Geometry Days

September 5 - 7, 2016

University of Fribourg


Please notice:

Participants should subscribe at the CUSO website [click here].

Room 2.52, Chemin du Musée 3 (Department of Physics).
Building number 8 on
this map.

Click here to download the poster.

You can find
the program here.


Corina Ciobotaru (Fribourg)
Ivan Izmestiev (Fribourg)
Patrick Ghanaat (Fribourg)
Marc Troyanov (EPFL)


Supported by:
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Main speakers:

  • Vladimir Matveev (Jena)
    Title: Metric projective geometry
    Abstract: My goal is to explain modern geometric techniques using simple examples. Each lecture introduces a new trick and shows its effectiveness by proving a nontrivial interesting statements from the theory of projectively equivalent metrics, that are metrics having the same geodesics considered as unparameterised curves. 

    I will start with definition and basic properties of projective structure and give an application to isometries of Hilbert metrics.
    I will continue with weighted metric tensors and projective invariant equations. As an application I describe topology of closed manifold admitting projectively equivalent metrics.
    The next topic is "local normal forms for projectively equivalent metrics". I mostly concentrate on dimension 2 and as an application I present a solution of the problems stated by Sophus Lie in 1882.

    Then I go to higher dimensions: I discuss special algebraic tricks related to metric projective structures in dimensions > 2. As an application I show that the degree of mobility on closed manifolds of nonconstant curvature is at most 2 and prove the classical projective Lichnerowicz conjecture.

    If the time allows I will also discuss open problems, possible applications and generalisations. 

              Lectures 1-3 pdf, Exercises 1-3 pdf.

  • Valentin Ovsienko (Reims)
    Title: Projective geometry and combinatorics
    Abstract: The goal of this mini-course is to explain relations between very classical invariants of projective geometry, such as the cross-ratio, the Schwarzian derivative, and several subjects of combinatorics that recently attracted much interest. Among the combinatorial notions that will be discussed, the most interesting is that of frieze pattern (due to Coxeter); the geometric notions related to this subject are the Grassmannians and moduli spaces of configurations of points in projective spaces.

    The course will be very elementary and accessible to everybody, no particular preparation is required.
              Most of the material covered in the course can be found in:

    1. Sophie Morier-Genoud, Valentin Ovsienko, Richard Evan Schwartz, Serge Tabachnikov, Linear difference equations, frieze patterns, and the combinatorial Gale transform.  Forum Math. Sigma 2 (2014), e22, 45 pp.
    2. Sophie Morier-Genoud, Valentin Ovsienko, Serge Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons. Ann. Inst. Fourier (Grenoble) 62 (2012), no. 3, 937-987.
    Classical articles:

    3. H. S. M. Coxeter, Frieze patterns. Acta Arith. 18 1971 297-310.

    4. J. H. Conway, H. S. M. Coxeter, Triangulated polygons and frieze patterns. Math. Gaz. 57 (1973), no. 401, 175-183. 

    5. J. H. Conway, H. S. M. Coxeter, Triangulated polygons and frieze patterns. Math. Gaz. 57 (1973), no. 400, 87-94.

    On connections with the cluster algebras: 

    6. Philippe Caldero, Frédéric Chapoton, Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81 (2006), no. 3, 595-616.

    A survey article that stresses a connection with the representation theory:

    7. Sophie Morier-Genoud, Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics. Bull. Lond. Math. Soc. 47 (2015), no. 6, 895-938.

    An application of the combinatorial Gale transform:

    8. Igor M. Krichever, Commuting difference operators and the combinatorial Gale transform . Translation of Funktsional. Anal. i Prilozhen. 49 (2015), no. 3, 22-40.

  • Athanase Papadopoulos (Strasbourg)
    Title: Teichmüller spaces

    Abstract: I will present several aspects of the metric theory of Teichmüller spaces, with an emphasis on the difference between the cases of surfaces of finite type and of infinite type.