Past

In recognition of a lifetime's contribution across the mathematical sciences, we are initiating a series of annual Public Lectures in honour of Roger Penrose. The first lecture will be given by his long-time collaborator and friend Stephen Hawking.

Registration will open at 10am on 4 January 2017. Please email:

@email from that time only.

When registering please give your name and affiliation - your position, department & organisation/institution as appropriate. Or if you are a member of the General Public, please say so. Places will be allocated on a first come, first served basis with only one place per person. We will only be able to respond if you have a place or are on the waiting list.

We will be podcasting the lecture live. More details to follow.

I'll show how a simple universal property attaches a category of derived manifolds to any category with finite products and some suitable notion of "topology". Starting with the category of real Euclidean spaces and infinitely differentiable maps yields the category of derived smooth manifolds studied by D. Spivak and others, while starting with affine spaces over some ring and polynomial maps produces a flavour of the derived algebraic geometry of Lurie and Toen-Vezzosi.

I'll motivate this from the differentiable setting by showing that the universal property easily implies all of D. Spivak's axioms for being "good for intersection theory on manifolds".

Consider the following particle system. We are given a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. Each node of the tree has space for a single particle (we think of them as cars). A number $m \le n$ of cars arrive one by one, and car $i$ wishes to park at node $S_i$, $1 \le i \le m$, where $S_1, S_2, \ldots, S_m$ are i.i.d. uniform random variables on $[n]$. If a car wishes to park at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Let $A_{n,m}$ denote the event that all $m$ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Set $m = \lfloor \alpha n \rfloor$. Then if $\alpha \le 1/2$, $\mathbb{P}(A_{n,\lfloor \alpha n \rfloor}) \to \frac{\sqrt{1-2\alpha}}{1-\alpha}$, whereas if $\alpha > 1/2$ we have $\mathbb{P}(A_{n,\lfloor \alpha n \rfloor}) \to 0$. In this talk, we will give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method.

Joint work with Christina Goldschmidt.

The local Langlands correspondence for classical groups gives a natural finite-to-one map between certain representations of p-adic classical groups and certain self-dual representations of the absolute Weil group of a p-adic field (and more). On both sides of the correspondence, the description of the representations involves a ``wild part'' of more arithmetic nature and a ``tame part'' of more geometric nature, and the notion of endo-parameter (due to Bushnell--Henniart for general linear groups) is designed to describe the ``wild part'' of the Langlands correspondence. I will explain what this means and the connection with representations of affine Hecke algebras. This is joint work with Blondel--Henniart, with Lust, and with Kurinczuk--Skodlerack.

Joint work with Jen Pestana.

Amplitudes in planar N=4 SYM are dual to light-like polygonal Wilson-loop expectation values. In many cases their perturbative expansion can be expressed in terms of multiple polylogarithms which also obey certain single-valuedness conditions or branch cut restrictions. The rigidity of this function space, together with a few other conditions, allows one to construct the six-point amplitude -- or hexagonal Wilson loop -- through at least five loops, and the seven-point amplitude through 3.5 loops. Then one can "fold" the polygonal Wilson loops into multiple degenerate configurations, expressing the limiting behavior in terms of simpler function spaces, and learning in the process about how amplitudes factorize.
 

In this talk we will motivate and discuss several problems and results in harmonic analysis that involve some arithmetic or discrete structure. We will focus on pioneering work of Bourgain on discrete restriction theorems and pointwise ergodic theorems for arithmetic sets, their modern developments and future directions for the field.

Coarse embeddings occur completely naturally in geometric group theory: every finitely generated subgroup of a finitely generated group is coarsely embedded. Since even very nice classes of groups - hyperbolic groups or right-angled Artin groups for example - are known to have 'wild' collections of subgroups, there are precious few invariants that one may use to prove a statement of the form '$H$ does not coarsely embed into $G$' for two finitely generated groups $G,H$.
The growth function and the asymptotic dimension are two coarse invariants which which have been extensively studied, and a more recent invariant is the separation profile of Benjamini-Schramm-Timar.

