Past

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.

Diffusion process models for evolution of neutral genes have a particle dual coalescent process underlying them. Models are reversible with transition functions having a diagonal expansion in orthogonal polynomial eigenfunctions of dimension greater than one, extending classical one-dimensional diffusion models with Beta stationary distribution and Jacobi polynomial expansions to models with Dirichlet or Poisson Dirichlet stationary distributions. Another form of the transition functions is as a mixture depending on the mutant and non-mutant families represented in the leaves of an infinite-leaf coalescent tree.

The one-dimensional Wright-Fisher diffusion process is important in a characterization of a wider class of continuous time reversible Markov processes with Beta stationary distributions originally studied by Bochner (1954) and Gasper (1972). These processes include the subordinated Wright-Fisher diffusion process.

I will go over my paper (arXiv:0902.2135v1) which explains how semi-flat Calabi-Yau / G$_2$ manifolds can be constructed from minimal 3-submanifolds in a signature (3,3) vector space.

This talk will mention methods of testing whether a given integer is prime. Included topics are Carmichael numbers, Fermat and Euler pseudo-primes and results contingent on the Generalised Riemann Hypothesis.

The notion of a sutured 3-manifold was introduced by Gabai. It is a powerful tool in 3-dimensional topology. A few years ago, Andras Juhasz defined an invariant of sutured manifolds called sutured Floer homology.

I'll discuss a simpler invariant obtained by taking the Euler characteristic of this theory. This invariant turns out to have many properties in common with the Alexander polynomial. Joint work with Stefan Friedl and Andras Juhasz.

This talk is based on a joint work with Zhan Shi: We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou (2005). Our method applies furthermore to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn (1988). Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

Let $\mathbb{E}$ be an elliptic curve defined over a number field $k$,

and let $a\in\mathbb{E}(\mathbb{C})$ be a complex point. Among the

possible choices of sequences of division points of $a$, $(a_n)_n$

such that $a_1 = a$ and $na_{nm} = a_m$, we can pick out those which

converge in the complex topology to the identity. We show that the

algebraic content of this effect of the complex topology is very

small, in the sense that any set of division sequences which shares

certain obvious algebraic properties with the set of those which

converge to the identity is conjugated to it by a field automorphism

of $\mathbb{C}$ over $k$.

As stated, this is a result of algebra and number theory. However, in

proving it we are led ineluctably to use model theoretic techniques -

specifically the concept of "excellence" introduced by Shelah for the

analysis of $L_{\omega_1,\omega}$ categoricity, which reduces the

question to that of proving certain unusual versions of the theorems

of Mordell-Weil and Kummer-Bashmakov. I will discuss this and other

aspects of the proof, without assuming any model- or number-theoretic

knowledge on the part of my audience.