Past

If $A$ is a set of $n$ positive integers, how small can the set

$\{ x/(x,y) : x,y \in A \}$ be? Here, as usual, $(x,y)$ denotes the highest common factor of

$x$ and $y$. This elegant question was raised by Granville and Roesler, who

also reformulated it in the following way: given a set $A$ of $n$ points in

the integer grid ${\bf Z}^d$, how small can $(A-A)^+$, the projection of the difference

set of $A$ onto the positive orthant, be?

Freiman and Lev gave an example to show that (in any dimension) the size can

be as small as $n^{2/3}$ (up to a constant factor). Granville and Roesler

proved that in two dimensions this bound is correct, i.e. that the size is

always at least $n^{2/3}$, and they asked if this holds in any dimension.

After some background material, the talk will focus on recent developments.

Joint work with B\'ela Bollob\'as.

 When modelling the motion of a sperm cell in the female reproductive tract, the Reynolds number is found to be very small, thus allowing the nonlinear Navier-Stokes equations to simplify to the linear Stokes equations stating that pressure, viscous and body forces balance each other at any instant in time. A wide range of analytical techniques can be applied to investigate the Stokes flow past a moving body. In this talk, we introduce various Stokes flow singularities and illustrate how they can provide a handy starting point (ansatz) when trying to determine the form of the flow field around certain bodies, from simple translating spheres to beating sperm tails.

The infamous inverse Galois problem asks whether or not every finite group can be realised as a Galois group over Q. We will see some techniques that have been developed to attack it, and will soon end up in the realms of class field theory, étale fundamental groups and modular representations. We will give some concrete examples and outline the so called 'rigidity method'. 

Morgan and Culler proved in the 1980’s that a minimal action of a free group on a tree is

completely determined by its length function. This theorem has been of fundamental importance in the

study of automorphisms of free groups. In particular, it gives rise to a compactification of Culler-Vogtmann's

Outer Space. We prove a 2-dimensional analogue of this theorem for right-angled Artin groups acting on

CAT(0) rectangle complexes. (Joint work with M. Margolis)

I will describe the recent construction of new supersymmetric compactifications of the heterotic string which yield just the spectrum of the MSSM at low energies. The starting point is the standard embedding solution on a Calabi-Yau manifold with Euler number -6 with various choices of Wilson lines, i.e., various choices of discrete holonomy for the gauge bundle. Although they yield three net generations of standard model matter, such models necessarily have a larger gauge group than the standard model, as well as exotic matter content. Families of stable bundles can be obtained by deformation of these simple solutions, the deformation playing the dual role of partially breaking the gauge group and reducing the matter content, and in this way we construct more realistic models. The moduli space breaks up into various branches depending on the initial choice of Wilson lines, and on eight of these branches we find models with exactly the standard model gauge group, three generations of quarks and leptons, two Higgs doublets, and no other massless charged states. I will also comment on why these are possibly the unique models of this type.

The Eilenberg-Ganea conjecture is the statement that every group of cohomological dimension two admits a two-dimensional classifying space.  This problem is unsolved after 50 years.  I shall discuss the background to this question and negative answers to some other related questions.  This includes recent joint work with Martin Fluch.

In a market with one safe and one risky asset, an investor with a long

horizon and constant relative risk aversion trades with constant

investment opportunities and proportional transaction costs. We derive

the optimal investment policy, its welfare, and the resulting trading

volume, explicitly as functions of the market and preference parameters,

and of the implied liquidity premium, which is identified as the

solution of a scalar equation. For small transaction costs, all these

quantities admit asymptotic expansions of arbitrary order. The results

exploit the equivalence of the transaction cost market to another

frictionless market, with a shadow risky asset, in which investment

opportunities are stochastic. The shadow price is also derived

explicitly. (Joint work with Paolo Guasoni, Johannes Muhle-Karbe, and

Walter Schachermayer)

The seminar will take place in Lecture Theatre A, Department of Computer Science.

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Contextuality and non-locality are features of quantum mechanics which stand in sharp contrast to the realistic picture underlying classical physics. We shall describe a unified geometric perspective on these notions in terms of *obstructions to the existence of global sections*. This allows general results and structural notions to be uncovered, with quantum mechanics appearing as a special case. The natural language to use here is that of sheaves and presheaves; and cohomological obstructions can be defined which witness contextuality in a number of salient examples.

This is joint work with Adam Brandenburger
 http://iopscience.iop.org/1367-2630/13/11/113036/
 http://arxiv.org/abs/1102.0264

and Shane Mansfield and Rui Soares Barbosa
 http://arxiv.org/abs/1111.3620

This work builds on the foundation laid by Benney & Timson (1980), who

examined the flow near a contact line and showed that, if the contact

angle is 180 degrees, the usual contact-line singularity does not arise.

