Groups acting on metric spaces of non-positive curvature

(MaD 380)

Thibaut Dumont (University of Jyväskylä)

This colloquium-style seminar will be an opportunity for me to present my field Geometric Group Theory (GGT) and some of my personal interests therein. The influence of Klein and his Erlangen's program lead geometers to look at geometries and their transformations from a group theoretic viewpoint. The study of a metric space and its group of isometries are closely related. A metric property on one side can dictate an algebraic one on the other, and vice-versa. The central idea of Geometric Group Theory is to look at groups themselves as geometric objects by endowing them with a distance function. The notion of nonpositive curvature in a metric space provide a rich framework for isometric group actions. Some classical examples will be discussed like symmetric spaces of Lie groups and their non-Archimedean counterparts: Euclidean buildings.

11/September/2017

Existence and non-existence results for minimal graphic and p-harmonic functions

(MaD 380)

Esko Heinonen (University of Helsinki)

In the Euclidean space, by the celebrated result due to
Bombieri, De Giorgi, and Miranda, all positive entire solutions of the
minimal graph equation are constant. It turns out that on Riemannian
manifolds similar results can be obtained for solutions with at most
linear growth if the manifold has only one end and asymptotically
non-negative sectional curvature. In this talk I will discuss about
recent results concerning the existence and non-existence of entire
minimal graphic and p-harmonic functions. Talk is based on joint work
with Jean-Baptiste Casteras and Ilkka Holopainen.

18/September/2017

The Sard conjecture on Martinet surfaces

(MaD 380)

Andre Ricardo Belotto da Silva (Universite Paul Sabatier)

Given a totally nonholonomic distribution of rank two $\Delta$ on a three-dimensional manifold $M$, it is natural to investigate the size of the set of points $\mathcal{X}^x$ that can be reached by singular horizontal paths starting from a same point $x \in M$. In this setting, the Sard conjecture states that $\mathcal{X}^x$ should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero.
In this seminar, I present a recent work in collaboration with Ludovic Rifford where we show that the conjecture holds whenever the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces and show that the result holds true under an assumption of non-transversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and resolution of singularities.

25/September/2017

Higher rank hyperbolicity in spaces of non-positive curvature

(MaD 380)

Urs Lang (ETH)

The large scale geometry of Gromov hyperbolic metric spaces exhibits many
distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the linear isoperimetric filling inequality for 1-cycles, the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. After briefly reviewing these properties, I will describe a number of closely analogous results for spaces of rank n > 1 in an asymptotic sense, under some weak assumptions reminiscent of non-positive curvature.
A central role is played by a suitable notion of n-dimensional quasi-minimizing surfaces of polynomial growth of order n, which serve as a substitute for quasi-geodesics.