# Non-smooth Geometry

The group is specialized in some topics in Differential Geometry that go beyond classical Riemannian Geometry. In particular, we study SubRiemannian Geometry, Geometric Measure Theory and Optimal Transport on metric spaces. We are interested also in Geometric Mapping Theory, Geometric Topology, Hyperbolic Geometry, Minimal Surfaces, and Geometric Group Theory.

In the last 25 years there has been a surge of interest in the geometry of non-smooth spaces and in their corresponding analysis. This movement arose from the interaction between active areas of mathematics concerning the theory of analysis on metric spaces along with geometric group theory, rigidity, and quasi-conformal homeomorphisms. One of the purposes was to study mappings between non-Riemannian metric structures such as boundaries of hyperbolic groups and Carnot groups, equipped with their subRiemannian metrics.

**Members**

- Professor Pekka Koskela
- Professor Kai Rajala
- Associate Professor/Academy Research Fellow Enrico Le Donne
- Senior Lecturer/Academy Research Fellow Tapio Rajala
- Senior Lecturer Jouni Parkkonen
- Postdoctoral Researcher Augusto Gerolin
- Postdoctoral Researcher Thibaut Dumont
- Postdoctoral Researcher Francesca Tripaldi

**Ph.D Students**

- Fares Essebei, Eero Hakavuori, Anna Kausamo, Ville Kivioja, Terhi Moisala, Timo Schultz.

**Research**

**Metric measure spaces with Ricci curvature lower bounds**

*(T. Rajala, A. Kausamo, T. Schultz)*

We study metric measure spaces with synthetic notions of Ricci curvature lower bounds. Especially, we focus on properties of several variants of CD(K,N) spaces, existence of optimal maps in these spaces, the role of non-branching geodesics and local structure of the spaces.

**SubRiemanian** **Geometry**

*(E. Le Donne, E. Hakavuori, V. Kivioja, T. Moisala)*

The general goal of this group is to develop an adapted geometric measure theory in non-Riemannian settings. There is a wide range of activities, but there is a common denominator in all of them: subRiemannian geometry of Lie groups. The subject has an obvious interdisciplinary character, since these particular metric spaces appear in various areas of mathematics, such as control theory, harmonic and complex analysis, asymptotic geometry, subelliptic PDE's, and geometric group theory.

The definition of the subRiemannian distance goes back to some ideas of Carnot and Carathéodory. The distance between two points in a manifold is the shortest length among all curves joining the two points and tangent to a bracket-generating subbundle of the tangent bundle.

The goal of the research group in the department is to understand how smooth this distance is when the subbundle is smooth and it is a proper subset of the tangent bundle. On the one hand, a subRiemannian space is fractal in the sense that its Hausdorff dimension is greater than the topological dimension. Moreover, there are smooth curves that have locally infinite length. On the other hand, one might expect that other objects associated to the distance are regular. Indeed, the following problems have only been solved in some special cases:

- Are length-minimizing curves smooth?
- Are distance-preserving homeomorphisms smooth?
- Are boundaries of metric balls rectifiable?
- Are sets of finite perimeter intrinsically rectifiable?
- Are 1-quasi-conformal maps smooth?
- How can one characterize subRiemannian geometries?

**Geometry Seminar**

The Jyväskylä Geometry seminar is on **Mondays** afternoons in the Department of Mathematics and Statistics. The meeting is usually from **14:15 to 16:00** in room MaD 380.

For more information check: GeoMeG seminars.

**Events**

- Sub-Riemannian Geometry and Beyond (Jyväskylä, 19th-23th February 2018).
- 27th Jyväskylä Summer School and MAnET-workshop (Jyväskylä, 7th-18th August 2017)
- Optimal transport meets density functional theory (Jyväskylä, 31st May-7th June 2017).

**Some International Collaborators**

Ugo Boscain (Paris), Emmanuel Breuillard (Cambridge), Luca Capogna (WPI), Richard Montgomery (UC Santa Cruz), Roberto Monti (Padova), Alessandro Ottazzi (UNSW, Sydney), Pekka Pankka (Helsinki), Pierre Pansu (Paris Orsay), Severine Rigot (Nice), Davide Vittone (Padova)

**Publications**

Publications of the group can be found on the arXiv preprint server and the TUTKA database. Some recent publications can be found in this link.

**Funding and external links**

- Academy of Finland:
*Local and global structure of metric measure spaces with Ricci curvature lower bounds* - Academy of Finland:
*Geometry of subRiemannian groups: regularity of finite-perimeter sets, geodesics, spheres, and isometries with applications and generalizations to biLipschitz homogeneous spaces.* - ERC GeoMeG: Geometric of Metric Groups
- The Finnish Centre of Excellence (CoE) in Analysis and Dynamics Research