noise that originates from the microscopic Monte Carlo simulation. Third, we need to analyse stability of the extrapolation step. We will discuss these aspects on two types of model problems -- scale-separated SDEs and kinetic equations -- and show the efficacity of the resulting methods in diverse applications, ranging from tumor growth to fusion energy.

We formulate and solve a class of Backward Stochastic Differential Equations (BSDEs) driven by the compensated random measure associated to a given marked point process on a general state space. We present basic well-posedness results in L 2 and in L 1 . We show that in the setting of point processes it is possible to solve the equation recursively, by replacing the BSDE by an ordinary differential equation in between jumps. Finally we address applications to optimal control of marked point processes, where the solution of a suitable BSDE allows to identify the value function and the optimal control. The talk is based on joint works with Marco Fuhrman and Jean Jacod. 

Maintenance activities help prevent costly power generator breakdowns but because generators under maintenance are typically unavailable, the impact of maintenance schedules is significant and their cost must be accounted for when planning maintenance. In this paper we address the generator maintenance scheduling problem in hydropower systems. While this problem has been widely studied, specific operating conditions of hydroelectric systems have received less attention. We present a mixed-integer linear programming model that considers the time windows of the maintenance activities, as well as the nonlinearities and disjunctions of the hydroelectric production functions. Because the resulting model is hard to solve, we also propose an extended formulation, a set reduction approach that uses logical conditions for excluding unnecessary set elements from the model, and valid inequalities. Computational experiments using a variety of instances adapted from a real hydropower system in Canada support the conclusion that the extended formulation with set reduction achieves the best results in terms of computational time and optimality gap. This is joint work with Jesus Rodriguez, Pascal Cote and Guy Desaulniers.

Outer Space is an important object in Geometric Group Theory and can be described from two viewpoints: as a space of marked graphs and a space of actions on trees. The latter viewpoint can be used to prove that Outer Space is contractible; and this fact together with some arguments using the first viewpoint enables us to say something about the Outer Automorphism group of a free group - I will sketch both these proofs.

We discuss the connections and differences between the ZFC set theory and univalent foundations and answer the above question in the negative.
 

The birth of fixed-wing, powered flight in the first decade of the twentieth century brought with it significant potential for pilots to return to Earth by unintended, often fatal, means. I will discuss the nature of the contemporary mathematical and engineering debates associated with these facets of flight, and the practical steps taken to facilitate safer aircraft and more robust operating procedures.

I will introduce a cohomology theory which combines topological and algebraic concepts. Interpretations of certain cohomology groups will be given. We also generalise the construction of the second Stiefel-Whitney class of a line bundle. As I will explain in my talk, the refined Stiefel-Whitney class of the canonical bundle on certain moduli stacks provides an obstruction for the construction of cohomological Hall algebras.

The edge isoperimetric problem for a graph G is to find, for each n, the minimum number of edges leaving any set of n vertices.  Exact solutions are known only in very special cases, for example when G is the usual cubic lattice on Z^d, with edges between pairs of vertices at l_1 distance 1.  The most attractive open problem was to answer this question for the "strong lattice" on Z^d, with edges between pairs of vertices at l_infty distance 1.  Whilst studying this question we in fact solved the edge isoperimetric problem asymptotically for every Cayley graph on Z^d.  I'll talk about how to go from the specification of a lattice to a corresponding near-optimal shape, for both this and the related vertex isoperimetric problem, and sketch the key ideas of the proof. Joint work with Joshua Erde.

The design of shapes that are in some sense optimal is a task faced by engineers in a wide range of disciplines. In shape optimisation one aims to improve a given initial shape by iteratively deforming it - if the shape is represented by a mesh, then this means that the mesh has to deformed. This is a delicate problem as overlapping or highly stretched meshes lead to poor accuracy of numerical methods.

In the presented work we consider a novel mesh deformation method motivated by the Riemannian mapping theorem and based on conformal mappings.

