Past

In this talk, we will discuss the dimension of $J_0(N)[m]$ at an Eisenstein prime m for
square-free level N. We will also study the structure of $J_0(N)[m]$ as a Galois module.
This work generalizes Mazur’s work on Eisenstein ideals of prime level to the case of
arbitrary square-free level up to small exceptional cases.

Acto-myosin network growth and remodeling in vivo is based on a large number of chemical and mechanical processes, which are mutually coupled and spatially and temporally resolved. To investigate the fundamental principles behind the self-organization of these networks, we have developed detailed physico-chemical, stochastic models of actin filament growth dynamics, where the mechanical rigidity of filaments and their corresponding deformations under internally and externally generated forces are taken into account. Our work sheds light on the interplay between the chemical and mechanical processes, and also will highlights the importance of diffusional and active transport phenomena. For example, we showed that molecular transport plays an important role in determining the shapes of the commonly observed force-velocity curves. We also investigated the nonlinear mechano-chemical couplings between an acto-myosin network and an external deformable substrate.

From a theoretical point of view, theta is a relatively simple quantity: the rate of change in value of a financial derivative with respect to time. In a Black-Scholes world, the theta of a delta hedged option can be viewed as `rent’ paid in exchange for gamma. This relationship is fundamental to the risk-management of a derivatives portfolio. However, in the real world, the situation becomes significantly more complicated. In practice the model is continually being recalibrated, and whereas in the Black-Scholes world volatility is not a risk factor, in the real world it is stochastic and carries an associated risk premium. With the heightened interest in automation and electronic trading, we increasingly need to attempt to capture trading, marking and risk management practice algorithmically, and this requires careful consideration of the relationship between the risk neutral and historical measures. In particular these effects need to be incorporated in order to make sense of theta and the time evolution of a derivatives portfolio in the historical measure. 

Gradient-based optimisation techniques have become a common tool in the analysis of fluid systems. They have been applied to replace and extend large-scale matrix decompositions to compute optimal amplification and optimal frequency responses in unstable and stable flows. We will show how to efficiently extract linearised and adjoint information directly from nonlinear simulation codes and how to use this information for determining common flow characteristics. We also extend this framework to deal with the optimisation of less common norms. Examples from aero-acoustics and mixing will be presented.

Using the fact that certain exotic spheres do not admit Lagrangian embeddings into $T^*{\mathcal S}^{n+1}$, as proven by Abouzaid and Ekholm-Smith, we produce non-trivial homotopy classes of the group of compactly supported symplectomorphisms of $T^*{\mathcal S}^n$. In particular, we show that the Hamiltonian isotopy class of the symplectic Dehn twist depends on the parametrisation used in the construction.  Related results are also obtained for $T^*({\mathcal S}^n \times {\mathcal S}^1)$.

Joint work with Jonny Evans.

We present advances in several fundamental fields of computer vision: image segmentation, object tracking, stereo reconstruction for depth map estimation and full 3D multi-view reconstruction. The basic method applied to these fields is convex relaxation. Convex relaxation methods allow for global optimization of numerous energy functionals and provide a step towards less user input and more automation. We will show how the respective computer vision problems can be formulated in this convex optimization framework. Efficient parallel implementations of the arising numerical schemes using graphics processing units allow for interactive applications.

We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to some fixed graph $F$. (Such a graph is said to be $r$-locally-$F$.) This is a natural extension of the study of regular graphs, and of the study of graphs of constant link. We focus on the case where $F$ is $\mathbb{L}^d$, the $d$-dimensional integer lattice. We obtain a characterisation of all the finite graphs in which the ball of radius $3$ around each vertex is isomorphic to the ball of radius $3$ in $\mathbb{L}^d$, for each integer $d$. These graphs have a very rigidly proscribed global structure, much more so than that of $(2d)$-regular graphs. (They can be viewed as quotient lattices in certain 'flat orbifolds'.) Our results are best possible in the sense that '3' cannot be replaced with '2'. Our proofs use a mixture of techniques and results from combinatorics, algebraic topology and group theory. We will also discuss some results and open problems on the properties of a random n-vertex graph which is $r$-locally-$F$. This is all joint work with Itai Benjamini (Weizmann Institute of Science). 

We investigate an X-ray imaging system that fires multiple point sources of radiation simultaneously from close proximity to a probe. Radiation traverses the probe in a non-parallel fashion, which makes it necessary to use tomosynthesis as a preliminary step to calculating a 2D shadowgraph. The system geometry requires imaging techniques that differ substantially from planar X-rays or CT tomography. We present a proof of concept of such an imaging system, along with relevant artefact removal techniques.  This work is joint with Kishan Patel.

