Past

High-frequency realized variance approaches offer great promise for 
estimating asset prices’ covariation, but encounter difficulties 
connected to the Epps effect. This paper models the Epps effect in a 
stochastic volatility setting. It adds dependent noise to a factor 
representation of prices. The noise both offsets covariation and 
describes plausible lags in information transmission. Non-synchronous 
trading, another recognized source of the effect, is not required. A 
resulting estimator of correlations and betas performs well on LSE 
mid-quote data, lending empirical credence to the approach.

The peeling of an elastic sheet away from thin layer of viscous fluid is a simply-stated and generic problem, that involves complex interactions between flow and elastic deformation on a range of length scales. 

I will illustrate the possibilities by considering theoretically and experimentally the injection and spread of viscous fluid beneath a flexible elastic lid; the injected fluid forms a blister, which spreads by peeling the lid away at the  perimeter of the blister. Among the many questions to be considered are the mechanisms for relieving the elastic analogue of the contact-line problem, whether peeling is "by bending" or "by pulling", the stability of the peeling front, and the effects of a capillary meniscus when peeling is by air injection. The result is a plethora of dynamical regimes and asymptotic scaling laws.

Advances in manufacturing technologies, most prominently in additive manufacturing or 3d printing, are making it possible to fabricate highly optimised products with increasing geometric and hierarchical complexity. This talk will introduce our ongoing work on design optimisation that combines CAD-compatible geometry representations, multiresolution geometry processing techniques and immersed finite elements with classical shape and topology calculus. As example applications,the shape optimisation of mechanical structures and electromechanical components, and the topology optimisation of lattice-skin structures will be discussed.

Thompson's group F is a group of homeomorphisms of the unit interval which exhibits a strange mix of properties; on the one hand it has some self-similarity type properties one might expect of a really big group, but on the other hand it is finitely presented. I will give a proof of finite generation by expressing elements as pairs of binary trees.

The Fujita equation $u_{t}=\Delta u+u^{p}$, $p>1$, has been a canonical blow-up model for more than half a century. A great deal is known about the singularity formation under a variety of conditions. In particular we know that blow-up behaviour falls broadly into two categories, namely Type I and Type II. The former is generic and stable while the latter is rare and highly unstable. One of the central results in the field states that in the Sobolev subcritical regime, $1<p<\frac{n+2}{n-2}$, $n\geq 3$, only type I is possible whenever the domain is \emph{convex} in $\mathbb{R}^n$. Despite considerable effort the requirement of convexity has not been lifted and it is not clear whether this is an artefact of the methodology or whether the geometry of the domain may actually affect the blow-up type. In my talk I will discuss how the question of the blow-up type for non-convex domains is intimately related to the validity of some Li-Yau-Hamilton inequalities.

In 2016, Facebook added an optional end-to-end (E2E) encryption feature called Secret Conversations to Messenger. This was challenging to design, as many of Messenger's key properties and features don't fit the typical model of E2E apps. Additionally, Messenger is already one of the world's most popular messaging apps, supporting nearly a billion people across a variety of technical and cultural environments. Because of this, Messenger's deployment of E2E encryption provides attendees with a valuable case study on how to build usable, secure products. 

We will discuss the core properties of a typical E2E app, the core features of Messenger, the distance between the two, and the approach we took to close the gap. We'll examine how minimizing the distance shaped the current E2E experience within Messenger. Through discussion of the key decisions in this process, we'll address the implications for alternative designs with real world comparisons where they exist. 

Although Secret Conversations in Messenger use off-the-shelf Signal Protocol for message encryption, Facebook also wanted to ensure a safe communication channel for community members who may be victims of online abuse. To this end, we created a way for people to report secret conversations that violate our Community Standards, without breaking any E2E guarantees for other messages.

Developing a reporting protocol created an interesting challenge: the potential of fake reports with no intermediary to invalidate them. To pre-empt the possibility of Bob forging a report to incriminate Alice, we added a method that uses two HMACs - one added by the sender and one by Facebook - to “cryptographically frank” messages as we forward them from one party to the other (physical mail uses a similar franking). This technique ensures similar confidence that a report is genuine as we have for messages stored in plaintext on our servers. Additionally, the frank is only verifiable by Facebook after receiving a report from the recipient, thus preventing a third party from using it as evidence against the sender.

We hope that this talk will provide an insight into the intricacies of deploying security features at scale, and the additional considerations necessary when developing an existing product.

