Introductory Winter School "Analysis and geometry on groups and spaces" - Videos

Monday

Regina Rotman

David Bate

Daniele Semola

Panos Papasoglu

Tuesday

Daniele Semola II

Panos Papasoglu II

David Bate II

Regina Rotman II

Wednesday

Daniele Semola III

David Bate III

Thursday

Daniele Semola IV

Panos Papasoglu III

David Bate IV

Regina Rotman III

Friday

Panos Papasoglu IV

Regina Rotman IV

David Bate (University of Warwick): An introduction to rectifiability in metric spaces

Geometric measure theory studies geometric properties of non-smooth sets. The key concept is that of an n-rectifiable set, which can be parametrised by countably many Lipschitz images of n-dimensional Euclidean space. Characterisations of rectifiable subsets of Euclidean space have important consequences in the theory of partial differential equations, harmonic analysis and fractal geometry. The recent interest in analysis in non-Euclidean metric spaces naturally leads to questions regarding geometric measure theory in this setting. This talk will give an overview of work in this direction. After introducing the necessary background, we will present recent characterisations of rectifiable subsets of an arbitrary metric space in terms of non-linear projections and tangent spaces.

Panos Papasoglu (University of Oxford): An Introduction to Systolic Geometry

I will give a quick introduction to systolic geometry. The systole of a Riemannian manifold is the length of the shortest non-contractible loop. One seeks an upper bound in terms of the volume of the manifold. The first such result was obtained by Loewner in 1949 in the case of the torus, and Berger in the 60's asked for a generalisation for aspherical manifolds. I plan to explain two results of Gromov in some detail:

1. The  proof of the upper bound for the systole of high genus surfaces. I will give Kodani's proof.
2. The recent proof by Nabutovsky for the upper bound of the systole of an aspherical manifold -- which also significantly improves the constant in the original proof.

Regina Rotman (Universtity of Toronto): Quantitative Topology and Geometric Inequalities

I will discuss the quantitative versions of some existence theorem in Geometric Calculus of Variations, originally proven using the methods of Algebraic Topology. Examples of such theorems are the following: a theorem of A. Fet and L. Lyusternik, stating that on any closed Riemannian manifold there exists a periodic geodesic; a theorem of J. P. Serre, stating that for any pair of points on a closed Riemannian manifold there exist infinitely many geodesics, connecting them; a recent result of X. Zhou about the existence of a geodesic chord in a complete manifold M orthogonal to a closed submanifold N, modulo the existence of a relevant sweepout. In particular, the quantitative versions of these theorems result in many geometric inequalities, relating the "size" of the minimal object to the geometric properties of the ambient space, such as volume, diameter, curvature.

Daniele Semola (University of Vienna): Ricci curvature and fundamental group

It has been known since the Forties that fundamental groups of complete Riemannian manifolds with nonnegative Ricci curvature satisfy certain restrictions. The goal of this mini-course will be to review some of these restrictions and to discuss their optimality. The topics will include:

• ˆStructure of fundamental groups of manifolds with nonnegative Ricci curvature (polynomial growth, virtual nilpotency, local uniform nite generation, . . . )
• Discussion about some motivating examples
• Methods to prove nite generation of fundamental groups (Gromov's short generators, Sormani's linear growth theorem, equivariant analysis of blow-downs, . . . )
• ˆSpaces/manifolds with innitely generated fundamental groups
• ˆGluing constructions preserving nonnegative Ricci curvature
• Construction of counterexamples to the Milnor conjecture