Past

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.