Several works (by Kontsevich, Soibelman, Berkovich, Nicaise, Boucksom, Jonsson...) have shown that the limit behavior of a one-parameter family $(X_t)$ of complex algebraic varieties can often be described using the associated Berkovich t-adic analytic space $X^b$. In a work in progress with E. Hrushovski and F. Loeser, we provide a new instance of this general phenomenon. Suppose we are given for every t an  $(n,n)$-form $ω_t$ on $X_t$ (for n= dim X). Then under some assumptions on the formula that describes $ω_t$, the family $(ω_t)$ has a "limit" ω, which is a real valued  (n,n)-form in the sense of Chambert-Loir and myself on the Berkovich space $X^b$, and the integral of $ω_t$ on $X_t$ tends to the integral of ω on $X^b$. 
In this talk I will first make some reminders about Berkovich spaces and (n,n)-forms in this setting, and then discuss the above result. 
In fact, as I will explain, it is more convenient to formulate it with  $(X_t)$ seen as a single algebraic variety over a non-standard model *C of C and (ω_t) as a (n,n) differential form on this variety. The field *C also carries a t-adic real valuation which makes it a model of ACVF (and enables to do Berkovich geometry on it), and our proof uses repeatedly RCF and ACVF theories. 
 

Part of the series 'What do historians of mathematics do?'

Both as a canonical mathematical text and as a representative of ancient thought, Euclid's Elements of Geometry has been a subject of study since its creation c. 300 BCE. It has been read as a practical and a theoretical text; it has been studied for its philosophical ramifications and for its perceived potential to inculcate logical thought. For the historian, it is where the history of mathematics meets the history of ideas; where the history of the book meets the history of practice. The study of the Elements enjoyed a particular resurgence during the Early Modern period, when around 200 editions of the text appeared between 1482 and 1700.  Depending on their theoretical and practical functions, they ranged between elaborate folios and pocket-size compendia, and were widely studied by scholars, natural philosophers, mathematical practitioners, and schoolchildren alike.

In this talk, I will present some of the preliminary results of the research we have been conducting for the AHRC-funded project based at the History Faculty 'Reading Euclid: Euclid's Elements of Geometry in Early Modern Britain', paying particular attention to how the books were printed, collected, and annotated. I will concentrate on our methodologies and introduce the database we have been building of all the early modern copies of the text in the British Isles, as well as the 'catalogue of book catalogues'.

The Schrödinger equation with a time-dependent potential occurs in a wide range of applications in theoretical chemistry, quantum physics and quantum computing. In this talk I will discuss a variety of Magnus expansion based schemes that have been found to be highly effective for numerically solving these equations since the pioneering work of Tal Ezer and Kosloff in the early 90s. Recent developments in the field focus on approximation of the exponential of the Magnus expansion via exponential splittings including some asymptotic splittings and commutator-free splittings that are designed specifically for this task.

I will also present a very recently developed methodology for the case of laser-matter interaction. This methodology allows us to extend any fourth-order scheme for Schrödinger equation with time-independent potential to a fourth-order method for Schrödinger equation with laser potential with little to no additional cost. These fourth-order methods improve upon many leading schemes of order six due to their low costs and small error constants.

Consider a family of k-element subsets of an n-element set, and assume that the family does not contain s pairwise disjoint sets. The well-known Erdos Matching Conjecture suggests the maximum size of such a family. Finding the maximum is trivial for n<(s+1)k and is relatively easy for n large in comparison to s,k. There was a splash of activity around the conjecture in the recent years, and, as far as the original question is concerned, the best result is due to Peter Frankl, who verified the conjecture for all n>2sk. In this work, we improve the bound of Frankl for any k and large enough s. We also discuss the connection of the problem to an old question on deviations of sums of random variables going back to the work of Hoeffding and Shrikhande.
 

Many problems that involve the propagation of time-harmonic waves are naturally posed in unbounded domains. For instance, a common problem in the are a of acoustic scattering is the determination of the sound field that is generated when an incoming time-harmonic wave (which is assumed to arrive ``from infinity'') impinges onto a solid body (the scatterer). The boundary
conditions to be applied on the surface of the scatterer (most often of Dirichlet, Neumann or Robin type) tend to be easy to enforce in most numerical solution schemes. Conversely, the imposition of a suitable decay condition (typically a variant of the Sommerfeld radiation condition), which is required to ensure the well-posedness of the solution, is considerably more involved. As a result, many numerical schemes generate spurious reflections from the outer boundary of the finite computational domain.

