I will introduce simple homotopy theory and then discuss relations between some conjectures in 2 dimensional simple homotopy theory and the 3 and 4 dimensional Poincaré conjectures.

# Past Junior Geometry and Topology Seminar

Since its genesis in 1915, General Relativity has proven to be one of the most successful physical theories ever invented. Providing a description of the large scale structure of the universe it continues to be in agreement with all experimental tests to high accuracy. By merging Classical Mechanics and Electrodynamics to a consistent geometrical theory of space-time it has become one of the two pillars of modern theoretical physics alongside Quantum Mechanics. This talk aims to give an introduction to the ideas and concepts of General Relativity. After briefly reviewing Classical (Newtonian) Mechanics and experiments in contradiction with it the framework and axioms of General Relativity will be introduced. This will be followed by a survey on major implications of the (new) geometrical description of gravity. Finally an outlook on physics beyond General Relativity will be provided.

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.

I will recall basic notions of operator K-theory as a non-commutative (C*-algebra) generalisation of topological K-theory. Twisted crossed products will be introduced as generalisations of group C*-algebras, and a model of Karoubi's K-theory, which makes sense for super-algebras, will be sketched. The motivation comes from physics, through the study of quantum mechanical symmetries, charged free quantum fields, and topological insulators. The relevant theorems, which are interesting in their own right but scattered in the literature, will be consolidated.

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.