      I will show uniqueness result for BV solutions of scalar conservation laws with discontinuous flux in several space dimensions. The proof is based on the notion of kinetic solution and on a careful analysis of the entropy dissipation along the discontinuities of the flux.
 

The idea of A-free group, where A is a discrete ordered abelian group, has been introduced by Myasnikov, Remeslennikov and Serbin. It generalises the construction of free groups. A proof will be outlined that a group is A-free for some A if and only if it acts freely and without inversions on a \lambda-tree, where \lambda is an arbitrary ordered abelian group.

Vector bundles over a compact manifold can be defined via transition 
functions to a linear group. Often one imposes 
conditions on this structure group. For example for real vector bundles on 
may  ask that all 
transition functions lie in the special orthogonal group to encode 
orientability. Commutative K-theory arises when we impose the condition 
that the transition functions commute with each other whenever they are 
simultaneously defined.

We will introduce commutative K-theory and some natural variants of it, 
and will show that they give rise to  new generalised 
cohomology theories.

This is joint work with Adem, Gomez and Lind building on previous work by 
Adem, F. Cohen, and Gomez.

The Taylor expansion of a controlled differential equation suggests that the solution at time 1 depends on the driving path only through the latter's iterated integrals up to time 1, if the vector field is infinitely differentiable. Hambly and Lyons proved that this remains true for Lipschitz vector fields if the driving path has bounded total variation. We extend the Hambly-Lyons result for weakly geometric rough paths in finite dimension. Joint work with X. Geng, T. Lyons and D. Yang.    

Mathematical imaging is not only a multidisciplinary research area but also a major cross-discipline subject within mathematical sciences as  image analysis techniques involve differential geometry, optimization, nonlinear partial differential equations (PDEs), mathematical analysis, computational algorithms and numerical analysis. Segmentation refers to the essential problem in imaging and vision  of automatically detecting objects in an image.

In this talk I first review some various models and techniques in the variational framework that are used for segmentation of images, with the purpose of discussing the state of arts rather than giving a comprehensive survey. Then I introduce the practically demanding task of detecting local features in a large image and our recent segmentation methods using energy minimization and  PDEs. To ensure uniqueness and fast solution, we reformulate one non-convex functional as a convex one and further consider how to adapt the additive operator splitting method for subsequent solution. Finally I show our preliminary work to attempt segmentation of blurred images in the framework of joint deblurring and segmentation.

This talk covers joint work with Jianping Zhang, Lavdie Rada, Bryan Williams, Jack Spencer (Liverpool, UK), N. Badshah and H. Ali (Pakistan). Other collaborators in imaging in general include T. F. Chan, R. H. Chan, B. Yu,  L. Sun, F. L. Yang (China), C. Brito (Mexico), N. Chumchob (Thailand),  M. Hintermuller (Germany), Y. Q. Dong (Denmark), X. C. Tai (Norway) etc. [Related publications from   http://www.liv.ac.uk/~cmchenke ]

Mathematical imaging is not only a multidisciplinary research area but also a major cross-discipline subject within mathematical sciences as  image analysis techniques involve differential geometry, optimization, nonlinear partial differential equations (PDEs), mathematical analysis, computational algorithms and numerical analysis. Segmentation refers to the essential problem in imaging and vision  of automatically detecting objects in an image.

In this talk I first review some various models and techniques in the variational framework that are used for segmentation of images, with the purpose of discussing the state of arts rather than giving a comprehensive survey. Then I introduce the practically demanding task of detecting local features in a large image and our recent segmentation methods using energy minimization and  PDEs. To ensure uniqueness and fast solution, we reformulate one non-convex functional as a convex one and further consider how to adapt the additive operator splitting method for subsequent solution. Finally I show our preliminary work to attempt segmentation of blurred images in the framework of joint deblurring and segmentation.

This talk covers joint work with Jianping Zhang, Lavdie Rada, Bryan Williams, Jack Spencer (Liverpool, UK), N. Badshah and H. Ali (Pakistan). Other collaborators in imaging in general include T. F. Chan, R. H. Chan, B. Yu,  L. Sun, F. L. Yang (China), C. Brito (Mexico), N. Chumchob (Thailand),  M. Hintermuller (Germany), Y. Q. Dong (Denmark), X. C. Tai (Norway) etc.

[Related publications from   http://www.liv.ac.uk/~cmchenke ]