Perfectly matched layers (PMLs) are in this context a versatile alternative to the usage of classical approaches such as employing absorbing boundary conditions or Dirichlet-to-Neumann mappings, but unfortunately most PML formulations contain adjustable parameters which have to be optimised to give the best possible performance for a particular problem. In this talk I will present a parameter-free PML formulation for the case of the two-dimensional Helmholtz equation. The performance of the proposed method is demonstrated via extensive numerical experiments, involving domains with smooth and polygonal boundaries, different solution types (smooth and singular, planar and non-planar waves), and a wide range of wavenumbers (R. Cimpeanu, A. Martinsson and M.Heil, J. Comp. Phys., 296, 329-347 (2015)). Possible extensions and generalisations will also be touched upon.

It is known that in steady-state potential flows, the separation of a gravity-driven free-surface from a solid exhibits a number of peculiar characteristics. For example, it can be shown that the fluid must separate from the body so as to form one of three possible in-fluid angles: (i) 180°, (ii) 120°, or (iii) an angle such that the surface is locally perpendicular to the direction of gravity. These necessary separation conditions were notably remarked by Dagan & Tulin (1972) in the context of ship hydrodynamics [J. Fluid Mech., 51(3) pp. 520-543], but they are of crucial importance in many potential flow applications. It is not particularly well understood why there is such a drastic change in the local separation behaviours when the global flow is altered. The question that motivates this work is the following: outside a formal balance-of-terms arguments, why must (i) through (iii) occur and furthermore, what is the connections between them?

              In this work, we seek to explain the transitions between the three cases in terms of the singularity structure of the associated solutions once they are extended into the complex plane. A numerical scheme is presented for the analytic continuation of a vertical jet (or alternatively a rising bubble). It will be shown that the transition between the three cases can be predicted by observing the coalescence of singularities as the speed of the jet is modified. A scaling law is derived for the coalescence rate of singularities.

Several scholars of evolutionary biology have suggested that functional redundancy (also known as "biological degener-
acy") is important for robustness of biological networks. Structural redundancy indicates the existence of structurally
similar subsystems that can perform the same function. Functional redundancy indicates the existence of structurally
di erent subsystems that can perform the same function. For networks with Ornstein--Uhlenbeck dynamics, Tononi et al.
[Proc. Natl. Acad. Sci. U.S.A. 96, 3257{3262 (1999)] proposed measures of structural and functional redundancy that are
based on mutual information between subnetworks. For a network of n vertices, an exact computation of these quantities
requires O(n!) time. We derive expansions for these measures that one can compute in O(n3) time. We use the expan-
sions to compare the contributions of di erent types of motifs to a network's functional redundancy.

In this talk, we will discuss some recent progress on the study of six-dimensional S-matrices as well as their applications. Six-dimensional theories we are interested include the world-volume theories of single probe M5-brane and D5-brane, as well as 6D super Yang-Mills and supergravity. We will present twistor-string-like formulas for all these theories, analogue to that of Witten’s twistor string formulation for 4D N=4 SYM. 
As the applications, from the 6D results we also deduce new formulas for scattering amplitudes of theories in lower dimensions, such as SYM and supergravity in five dimensions, and 4D N=4 SYM on the Columbo branch. 
 

 I will discuss the quantum-field-theory origins of a classic result of Goresky-Kottwitz-MacPherson concerning the Koszul duality between the homology of G and the G-equivariant cohomology of a point. The physical narrative starts from an analysis of supersymmetric quantum mechanics with G symmetry, and leads naturally to a definition of the category of boundary conditions in two-dimensional topological gauge theory, which might be called the "G-equivariant Fukaya category of a point." This simple example illustrates a more general phenomenon (also appearing in C. Teleman's work in recent years) that pure gauge theory in d dimensions seems to control the structure of G-actions in (d-1)-dimensional QFT. This is part of joint work with C. Beem, D. Ben Zvi, M. Bullimore, and A. Neitzke.

In this talk I will discuss some recent constructions of blow-up solutions for a Fujita type problem for power related to the critical Sobolev exponent. Both finite type blow-up (of type II) and infinite time blow-up are considered. This research program is in collaboration with C. Cortazar, M. del Pino and J. Wei.

