The fundamental results about definable groups in o-minimal structures all suggested a deep connection between these groups and Lie groups. Pillay's conjecture explicitly formulates this connection in analogy to Hilbert's fifth problem for locally compact topological groups, namely, a definably compact group is, after taking a suitable the quotient by a "small" (type definable of bounded index) subgroup, a Lie group of the same dimension. In this talk we will report on the proof of this conjecture in the remaining open case, i.e. in arbitrary o-minimal structures. Most of the talk will be devoted to one of the required tools, the formalism of the six Grothendieck operations of o-minimals sheaves, which might be useful on it own. 

Lagrangian Floer cohomology categorifies the intersection number of (half-dimensional) Lagrangian submanifolds of a symplectic manifold. In this talk I will describe how and when can we define Lagrangian Floer cohomology. In the case when Floer cohomology cannot be defined I will describe an alternative invariant known as the Fukaya (A-infinity) algebra.

In this talk we want to discuss results by Dimca, Papadima, and Suciu about the finiteness properties of Kähler groups. Namely, we will sketch their proof that for every $2\leq n\leq \infty$ there is a Kähler group with finiteness property $\mathcal{F}_n$, but not $FP_{n+1}$. Their proof is by explicit construction of examples. These examples all arise as subgroups of finite products of surface groups and they are the first known examples of Kähler groups with arbitrary finiteness properties. The talk does not require any prior knowledge of finiteness properties or of Kähler groups.

Historically, the study of positively curved manifolds has always been challenging. There are many reasons for this, but among them is the fact that the existence of a metric of positive curvature on a manifold imposes strong topological restrictions. In this talk we will discuss some of these topological implications and we will introduce the main results in this area. We will also present some recent results that relate positive curvature to the smooth structure of the manifold.

The Calabi conjecture, posed in 1954 and proved by Yau in 1976, guaranties the existence of Ricci-flat Kahler metrics on compact Kahler manifolds with vanishing first Chern class, providing examples of the so called Calabi-Yau manifolds. The latter are of great importance to the fields of Riemannian Holonomy Groups, having Hol0 as a subgroup of SU; Calibrated Geometry, more precisely Special Lagrangian Geometry; and to String theory with the discovery of the phenomenon of Mirror Symmetry (to mention a few!). In the talk, we will discuss the necessary background to formulate the Calabi conjecture and explain some of the main ideas behind its proof by Yau, which itself is a jewel from the point of view of non-linear PDEs.

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 ]