Past

Let G be a compact semisimple Lie group. A classical paper of Atiyah and Bott (from 1982) studies the moduli space of flat G-bundles on a fixed Riemann surface S. Their approach completely determines the integral homology of this moduli space, using Morse theoretic methods. In the case where G is U(n), this moduli space is homotopy equivalent to the moduli space of holomorphic vector bundles on S which are "semi-stable". Previous work of Harder and Narasimhan determined the Betti numbers of this moduli space using the Weil conjectures. 20 years later, a Madsen and Weiss determined the homology of the moduli space of Riemann surfaces, in the limit where the genus of the surface goes to infinity.

My talk will combine these two spaces: I will describe the homology of the moduli space of Riemann surfaces S, equipped with a flat G-bundle E -> S, where we allow both the flat bundle and the surface to vary. I will start by reviewing parts of the Atiyah-Bott and Madsen-Weiss papers. Our main theorem will then be a rather easy consequence. This is joint work with Nitu Kitchloo and Ralph Cohen.

In this lecture, I will illustrate the cobordism hypothesis by presenting some examples. Exact content to be determined, depending on the interests of the audience.

In this lecture, I will give a more precise statement of the Baez-Dolan cobordism hypothesis, which gives a description of framed bordism (higher) categories by a universal mapping property. I'll also describe some generalizations of the cobordism hypothesis, which take into account the structure of diffeomorphism groups of manifolds and which apply to manifolds which are not necessarily framed.

In this lecture, I'll give an overview of some ideas from higher category theory which are needed to make sense of the Baez-Dolan cobordism hypothesis. If time permits, I'll present Rezk's theory of complete Segal spaces (a model for the theory of higher categories in which most morphisms are assumed to be invertible) and explain how bordism categories can be realized in this framework.

In this lecture, I will review Atiyah's definition of a topological quantum field theory. I'll then sketch the definition of a more elaborate structure, called an "extended topological quantum field theory", and describe a conjecture of Baez and Dolan which gives a classification of these extended theories.

PROBLEM STATEMENT:

Consider a set of measurements made by many sensors placed in a noisy environment, the noise is both temporally and spatially correlated and has time varying statistics. Given this environment, characterised by spatial and temporal scales of correlation, the challenge is to detect the presence of a weak, stationary signal described by smaller scales of temporal and spatial correlation.

Many current and future challenges involve detection of signals in the presence of other, similar, signals. The signal environment is extremely busy and thus the traditional process of detection of a signal buried in noise at reducing signal to noise ratio is no longer sufficient. Signals of interest may be at high SNR but need to be detected, classified, isolated and analysed as close to real time as is possible. All interfering signals are potentially signals of interest and all overlap in time and frequency.

Can the performance of signal detection algorithms be parameterised by some characteristic(s) of the signal environment?

A problem exists to detect and classify multiple signal types, but with a very low duty cycle for the receiver. In certain circumstances, very short windows of opportunity exist where the local signal environment can be sampled and the duty cycle of observation opportunities can be as low as 10%. The signals to be detected may be continuous or intermittent (burst) transmissions. Within these short windows, it is desirable to detect and classify multiple transmissions in terms of signal type (e.g. analogue or digital comms, navigation etc.) and location of transmitters. The low duty cycle of observations for the receiver makes this a challenging prospect.

Again, can the performance of signal detection algorithms be parameterised by some characteristic(s) of the signal environment?

The classic coating-flow problem first studied experimentally by Moffat and asymptotically by Pukhnachov in 1977 is reconsidered in the framework of multiple-timescale asymptotics. Two-timescale approximations of the height of the thin film coating a rotating horizontal circular cylinder are obtained from an asymptotic analysis, in terms of small gravitational and capillary parameters, of Pukhnachov's nonlinear evolution for the film thickness. The transition, as capillary effects are reduced, from smooth to shock-like solutions is examined, and interesting large-time dynamics in this case are determined via a multiple-timescale analysis of a Kuramoto-Sivashinsky equation. A pseudo-three-timescale method is proposed and demonstrated to improve the accuracy of the smooth solutions, and an asymptotic analysis of a modified Pukhnachov's equation, one augmented by inertial terms, leads to an expression for the critical Reynolds number above which the steady states first analysed by Moffatt and Pukhnachov cannot be realised. As part of this analysis, some interesting implications of the effects of different scalings on inertial terms is discussed. All theoretical results are validated by either spectral or extrapolated numerics.

In recent years, the interdisciplinary field of imaging science has been experiencing an explosive growth in research activities including more models being developed, more publications generated, and above all wider applications attempted.
In this talk I shall first give an overview of the various imaging work carried out in our Liverpool group, some with collaborations with UCLA (T F Chan), CUHK (R H Chan) and Bergen (X C Tai) and several colleagues from other departments in Liverpool. Then I shall focus on two pieces of recent work, denoising and segmentation respectively:
(i) Image denoising has been a research topic deeply investigated within the last two decades. Even algorithmically the well-known ROF model (1992) can be solved efficiently. However less work has been done on models using high order regularization. I shall describe our first and successful attempt to develop a working multilevel algorithm for a 4th order nonlinear denoising model, and our work on solving the combined denoising and deblurring problem, different from the reformulation approach by M N Ng and W T Yin (2008) et al.
(ii) the image active contour model by Chan-Vese (2001) can be solved efficiently both by a geometric multigrid method and by an optimization based multilevel method. Surprisingly the new multilevel methods can find a solution closer to the global minimize than the existing unilevel methods. Also discussed are some recent work (jointly with N Badshah) on selective segmentation that has useful medical applications.