Their local analysis, however, does not allow one to determine the

velocity of the contact line and their expression for the shape of the

free boundary involves undetermined constants - for which they have been

severely criticised by Ngan & Dussan V. (1984). As a result, the ideas

of Benny & Timson (1980) have been largely forgotten.

The present work shows that the criticism of Ngan & Dussan V. (1984)

was, in fact, unjust. We consider a two-dimensional steady Couette flow

with a free boundary, for which the local analysis of Benney & Timson

(1980) can be complemented by an analysis of the global flow (provided

the slope of the free boundary is small, so the lubrication

approximation can be used). We show that the undetermined constants in

the solution of Benney & Timson (1980) can all be fixed by matching

their local solution to the global one. The latter also determines the

contact line's velocity, which we compute among other characteristics of

the global flow.

We show that the steepest-descent and Newton's methods for unconstrained nonconvex optimization

under standard assumptions may both require a number of iterations and function evaluations

arbitrarily close to the steepest-descent's global worst-case complexity bound. This implies that

the latter upper bound is essentially tight for steepest descent and that Newton's method may be as

slow as the steepest-descent method in the worst case. Then the cubic regularization of Newton's

method (Griewank (1981), Nesterov & Polyak (2006)) is considered and extended to large-scale

problems, while preserving the same order of its improved worst-case complexity (by comparison to

that of steepest-descent); this improved worst-case bound is also shown to be tight. We further

show that the cubic regularization approach is, in fact, optimal from a worst-case complexity point

of view amongst a wide class of second-order methods. The worst-case problem-evaluation complexity

of constrained optimization will also be discussed. This is joint work with Nick Gould (Rutherford

Appleton Laboratory, UK) and Philippe Toint (University of Namur, Belgium).

I will give an introduction to the variational characterisation of the Ricci flow that was first introduced by G. Perelman in his paper on "The entropy formula for the Ricci flow and its geometric applications" http://arxiv.org/abs/math.DG/0211159. The first in a series of three papers on the geometrisation conjecture. The discussion will be restricted to sections 1 through 5 beginning first with the gradient flow formalism. Techniques from the Calculus of Variations will be emphasised, notably in proving the monotonicity of particular functionals. An overview of the local noncollapsing theorem (Perelman’s first breakthrough result) will be presented with refinements from Topping [Comm. Anal. Geom. 13 (2005), no. 5, 1039–1055.]. Some remarks will also be made on connections to implicit structures seen in the physics literature, for instance of those seen in D. Friedan [Ann. Physics 163 (1985), no. 2, 318–419].

Decision making in the presence of uncertainty is a mathematically delicate topic. In this talk, we consider coherent sublinear expectations on a measurable space, without assuming the existence of a dominating probability measure. By considering discrete-time `martingale' processes, we show that the classical results of martingale convergence and the up/downcrossing inqualities hold in a `quasi-sure' sense. We also give conditions, for a general filtration, under which an `aggregation' property holds, generalising an approach of Soner, Touzi and Zhang (2011). From this, we extend various results on the representation of conditional sublinear expectations to general filtrations under uncertainty.

In the talk I will mention two regularity results: the SBV regularity for strictly hyperbolic, genuinely nonlinear 1D systems of conservation laws and the characterization of intrinsic Lipschitz codimension 1 graphs in the Heisenberg groups. In both the contexts suitable scalar, 1D balance laws arise with very low regularity. I will in particular highlight the role of characteristics.

This seminar will be based on joint works with G. Alberti, S. Bianchini, F. Bigolin and F. Serra Cassano, and the main previous literature.

 I will first present Priestley style topological dualities for 
several categories of distributive meet-semilattices
and implicative semilattices developed by G. Bezhanishvili and myself. 
Using these dualities I will introduce a topological duality for Hilbert 
algebras, 
the algebras that correspond to the implicative reduct of intuitionistic logic.

In this talk I'll describe recent work with Henry Wilton (UCL) in which

we prove that there does not exist an algorithm that can determine which

finitely presented groups have a non-trivial finite quotient.

There is a beautiful classification of full (rational) CFT due to

Fuchs, Runkel and Schweigert. The classification says roughly the

following. Fix a chiral algebra A (= vertex algebra). Then the set of

full CFT whose left and right chiral algebras agree with A is

classified by Frobenius algebras internal to Rep(A). A famous example

to which one can successfully apply this is the case when the chiral

algebra A is affine su(2): in that case, the Frobenius algebras in

Rep(A) are classified by A_n, D_n, E_6, E_7, E_8, and so are the

corresponding CFTs.

Recently, Kapustin and Saulina gave a conceptual interpretation of the

FRS classification in terms of 3-dimentional Chern-Simons theory with

defects. Those defects are also given by Frobenius algebras in Rep(A).

Inspired by the proposal of Kapustin and Saulina, we will (partially)

construct the three-tier CFT associated to a given Frobenius algebra.

