Past

We present an algebraic formulation for the flow of a differential equation driven by a path in a Lie group. The formulation is motivated by formal differential equations considered by Chen.

Lagrangian Floer cohomology can be enriched by using local coefficients to record some homotopy data about the boundaries of the holomorphic disks counted by the theory. In this talk I will explain how one can do this under the monotonicity assumption and when the Lagrangians are equipped with local systems of rank higher than one. The presence of holomorphic discs of Maslov index 2 poses a potential obstruction to such an extension. However, for an appropriate choice of local systems the obstruction might vanish and, if not,
one can always restrict to some natural unobstructed subcomplexes. I will showcase these constructions with some explicit calculations for the Chiang Lagrangian in CP^3 showing that it cannot be disjoined from RP^3 by a Hamiltonian isotopy, answering a question of Evans-Lekili. Time permitting, I will also discuss some work-in-progress on the topology of monotone Lagrangians in CP^3, part of which follows from more general joint work with Jack Smith on the topology of monotone Lagrangians of maximal Maslov number in
projective spaces.

 I will summarise old and recent developments on the classification and solution of Rational Conformal Field Theories in 2 dimensions using the method of Modular Differential Equations. Novel and exotic theories are found with small numbers of characters and simple fusion rules, one of these being the Baby Monster CFT. Correlation functions for many of these theories can be computed using crossing-symmetric differential equations.

Details to follow

A Surstseyan eruption is a particular kind of volcanic eruption which involves the bulk interaction of water and hot magma. Surtsey Island was born during such an eruption process in the 1940s. I will talk about mathematical modelling of the flashing of water to steam inside a hot erupted lava ball called a Surtseyan bomb. The overall motivation is to understand what determines whether such a bomb will fragment or just quietly fizzle out...
Partial differential equations model transient changes in temperature and pressure in Surtseyan ejecta. We have used a highly simplified approach to the temperature behaviour, to separate temperature from pressure. The resulting pressure diffusion equation was solved numerically and asymptotically to derive a single parametric condition for rupture of ejecta. We found that provided the permeability of the magma ball is relatively large, steam escapes rapidly enough to relieve the high pressure developed at the flashing front, so that rupture does not occur. This rupture criterion is consistent with existing field estimates of the permeability of intact Surtseyan bombs, fizzlers that have survived.
I describe an improvement of this model that allows for the fact that pressure and temperature are in fact coupled, and that the process is not adiabatic. A more systematic reduction of the resulting coupled nonlinear partial differential equations that arise from mass, momentum and energy conservation is described. We adapt an energy equation presented in G.K. Batchelor's book {\em An Introduction to Fluid Dynamics} that allows for pressure-work. This is work in progress.  Work done with Emma Greenbank, Ian Schipper and Andrew Fowler 

The local volatility model is a celebrated model widely used for pricing and hedging financial derivatives. While the model’s main appeal is its capability of reproducing any given surface of observed option prices—it provides a perfect fit—the essential component of the model is a latent function which can only be unambiguously determined in the limit of infinite data. To (re)construct this function, numerous calibration methods have been suggested involving steps of interpolation and extrapolation, most often of parametric form and with point-estimates as result. We seek to look at the calibration problem in a probabilistic framework with a nonparametric approach based on Gaussian process priors. This immediately gives a way of encoding prior believes about the local volatility function, and a hypothesis model which is highly flexible whilst being prone to overfitting. Besides providing a method for calibrating a (range of) point-estimate, we seek to draw posterior inference on the distribution over local volatility to better understand the uncertainty attached with the calibration. Further, we seek to understand dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this gives us means of inferring predictive distributions not only locally, but also for entire surfaces forward in time.

This talk focuses on algebraic and combinatorial-topological problems motivated by neuroscience. Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus,where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets?

This talk describes how to use tools from combinatorics and commutative algebra to uncover a variety of signatures of convex and non-convex codes.

This talk is based on joint works with Aaron Chen and Florian Frick, and with Carina Curto, Elizabeth Gross, Jack Jeffries, Katie Morrison, Mohamed Omar, Zvi Rosen, and Nora Youngs.

Archimedes, who famously jumped out of his bath shouting "Eureka", also invented $\pi$. 

Euler invented $e$ and had fun with his formula $e^{2\pi i} = 1$

The world is full of important numbers waiting to be invented. Why not have a go ?

Michael Atiyah is one of the world's foremost mathematicians and a pivotal figure in twentieth and twenty-first century mathematics. His lecture will be followed by an interview with Sir John Ball, Sedleian Professor of Natural Philosophy here in Oxford where Michael will talk about his lecture, his work and his life as a mathematician.

Please email @email to register.