This talk concerns the relationships between Donaldson-Thomas, Pandharipande-Thomas, and Szendroi invariants established via analysis of the geometry of wall crossing phenomena of suitably general moduli spaces. I aim to give a reasonably detailed account of the simplest example, the conifold, where in fact all of the major ideas can be easily seen.

We analyze the impact of portfolio rebalancing frequency on the measurement of risk

over a moderately long horizon. This problem arises from an incremental capital charge recently

proposed by the Basel Committee on Banking Supervision. The new risk measure calculates

VaR over a one-year horizon at a high confidence level and assigns different

rebalancing frequencies to different types of assets to capture potential illiquidity.

We analyze the difference between discretely and continuously rebalanced portfolios in a simple model of asset dynamics by examining the limit as the rebalancing frequency increases. This leads to alternative approximations at moderate and extreme loss levels. We also show how to incorporate multiple scales of rebalancing frequency in the analysis

The aim of my talk is to present the work of my PhD Thesis and my current research. It is concerned with local/global bifurcation of standing wave solutions to some nonlinear Schr\"odinger equations in $\mathbb{R}^N \ (N\geq1)$ and with stability properties of these solutions. The equations considered have a nonlinearity of the form $V(x)|\psi|^{p-1}\psi$, where $V:\mathbb{R}^N\to\mathbb{R}$ decays at infinity and is subject to various assumptions. In particular, $V$ could be singular at the origin.

Local/global smooth branches of solutions are obtained for the stationary equation by combining variational techniques and the implicit function theorem. The orbital stability of the corresponding standing waves is studied by means of the abstract theory of Grillakis, Shatah and Strauss.

We classify certain linear representations of finite groups with a large orbit. This is motivated by a question on the number of conjugacy classes of a finite group.

There are many theorems concerning cycles in graphs for which it is natural to seek analogous results for directed graphs. I will survey

recent progress on certain questions of this type. New results include

(i) a solution to a question of Thomassen on an analogue of Dirac’s theorem

for oriented graphs,

(ii) a theorem on packing cyclic triangles in tournaments that “almost” answers a question of Cuckler and Yuster, and

(iii) a bound for the smallest feedback arc set in a digraph with no short directed cycles, which is optimal up to a constant factor and extends a result of Chudnovsky, Seymour and Sullivan.

These are joint work respectively with (i) Kuhn and Osthus, (ii) Sudakov, and (iii) Fox and Sudakov.

We consider problems of elastic plastic deformation with isotropic and  kinematic hardening.

A dual formulation with stresses as principal variables is used. 

We obtain several results on Sobolev space regularity of the stresses  and strains.

In particular, we obtain the existence of a full derivative of the  stress tensor up to the boundary of the basic domain.

Finally, we present an outlook for obtaining further regularity  results in connection with general nonlinear evolution problems.

The goal of this talk is to give sufficient conditions for the existence of points on certain varieties defned over finite fields.

I will outline the proof of two old conjectures of Cameron Gordon. The first states that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10. The second states the maximal distance between exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 8. The proof uses a combination of new geometric techniques and rigorous computer-assisted calculations.

This is joint work with Rob Meyerhoff.

A financial market model is considered on which agents (e.g. insurers) are subject to an exogenous financial risk, which they trade by issuing a risk bond. Typical risk sources are climate or weather. Buyers of the bond are able to invest in a market asset correlated with the exogenous risk. We investigate their utility maximization problem, and calculate bond prices using utility indi®erence. This hedging concept is interpreted by means of martingale optimality, and solved with BSDE and Malliavin's calculus tools. Prices are seen to decrease as a result of dynamic hedging. The price increments are interpreted in terms of diversification pressure.

GIBSON BUILDING COMMON ROOM 2ND FLOOR

(Coffee and Cakes in Gibson Meeting Room - opposite common room)

The effects of electric fields on nonlinear free surface flows are investigated. Both inviscid and Stokes flows are considered.

Fully nonlinear solutions are computed by boundary integral equation methods and weakly nonlinear solutions are obtained by using long wave asymptotics and lubrication theory. Effects of electric fields on the stability of the flows are discussed. In addition applications to coating flows are presented.

The "large sieve" was invented by Linnik in order to attack problems involving the distribution of integers subject to certain constraints modulo primes, for which earlier methods of sieve theory were not suitable. Recently, the arithmetic large sieve inequality has been found to be capable of much wider application, and has been used to obtain results involving objects not usually considered as related to sieve theory. A form of the general sieve setting will be presented, together with sample applications; those may involve arithmetic properties of random walks on discrete groups, zeta functions over finite fields, modular forms, or even random groups.

Geometric integration is the numerical integration of a differential equation, while preserving one or more of its geometric/physical properties exactly, i.e. to within round-off error.

Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase-space volume, symmetries, time-reversal symmetry, symplectic structure and dissipation are examples. The field has tantalizing connections to dynamical systems, as well as to Lie groups.

In this talk we first present a survey of geometric numerical integration methods for differential equations, and then exemplify this by discussing symplectic vs energy-preserving integrators for ODEs as well as for PDEs.