Pseudo-reductive groups are smooth connected linear algebraic groups over a field k whose k-defined unipotent radical is trivial. If k is perfect then all pseudo-reductive groups are reductive, but if k is imperfect (hence of characteristic p) then one gets a strictly larger collection of groups. They come up in a number of natural situations, not least when one wishes to say something about the simple representations of all smooth connected linear algebraic groups. Recent work by Conrad-Gabber-Prasad has made it possible to reduce the classification of the simple representations of pseudo-reductive groups to the split reductive case. I’ll explain how. This is joint work with Mike Bate.

Time parallelisation techniques provide an additional direction for the parallelisation of the solution of time-dependent PDEs or of systems of ODEs. In particular, the Parareal algorithm has imposed itself as the canonical choice to achieve parallelisation in time, also because of its simplicity and flexibility. The algorithm works by splitting the time domain in chunks, and iteratively alternating a prediction step (parallel), in which a "fine" solver is employed to achieve a high-accuracy solution within each chunk, to a correction step (serial) where a "coarse" solver is used to quickly propagate the update between the chunks. However, the stability of the method has proven to be highly sensitive to the choice of fine and coarse solver, even more so when applied to chaotic systems or advection-dominated problems.

In this presentation, an alternative formulation of Parareal is discussed. This aims to conduct the update by estimating directly the sensitivity of the solution of the integration with respect to the initial conditions, thus eliminating altogether the necessity of choosing the most apt coarse solver, and potentially boosting its convergence properties.

 

This will be a quick introduction to tropical algebra and the main results from the paper https://arxiv.org/pdf/1604.00113.pdf

Many species of insects adhere to vertical and inverted surfaces using footpads that secrete thin films of a mediating fluid. The fluid bridges the gap between the foot and the target surface. The precise role of this liquid is still subject to debate, but it is thought that the contribution of surface tension to the adhesive force may be significant. It is also known that the footpad is soft, suggesting that capillary forces might deform its surface. Inspired by these physical ingredients, we study a model problem in which a thin, deformable membrane under tension is adhered to a flat, rigid surface by a liquid droplet. We find that there can be multiple possible equilibrium states, with the number depending on the applied tension and aspect ratio of the system. The presence of elastic deformation significantly enhances the adhesion force compared to a rigid footpad. A mathematical model shows that the equilibria of the system can be controlled via two key parameters depending on the imposed separation of the foot and target surface, and the tension applied to the membrane. We confirm this finding experimentally and show that the system may transition rapidly between two states as the two parameters are varied. This suggests that different strategies may be used to adhere strongly and then detach quickly.

Temporal networks are increasingly being used to model the interactions of complex systems. 
Most studies require the temporal aggregation of edges (or events) into discrete time steps to perform analysis.
In this article we describe a static, behavioural representation of a temporal network, the temporal event graph (TEG).
The TEG describes the temporal network in terms of both inter-event time and two-event temporal motifs.
By considering the distributions of these quantities in unison we provide a new method to characterise the behaviour of individuals and collectives in temporal networks as well as providing a natural decomposition of the network.
We illustrate the utility of the TEG by providing examples on both synthetic and real temporal networks.

We consider two-dimensional chiral, first-order conformal field theories governing maps from the Riemann sphere to the projective light cone inside Minkowski space -- the natural setting for describing conformal field theories in two fewer dimensions. These theories have a SL(2) algebra of local bosonic constraints which can be supplemented by additional fermionic constraints depending on the matter content of the theory. By computing the BRST charge associated with gauge fixing these constraints, we find anomalies which vanish for specific target space dimensions. These critical dimensions coincide precisely with those for which (biadjoint) cubic scalar theory, gauge theory and gravity are classically conformally invariant. Furthermore, the BRST cohomology of each theory contains vertex operators for the full conformal multiplets of single field insertions in each of these space-time CFTs. We give a prescription for the computation of three-point functions, and compare our formalism with the scattering equations approach to on-shell amplitudes.