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

It is an old problem in number theory to count number fields of a fixed degree and having a fixed Galois group for its Galois closure, ordered by their absolute discriminant, say. In this talk, I shall discuss some background of this problem, and then report a recent work with Stanley Xiao. In our paper, we considered quartic $D_4$-fields whose ring of integers has a certain nice algebraic property, and we counted such fields by their conductor.

The Grojnowski-Nakajima theorem states that the direct sum of the homologies of the Hilbert schemes on n points on an algebraic surface is an irreducible highest weight representation of an infinite-dimensional Heisenberg superalgebra. We present an idea to rederive the Grojnowski-Nakajima theorem using Halpern-Leistner's categorical Kirwan surjectivity theorem and Joyce's theorem that the homology of a moduli space of sheaves is a vertex algebra. We compute the homology of the moduli stack of perfect complexes of coherent sheaves on a smooth quasi-projective variety X, identify it as a (modified) lattice vertex algebra on the Lawson homology of X, and explain its relevance to the aforementioned problem.

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.

How can we adapt the Topological Data Analysis (TDA) pipeline to use several filter functions at the same time? Two orthogonal approaches can be considered: (1) running the standard 1-parameter pipeline and doing statistics on the resulting barcodes; (2) running a multi-parameter version of the pipeline, still to be defined. In this talk I will present two recent contributions, one for each approach. The first contribution considers intrinsic compact metric spaces and shows that the so-called Persistent Homology Transform (PHT) is injective over a dense subset of those. When specialized to metric graphs, our analysis yields a stronger result, namely that the PHT is injective over a subset of full measurem which allows for sufficient statistics. The second contribution investigates the bi-parameter version of the TDA pipeline and shows a decomposition result "à la Crawley-Boevey" for a subcategory of the 2-parameter persistence modules called "exact modules". This result has an impact on the study of interlevel-sets persistence and on that of sheaves of vector spaces on the real line. 

This is joint work with Elchanan Solomon on the one hand, with Jérémy Cochoy on the other hand.

When dealing with geometric structures one natural question that arise is "when does a subset inherit the geometry of the ambient space"? In the case of hyperbolic space, the concept of quasi-convexity provides answer to this question. However, for a general metric space, being quasi-convex is not a quasi-isometric invariant. This motivates the notion of Morse subsets. In this talk we will motivate the definition and introduce some examples. Then we will introduce the class of hierarchically hyperbolic groups (HHG), and furnish a complete characterization of Morse subgroups of HHG. If time allows, we will discuss the relationship between Morse subgroups and hyperbolically-embedded subgroups. This is a joint work with Hung C. Tran and Jacob Russell.

Exponents of Diophantine approximation are defined to study specific sets of real numbers for which Dirichlet's pigeonhole principle can be improved. Khintchine stated a transference principle between the two exponents in the cases  of simultaneous approximation and approximation by linear forms. This shows that exponents of Diophantine approximation are related, and these relations can be studied via so called spectra. In this talk, we provide an optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation for both simultaneous approximation and approximation by linear forms. This is joint work with Nikolay Moshchevitin.

Abstract:
Highlights:

•We increasingly live in a digital world and commercial companies are not the only beneficiaries. The public sector can also use data to tackle pressing issues.
•Machine learning is starting to make an impact on the tools regulators use, for spotting the bad guys, for estimating demand, and for tackling many other problems.
•The speech uses an array of examples to argue that much regulation is ultimately about recognising patterns in data. Machine learning helps us find those patterns.
 
Just as moving from paper maps to smartphone apps can make us better navigators, Stefan’s speech explains how the move from using traditional analysis to using machine learning can make us better regulators.
 
Mini Biography:
 
Stefan Hunt is the founder and Head of the Behavioural Economics and Data Science Unit. He has led the FCA’s use of these two fields and designed several pioneering economic analyses. He is an Honorary Professor at the University of Nottingham and has a PhD in economics from Harvard University.
 

Superhydrophobic surfaces, formed by air entrapment within the cavities of a hydrophobic solid substrate, offer a promising potential for drag reduction in small-scale flows. It turns out that low-drag configurations are associated with singular limits, which to date have typically been addressed using numerical schemes. I will discuss the application of singular perturbations to several of the canonical problems in the field. 

The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed.

However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large-scale applications, in particular for nonlinear and/or time-dependent problems.

After giving a brief general introduction to reduced order models, we discuss developments in two different directions. In the first part, we discuss recent developments of reduced methods that conserve chosen invariants for nonlinear time-dependent problems. We pay particular attention to the development of reduced models for Hamiltonian problems and propose a greedy approach to build the basis. As we shall demonstrate, attention to the construction of the basis must be paid not only to ensure accuracy but also to ensure stability of the reduced model. Time permitting, we shall also briefly discuss how to extend the approach to include more general dissipative problems through the notion of port-Hamiltonians, resulting in reduced models that remain stable even in the limit of vanishing viscosity and also touch on extensions to Euler and Navier-Stokes equations.

