Geometry Days
September
5  7, 2016
University
of Fribourg
Switzerland
Please notice:
Participants should subscribe at the CUSO website [click
here].
Location: Room 2.52, Chemin du Musée 3
(Department of Physics).
Building number 8 on
this map.
Poster: Click
here to download the poster.
Program: You can find the
program here.
Organizers:
Corina Ciobotaru (Fribourg)
Ivan Izmestiev (Fribourg)
Patrick Ghanaat (Fribourg)
Marc Troyanov (EPFL)
Contact: corina.ciobotaru@unifr.ch; ivan.izmestiev@unifr.ch
Supported by:


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
13 pdf, Exercises
13 pdf.
Valentin
Ovsienko (Reims)
 _{Title:
Projective geometry and combinatorics}
 _{Abstract:
The goal of this minicourse is to explain
relations between very classical invariants of
projective geometry, such as the crossratio, 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 MorierGenoud, 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 MorierGenoud, Valentin Ovsienko, Serge
Tabachnikov, 2frieze
patterns and the cluster structure of the space of
polygons. Ann. Inst. Fourier (Grenoble) 62
(2012), no. 3, 937987.}
 _{}_{Classical
articles:}
_{}
 _{3.
H. S. M. Coxeter, Frieze
patterns. Acta Arith. 18 1971 297310.
}
 _{http://matwbn.icm.edu.pl/ksiazki/aa/aa18/aa18132.pdf}
_{}
 _{4.
J. H. Conway, H. S. M. Coxeter, Triangulated
polygons and frieze patterns. Math. Gaz. 57
(1973), no. 401, 175183. }
 _{5.
J. H. Conway, H. S. M. Coxeter, Triangulated
polygons and frieze patterns. Math. Gaz. 57
(1973), no. 400, 8794.}_{}
 _{http://www.link.cs.cmu.edu/15859s11/notes/friezepatternsgazette.pdf}_{}
_{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, 595616.}
 _{A
survey article that stresses a connection with the
representation theory:}
 _{7.
Sophie MorierGenoud, Coxeter's
frieze patterns at the crossroads of algebra,
geometry and combinatorics. Bull. Lond.
Math. Soc. 47 (2015), no. 6, 895938.}
 _{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, 2240.}
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.}