In this talk I will describe a new spectrum of coarse invariants, which include both the separation profile and the growth function, and can be used to tackle many interesting problems, for instance: Does there exist a coarse embedding of the Baumslag-Solitar group $BS(1,2)$ or the lamplighter group $\mathbb{Z}_2\wr\mathbb{Z}$ into a hyperbolic group?

This is part of an ongoing collaboration with John Mackay and Romain Tessera.
 

What is the correct combinatorial object to encode a linear representation?  Many shadows of this problem have been studied:moment polytopes, Duistermaat-Heckman measures, Okounkov bodies.  We suggest that already in very simple cases these miss a crucial feature.  The ring theory, as opposed to just the linear algebra, of the group action on the coordinate ring, depends on some non-trivial lattice geometry and an associated filtration.  Some striking similarities to, and key differences from, the theory of toric varieties ensue.  Finite and non-finite generation phenomena emerge naturally.  We discuss motivations from, and applications to, questions in the effective geometry of moduli of curves.

The ambitwistor string of Mason and Skinner has been very successful in describing field theory amplitudes, at both loop and tree-level for a variety of theories. But the original action given by Mason and Skinner is already partially gauge-fixed, which obscures some issues related to modular invariance and the connection to conventional string theories. In this talk I will argue that the Null string is the ungauge-fixed version of the Ambitwistor string. This clarifies the geometry of the original Ambitwistor string and gives a road map to understanding modular invariance, and gives new formulas for loop amplitudes in which we expect that UV divergences will be easier to analyse.

 Functions defined by evaluation programs involving smooth  elementals and absolute values as well as max and min are piecewise smooth. For this class we present first and second order, necessary and sufficient conditions for the functions to be locally optimal, or convex, or at least possess a supporting hyperplane. The conditions generalize the classical KKT and SSC theory and are constructive; though in the case of convexity they may be combinatorial to verify. As a side product we find that, under the Mangasarin-Fromowitz-Kink-Qualification, the well established nonsmooth concept of subdifferential regularity is equivalent to first order convexity. All results are based on piecewise linearization and suggest corresponding optimization algorithms.

Puzzling things happen in human perception when ambiguous or incomplete information is presented to the eyes. Rivalry occurs when two different images, presented one to each eye, lead to alternating percepts, possibly of neither image separately. Illusions, or multistable figures, occur when a single image can be perceived in several ways. The Necker cube is the most famous example. Impossible objects arise when a single image has locally consistent but globally inconsistent geometry. Famous examples are the Penrose triangle and etchings by Maurits Escher.

In this lecture Ian Stewart will demonstrate how these phenomena provide clues about the workings of the visual system, with reference to recent research in the field which has modelled simplified, systematic methods by which the brain can make decisions. In these models a neural network is designed to interpret incoming sensory data in terms of previously learned patterns. Rivalry occurs when different interpretations are confused, and illusions arise when the same data have several interpretations.

The lecture will be non-technical and highly illustrated, with plenty of examples.

Please email @email to register

Tbd

As part of a suite of products that provide a drive thorough vehicle tyre inspection system the assessment of wheel alignment would be useful to drivers in maintaining their vehicles and reducing tyre wear.  The current method of assessing wheel alignment involves fitting equipment to the tyre and assessment within a garage environment. 

The challenge is to develop a technique that can be used in the roadway with no equipment fitted to the vehicle.  The WheelRight equipment is already capturing images of tyres from both  front and side views.  Pressure sensors in the roadway also allow a tyre pressure footprint to be created.  Using the existing data to interpret the alignment of the wheels on each axle is a preferred way forward.

The stability of structures continues to be scientifically fascinating and technically important.  Shell buckling emerged as one of the most challenging nonlinear problems in mechanics more than fifty years ago when it was intensively studied.  It has returned to life with new challenges motivated not only by structural applications but also by developments in the life sciences and in soft materials.  It is not at all uncommon for slightly imperfect thin cylindrical shells under axial compression or spherical shells under external pressure to buckle at 20% of the buckling load of the perfect shell.  A historical overview of shell buckling will be presented followed by a discussion of recent work by the speaker and his collaborators on the buckling of spherical shells.  Experimental and theoretical work will be described with a focus on imperfection-sensitivity and on viewing the phenomena within the larger context of nonlinear stability. 