The second part of the talk discusses the combination of reduced order modeling for nonlinear problems with the use of neural networks to overcome known problems of on-line efficiency for general nonlinear problems. We discuss the general idea in which training of the neural network becomes part of the offline part and demonstrate its potential through a number of examples, including for the incompressible Navier-Stokes equations with geometric variations.

This work has been done with in collaboration with B.F. Afkram (EPFL, CH), N. Ripamonti EPFL, CH) and S. Ubbiali (USI, CH).

Given $d \ge 1$, $T>0$ and a vector field $\mathbf b \colon [0,T] \times \mathbb R^d \to \mathbb R^d$, we study the problem of uniqueness of weak solutions to the associated transport equation $\partial_t u + \mathbf b \cdot \nabla u=0$ where $u \colon [0,T] \times \mathbb R^d \to \mathbb R$ is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field $\mathbf b$. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost.
In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. We will finally show that if $\mathbf b$ is locally of class $BV$ in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible $BV$ vector fields and thus gives a positive answer to the (weak) Bressan's Compactness Conjecture. This is a joint work with S. Bianchini.
 

Tight security is increasingly gaining importance in real-world
cryptography, as it allows to choose cryptographic parameters in a way
that is supported by a security proof, without the need to sacrifice
efficiency by compensating the security loss of a reduction with larger
parameters. However, for many important cryptographic primitives,
including digital signatures and authenticated key exchange (AKE), we
are still lacking constructions that are suitable for real-world deployment.

This talk will present the first first practical AKE protocol with tight
security. It allows the establishment of a key within 1 RTT in a
practical client-server setting, provides forward security, is simple
and easy to implement, and thus very suitable for practical deployment.
It is essentially the "signed Diffie-Hellman" protocol, but with an
additional message, which is crucial to achieve tight security. This
message is used to overcome a technical difficulty in constructing
tightly-secure AKE protocols.

The second important building block is a practical signature scheme with
tight security in a real-world multi-user setting with adaptive
corruptions. The scheme is based on a new way of applying the
Fiat-Shamir approach to construct tightly-secure signatures from certain
identification schemes.

For a theoretically-sound choice of parameters and a moderate number of
users and sessions, our protocol has comparable computational efficiency
to the simple signed Diffie-Hellman protocol with EC-DSA, while for
large-scale settings our protocol has even better computational per-
formance, at moderately increased communication complexity.

Given an endomorphism f:X --> X of a 'dualisable' object in a symmetric monoidal category, one can define its trace Tr(f). It turns out that the trace is 'universal' among the scalars we can produce from f. To prove this we will think of the 1d framed bordism category as the 'walking dualisable object' (using the cobordism hypothesis) and then apply the Yoneda lemma.
Employing similar techniques we can define 'hermitian' objects (generalising hermitian vector spaces) and prove that there is a 1-1 correspondence between Hermitian structures on a fixed object X and self-adjoint automorphisms of X. If time permits I will sketch how this relates to hermitian K-theory.

While all results of the talk hold for infinity-categories, they work equally well for ordinary categories. Therefore no knowledge of higher category theory is needed to follow the talk.

I will report on two recent papers with D. Ghioca and U. Zannier (joined by P. Corvaja and F. Hu, respectively) in which we consider variants of the Mordell-Lang conjecture.  In the first of these, we study the dynamical Mordell-Lang conjecture in positive characteristic, proving some instances, but also showing that in general the problem is at least as hard as a difficult diophantine problem over the integers.  In the second paper, we study the Mordell-Lang problem for extensions of abelian varieties by the additive group.  Here we have positive results in the function field case obtained by using the socle theorem in the form offered as an aside in Hrushovski's 1996 paper and in the number field case we relate this problem to the Bombieri-Lang conjecture.

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

In 1873 the Personal Recollections from Early Life to Old Age of Mary Somerville were published, containing detailed descriptions of her life as a 19th century philosopher, mathematician and advocate of women's rights. In an early draft of this work, Somerville reiterated the widely held view that a fundamental difference between men and women was the latter's lack of originality, or 'genius'.

In my talk I will examine how Somerville's view was influenced by the historic treatment of women, both within scientific research, scientific institutions and wider society. By building on my doctoral research I will also suggest an alternative viewpoint in which her work in the differential calculus can be seen as original, with a focus on her 1834 treatise On the Theory of Differences.