Recently, Deya, Gubinelli, Hofmanova and Tindel ('16) (also Bailleul-Gubinelli '15) have provided a general approach in order to obtain a priori estimates for rough partial differential equations of the form
(*)    du = Au dt + Bu dX
where X is a two-step rough path, A is a second order operator (elliptic), while B is first order. We will pursue the line of this work by presenting an L^p theory "à la Krylov" for generalized versions of (*). We will give an application of this theory by proving boundedness of solutions for a certain class

My talk is based on joint work with Claire Debord (Univ. Auvergne).
We will explain why Lie groupoids are very naturally linked to Atiyah-Singer index theory.
In our approach -originating from ideas of Connes, various examples of Lie groupoids
- allow to generalize index problems,
- can be used to construct the index of pseudodifferential operators without using the pseudodifferential calculus,
- give rise to proofs of index theorems, 
- can be used to construct the pseudodifferential calculus.

The celebrated Black-Scholes theory shows that one can get a risk-neutral option price through hedging. The Cameron-Martin-Girsanov theorem for diffusion processes plays a key role in this theory. We show that one can get some statistical arbitrage from a sequence of well-designed repeated trading at their prices according to the ergodic theorem for stationary process. Our result is based on both theoretical model and the real market data. 

Stability conditions on derived categories of algebraic varieties and their wall-crossings have given algebraic geometers a number of new tools to study the geometry of moduli spaces of stable sheaves. In work in progress with Macri, Lahoz, Nuer, Perry and Stellari, we are extending this toolkit to a the "relative" setting, i.e. for a family of varieties. Our construction comes with relative moduli spaces of stable objects; this gives additional ways of constructing new families of varieties from a given family, thereby potentially relating different moduli spaces of varieties.

The central charges “c” and “a” in two and four dimensional conformal field theories (CFTs) have a central organizing role in our understanding of quantum field theory (QFT) more generally.  Appearing as coefficients of curvature invariants in the anomalous trace of the stress tensor, they constrain the possible relationships between QFTs under renormalization group flow.  They provide important checks for dualities between different CFTs.  They even have an important connection to a measure of quantum entanglement, the entanglement entropy.  Less well known is that additional central charges appear when there is a boundary, four new coefficients in total in three and four dimensional boundary CFTs.   While largely unstudied, these boundary charges hold out the tantalizing possibility of being as important in the classification of quantum field theory as the bulk central charges “a” and “c”.   I will show how these charges can be computed from displacement operator correlation functions.  I will also demonstrate a boundary conformal field theory in four dimensions with an exactly marginal coupling where these boundary charges depend on the marginal coupling.  The talk is based on arXiv:1707.06224, arXiv:1709.07431, as well as work to appear shortly.  

Research suggests that students with a 'growth mindset' may do better than those with a 'fixed mindset'.

• What does that mean for our teaching?
• How can we support students to develop a growth mindset?
• What sorts of mindsets do we ourselves have?
• And how does that affect our teaching and indeed the rest of our work?

Oscar García-Prada - The Mathematics of Kolam

In Tamil Nadu, a state in southern India, it is an old tradition to decorate the entrance to the home with a geometric figure called ``Kolam''. A kolam is a geometrical line drawing composed of curved loops, drawn around a grid pattern of dots. This is typically done by women using white rice flour. Kolams have connections to discrete mathematics, number theory, abstract algebra, sequences, fractals and computer science. After reviewing a bit of its history, Oscar will explore some of these connections. 

Claudia Silva - Kolam: An Ephemeral Women´s art of South India

Kolam is a street drawing, performed by women in south India. This daily ritual of "putting" the kolam on the ground represents a time of intimacy, concentration and creativity. Through some videos, Claudia will explain some basic features of kolam, focusing on anthropological, religious, educational and artistic aspects of this beautiful female art expression.

The lectures are accompanied by a photography exhibition at Wolfson College.

Since the invention of the microscope, scientists have known that pond-dwelling algae can actually swim – powering their way through the fluid using tiny limbs called cilia and flagella. Only recently has it become clear that the very same structure drives important physiological and developmental processes within the human body. Motivated by this connection, we explore flagella-mediated swimming gaits and stereotyped behaviours in diverse species of algae, revealing the extent to which control of motility is driven intracellularly. These insights suggest that the capacity for fast transduction of signal to peripheral appendages may have evolved far earlier than previously thought.