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. Although the evolution of such `local' modifications of the Erdös-Rényi random graph processes has received considerable attention during the last decade, so far only rather `simple' rules are well-understood. Indeed, the main focus has been on bounded size rules (where all component sizes larger than some constant B are treated the same way), and for more complex rules hardly any rigorous results are known. In this talk we will discuss a new approach that applies to many involved Achlioptas processes: it allows us to prove that certain key statistics are tightly concentrated during the early evolution of e.g. the sum and product rule.

Joint work with Oliver Riordan.

We still don't know what dark matter is but a class of leading candidates

are weakly interacting massive particles or WIMPs. These WIMP models are

falsifiable, which is why we like them. However, the epoch of their

falsifiability is upon us and a slew of data from different directions is

placing models for WIMPs under pressure. I will try and present an updated

overview of the different pieces of evidence, false (?) alarms and

controversies that are making this such an active area of research at the

moment.

In this talk we aim to filter the Majda-McLaughlin-Tabak(MMT) model, which is a one-dimensional prototypical turbulence system. Due to its inherent high dimensionality, we first try to find a low dimensional dynamical system whose statistical property is similar to the original complexity system. This dimensional reduction, called stochastic parametrization, is clearly well-known method but the value of current work lies in the derivation of an analytic closure for the parameters. We then discuss the necessity of the accurate filtering algorithm for this effective dynamics, and introduce the particle filter using the cubature on Wiener space and the recombination skill.

In the first part, a variational model for composition of finitely many strongly elliptic

homogenous elastic materials in linear elasticity is considered. The notion of`universal coercivity' for the variational integrals is introduced which is independent of particular compositions of materials involved. Examples and counterexamples for universal coercivity are presented.

In the second part, some results of recent work with colleagues on image processing and feature extraction will be displayed.

I will discuss the structure of the Selberg class - in which certain expected properties of Dirichlet series and L-functions are axiomatised - along with the numerous interesting conjectures concerning the Dirichlet series in the Selberg class. Furthermore, results regarding the degree of the elements in the Selberg class shall be explored, culminating in the recent work of Kaczorowski and Perelli in which they prove the absence of elements with degree between one and two.

I will discuss quasi-isometries of the free group that preserve an

equivariant pattern of lines.

There is a type of boundary at infinity whose topology determines how

flexible such a line pattern is.

For sufficiently complicated patterns I use this boundary to define a new

metric on the free group with the property that the only pattern preserving

quasi-isometries are actually isometries.

We study pure Seiberg-Witten theories with $SU(r+1)$ and $Sp(2r)$ gauge groups with no flavors. We study singularity loci of moduli space of the Seiberg-Witten curve. Using exterior derivative and discriminant operators, we can find Argyres-Douglas loci of the SW theory. We also compute BPS charges of the massless dyons of $SU$ and $Sp$ SW theory. In a detailed example of $C_2=Sp(4)$, we find 6 points in the moduli space where we have 2 massless BPS dyons, and 3 of them give Argyres-Douglas loci. We show that BPS charges of the massless dyons jump as we go across Argyres-Douglas loci, giving an explicit example of Argyres-Douglas loci living inside the wall of marginal stability. (Based on work in progress with Keshav Dasgupta)

The idea of three-tier conformal field theory (CFT) was first proposed by Greame Segal. It is an extension of the functorial approach to CFT, where one replaces the bordism category of Riemann surfaces by a suitable bordism 2-category, whose objects are points, whose morphism are 1-manifolds, and whose 2-morphisms are pieces of Riemann surface. The Baez-Dolan cobordism hypothesis is a meta-mathematical principle. It claims that functorial quantum field theory (i.e. quantum field theory expressed as a functor from some bordism category) becomes simper once "you go all the way down to points", i.e., once you replace the bordism category by a higher category. Three-tier CFT is an example of "going all the way down to points". We will apply the cobordism hypothesis to the case of three-tier CFT, and show how the modular invariance of the partition function can be derived as a consequence of the formalism, even if one only starts with genus-zero data.

Data assimilation aims to correct a forecast of a physical system, such as the atmosphere or ocean, using observations of that system, in order to provide a best estimate of the current system state. Since it is not possible to observe the whole state it is important to know how errors in different variables of the forecast are related to each other, so that all fields may be corrected consistently. In the first part of this talk we consider how we may impose constraints between different physical variables in data assimilation. We examine how we can use our knowledge of atmospheric physics to pose the assimilation problem in variables that are assumed to be uncorrelated. Using a shallow-water model we demonstrate that this is best achieved by using potential vorticity rather than vorticity to capture the balanced part of the flow. The second part of the talk will consider a further transformation of variables to represent spatial correlations. We show how the accuracy and efficiency with which we can solve the transformed assimilation problem (as measured by the condition number) is affected by the observation distribution and accuracy and by the assumed correlation lengthscales. Theoretical results will be illustrated using the Met Office variational data assimilation scheme.

