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.