We prove higher differentiability of bounded local minimizers to some degenerate functionals satisfying anisotropic growth conditions. In the two-dimensional case we also study the Lipschitz regularity of such minimizers without any limitation on the exponents of anisotropy.

I will discuss on the existence and regularity results for the heat flow of the so called H-systems and for more general parabolic p-laplacian problems with critical growth.

In two dimensional topological phases of matter, processes depend on gross topology rather than detailed geometry. Thinking in 2+1 dimensions, the space-time histories of particles can be interpreted as knots or links, and the amplitude for certain processes becomes a topological invariant of that link. While sounding rather exotic, we believe that such phases of matter not only exist, but have actually been observed (or could be soon observed) in experiments. These phases of matter could provide a uniquely practical route to building a quantum computer. Experimental systems of relevance include Fractional Quantum Hall Effects, Exotic superconductors such as Strontium Ruthenate, Superfluid Helium, Semiconductor-Superconductor-Spin-Orbit systems including Quantum Wires. The physics of these systems, and how they might be used for quantum computation will be discussed.

The mathematical design of the table flood demonstrator Wetropolis will be presented. Wetropolis illustrates the concepts of extreme rainfall and flooding.

It shows how extreme rainfall events  can cause flooding of a city due to groundwater and river flood peaks. Rainfall is supplied randomly in space using four outcomes (in a reservoir, on a moor, at both places or nowhere) and randomly in time using four rainfall intensities (1s, 2s, 4s, or 9s during a 10s Wetropolis day), including one extreme event, via two skew-symmetric discrete probability distributions visualised by two Galton boards. Wetropolis can be used for both public outreach and as scientific testing environment for flood mitigation and data assimilation.

More information: https://www.facebook.com/resurging.flows

In this talk, we present and analyse a class of “filtered” numerical schemes for second order Hamilton-Jacobi-Bellman (HJB) equations, with a focus on examples arising from stochastic control problems in financial engineering. We start by discussing more widely the difficulty in constructing compact and accurate approximations. The key obstacle is the requirement in the established convergence analysis of certain monotonicity properties of the schemes. We follow ideas in Oberman and Froese (2010) to introduce a suitable local modification of high order schemes, which are necessarily non-monotone, by “filtering” them with a monotone scheme. Thus, they can be proven to converge and still show an overall high order behaviour for smooth enough value functions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests. 

This talk is based on joint work with Olivier Bokanowski and Athena Picarelli.

Hotels.com is one of the world’s leading accommodation booking websites featuring an inventory of around 300.000 hotels and 100s of millions of users. A crucial part of our business is to act as an agent between these two sides of the market, thus reducing search costs and information asymmetries to enable our visitors to find the right hotel in the most efficient way.

From this point of view selling hotels is one large recommendation challenge: given a set of items and a set of observed choices/ratings, identify a user’s preference profile.  Over the last years this particular problem has been intensively studied by a strongly interdisciplinary field based on ideas from choice theory, linear algebra, statistics, computer science and machine learning. This pluralism is reflected in the broad array of techniques that are used in today’s industry applications, i.e. collaborative filtering, matrix factorization, graph-based algorithms, decision trees or generalized linear models.

The aim of this workshop is twofold.

Firstly we want to give some insight into the statistical modelling techniques and assumptions employed at hotels.com, the practical challenges one has to face when designing a flexible and scalable recommender system and potential gaps between current research and real-world applications.

Secondly we are going to consider some more advanced questions around learning to rank from partial/incomplete feedback (1), dealing with selection-bias correction (2) and how econometrics and behavioral theory (eg Luce, Kahneman /Tversky) can be used to complement existing techniques (3).

After giving some motivation, I will discuss work in progress with Harry Schmidt in which we give a pfaffian definition of Weierstrass elliptic functions, refining a result due to Macintyre. The complexity of our definition is bounded by an effective absolute constant. As an application we give an effective version of a result of Corvaja, Masser and Zannier on a sharpening of Manin-Mumford for non-split extensions of elliptic curves by the additive group. We also give a higher dimensional version of their result.