We propose a dynamic insurance network model that allows to deal with reinsurance counter-party default risks with a particular aim of capturing cascading effects at the time of defaults. We capture these effects by finding an equilibrium allocation of settlements which can be found as the unique optimal solution of a linear programming problem. This equilibrium allocation recognizes 1) the correlation among the risk factors, which are assumed to be heavy-tailed, 2) the contractual obligations, which are assumed to follow popular contracts in the insurance industry (such as stop-loss and retro-cesion), and 3) the interconnections of the insurance-reinsurance network. We are able to obtain an asymptotic description of the most likely ways in which the default of a specific group of insurers can occur, by means of solving a multidimensional Knapsack integer programming problem. Finally, we propose a class of provably strongly efficient estimators for computing the expected loss of the network conditioning the failure of a specific set of companies. Strong efficiency means that the complexity of computing large deviations probability or conditional expectation remains bounded as the event of interest becomes more and more rare.

Introductory talk on topological quantum field theories (TQFTs) and the cobordism hypothesis, focusing on the conceptual issues involved.

The lecture will take place this Friday at 11am in Lecture Theatre A of the Department of Computer Science

DNA double strand breaks (DSB) are the most deleterious type of DNA damage induced by ionizing radiation and cytotoxic agents used in the treatment of cancer. When DSBs are formed, the cell attempts to repair the DNA damage through activation of a variety of molecular repair pathways. One of the earliest events in response to the presence of DSBs is the phosphorylation of a histone protein, H2AX, to form γH2AX. Many hundreds of copies of γH2AX form, occupying several mega bases of DNA at the site of each DSB. These large collections of γH2AX can be visualized using a fluorescence microscopy technique and are called ‘γH2AX foci’. γH2AX serves as a scaffold to which other DNA damage repair proteins adhere and so facilitates repair. Following re-ligation of the DNA DSB, the γH2AX is dephosphorylated and the foci disappear.

We have developed a contrast agent, 111In-anti-γH2AX-Tat, for nuclear medicine (SPECT) imaging of γH2AX which is based on an anti-γH2AX monoclonal antibody. This agent allows us to image DNA DSB in vitro in cells, and in in vivo model systems of cancer. The ability to track the spatiotemporal distribution of DNA damage in vivo would have many potential clinical applications, including as an early read-out of tumour response or resistance to particular anticancer drugs or radiation therapy.

The imaging tracer principle states that a contrast agent should not interfere with the physiology of the process being imaged. Therefore, we have investigated the influence of the contrast agent itself on the kinetics of DSB formation, repair and on γH2AX foci formation and resolution and now wish to synthesise these data into a coherent kinetic-dynamic model.

Dualities of various types have been used by different authors to 
describe free and projective objects in a large
  number of classes of algebras. Particularly, natural dualities provide a 
general tool to describe free objects. In
  this talk we present two interesting applications of this fact. 
  We first provide a combinatorial classification of unification problems 
by their unification type for the
varieties of Bounded Distributive Lattices, Kleene algebras, De Morgan 
algebras. Finally we provide axiomatizations forsingle
and multiple conclusion admissible rules for the varieties of Kleene 
algebras, De Morgan algebras, Stone algebras.

Understanding the fatigue of metals under cyclic loads is crucial for some fields in mechanical engineering, such as the design of wheels of high speed trains and aero-plane engines. Experimentally it has been found that metal fatigue induced by cyclic loads is closely related to a ladder shape pattern of dislocations known as a persistent slip band (PSB). In this talk, a quantitative description for the formation of PSBs is proposed from two angles: 1. the motion of a single dislocation analised by using asymptotic expansions and numerical simulations; 2. the collective behaviour of a large number of dislocations analised by using a method of multiple scales.

De Concini-Kac-Procesi conjecture gives a good estimate for the dimensions of finite--dimensional non-restricted representations of quantum groups at m-th root of unity. According to De Concini, Kac and Procesi such representations can be split into families parametrized by conjugacy classes in an algebraic group G, and the dimensions of representations corresponding to a conjugacy class O are divisible by m^{dim O/2}. The talk will consist of two parts. In the first part I shall present an approach to the proof of De Concini-Kac-Procesi conjecture based on the use of q-W algebras and Bruhat decomposition in G. It turns out that properties of representations corresponding to a conjugacy class O depend on the properties of intersection of O with certain Bruhat cells. In the second part, which is more technical, I shall discuss q-W algebras and some related results in detail.

We present progress on an Interior Point based multi-step solution approach for stochastic programming problems. Our approach works with a series of scenario trees that can be seen as successively more accurate discretizations of an underlying probability distribution and employs IPM warmstarts to "lift" approximate solutions from one tree to the next larger tree.