Reading Seminar on Geometry

The Reading Seminar on Geometry aims to introduce and study interesting current topics in geometry which is interpreted quite broadly. It is open to anyone interested in related fields and encourage informal discussions between students, researchers and professors. It is organized by the research team of Prof. Jean-Marc Schlenker.

Fall Semester sessions of the Academic year 2017-2018

Organizers: Andrew Yarmola

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There will be two parallel reading groups this semester. One group will meet at 11:00 on Tuesdays to go over Random walks on the mapping class group by Maher. The second groupp will meet after lunch on Tuesdays at 13:30 for a minicourse on symmetric spaces.

References

• Maher, Joseph Random walks on the mapping class group, Duke Math. J. Volume 156, Number 3 (2011), 429-468.
• Paulin, Frédéric Groupes et géométries, 2013-2014
• Pecastaing, Vincent Riemannian Symmetric Spaces
• Eberlein, Patrick Geometry of Nonpositively Curved Manifolds, University of Chicago Press, 1996

Upcoming sessions

• Tuesday 24 October, 11h00 to 12h50, Venue: Room MNO 6B
  Funda Gultepe (University of Luxembourg).
  Funda will be continuing our seminar on the paper Random walks on the mapping class group by Maher. The focus this week will be on the outline of the proof.
  Tuesday 24 October, 13h00 to 14h00, Venue: Room MNO 6B
  Vincent Pecastaing (University of Luxembourg).

  Abstract : In this talk, I will try to illustrate what we have done so far with different examples. I will focus on Weyl chambers in symmetric spaces, and give illustration of classical reduced abstract root systems as restricted root systems of classical real-split simple Lie algebras, and also describe some non-split examples. We then will examine the action of the isotropy on Weyl chambers and relate it to the Weyl group, and we will see it explicitly on the examples.

Past sessions

• Tuesday 16 October, 11h00 to 12h30, Venue: Room MNO 6B
  Funda Gultepe (University of Luxembourg).
  Funda will be continuing our seminar on the paper Random walks on the mapping class group by Maher. The focus this week will be on the the curve complex and relative hyperbolicity.
  Tuesday 316 October, 13h30 to 14h00, Venue: Room MNO 6B
  Vincent Pecastaing (University of Luxembourg).
  Vincent will be continuing a reading course on symmetric spaces following the note Groupes et géométries of Frédéric Paulin and Geometry of Nonpositively Curved Manifolds by Patrick Eberlein.
• Monday 9 October, 15h00 to 16h00, Venue: Room MNO 6A
  Olivier Glorieux (University of Luxembourg).
  Olivier will give the second talk for the minicourse on symmetric spaces. His talk will borrow from Geometry of Nonpositively Curved Manifolds by Partick Eberlein.
  Tuesday 3 October, 11h00 to 12h30, Venue: Room MNO 6B
  Binbin Xu (University of Luxembourg).
  Binbin will give the second talk on the paper Random walks on the mapping class group.
• Tuesday 3 October, 11h00 to 12h30, Venue: Room MNO 6B
  Binbin Xu (University of Luxembourg).
  Binbin will give the first talk on the paper Random walks on the mapping class group.
  Tuesday 3 October, 13h30 to 14h00, Venue: Room MNO 6B
  Vincent Pecastaing (University of Luxembourg).
  Vincent will give the first talk for the minicourse on symmetric spaces. His talk will follow the notes Groupes et géométries by Frédéric Paulin.

Spring Semester sessions of the Academic year 2016-2017

Organizers: Andrew Yarmola and Vincent Pecastaing

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The first half of the semester will be devoted to the proof of the Zimmer Conjecture.

References

• Aaron Brown, David Fisher, Sebastian Hurtado Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T), 2016.

Past sessions

• Thursday 6 April, 11h00 to 12h30, Venue: Room S.5A
  Olivier Glorieux (University of Luxembourg).
  Olivier Glorieux will give the second talk on uniform subexponential growth of derivatives. A first step in the proof of the Zimmer Conjecture.
• Thursday 30 March, 11h00 to 12h30, Venue: Room S.5A
  Olivier Glorieux (University of Luxembourg).
  Olivier Glorieux will give the first talk on uniform subexponential growth of derivatives. A first step in the proof of the Zimmer Conjecture.
• Thursday 23 March, 11h00 to 12h30, Venue: Room S.5A
  Vincent Pecastaing (University of Luxembourg).
  Vincent Pecastaing will give the second talk in a series on the Zimmer Conjecture. He will talk about super rigidity of cocycles.
• Thursday 09 March, 11h00 to 12h30, Venue: Room S.5A
  Vincent Pecastaing (University of Luxembourg).
  Vincent Pecastaing will give the first talk in a series on the Zimmer Conjecture. He will provide background and talk about super rigidity of cocycles.

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Autumn Semester sessions of the Academic year 2016-2017

Organizers: Daniel Monclair and Andrew Yarmola

Unless otherwise specified, the reading group will meet on Tuesday, 12h30 - 14h00, Room G002
Contact: Daniel Monclair (daniel.monclair -at- uni.lu) or Andrew Yarmola (andrew.yarmola -at- uni.lu)

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The first sessions will be devoted to the pressure metric on Teichmüller space, finishing the series of talks on the pressure metric for Anosov representations.

References

• Curtis McMullen , Thermodynamics, dimension and the Weil-Petersson metric, 2006.

After this, we will study the recent solution by Benoist and Hulin of the Schoen-Li-Wang conjecture: every quasiisometric map between rank one symmetric spaces is within bounded distance of a (unique) harmonic map.

References

• Yves Benoist, Domique Hulin , Quasiisometric harmonic maps between rank one symmetric spaces, 2015.
• P. Li and J. Wang. Harmonic rough isometries into Hadamard space. Asian J. Math., 2:419–442, 1998
• P. Pansu. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math., 129:1–60, 1989.

Past sessions

• Tuesday 17 January, 12h30 to 14h00, Venue: Room G002
  Daniel Matei (IMAR Bucharest).
  Analytic and topological torsion

  Abstract: We discuss the Ray-Singer analytic torsion and the Reidemeister-Franz topological torsion of odd-dimensional manifolds with a focus on dimension 3, and their connections to volume, homology growth and various zeta functions.

• Tuesday 20 December, 12h30 to 14h00, Venue: Room G002
  Qiyu Chen (University of Luxembourg).
  Heat kernel and determinant of the Laplacian on finite area Euclidean sectors.

  Abstract: The Thurston metric was originally introduced by Thurston for the study of hyperbolic surfacesof finite type without boundary. For surfaces of finite type with boundary, a modification of the Thurstonmetric, the so-called arc metric, was studied.It is a naturalproblem whether the Thurston metric is well-defined in Teichmüller spaces of surfaces of infinite type. We will show how to generalize thearc metric to hyperbolic surfaces of infinite type with boundary, under a geometric condition ( introduced by K. Matsuzaki). We will also present several examples for these surfaces anddiscuss the relationbetween this geometric condition andthe Shiga's condition.This is a joint work with L. Liu.

• Tuesday 13 December, 12h30 to 14h00, Venue: Room G002
  Andrea Tamburelli (University of Luxembourg).
  The volume of the convex core of GHMC Anti-de Sitter 3-manifolds.

  Abstract: Motivated by the work of Brock that provides estimates for the volume of the convex core of quasi-Fuchsian manifolds in terms of the Weil-Petersson distance of the conformal structures at infinity, we study the analogous question for the convex core of GHMC AdS manifolds. We will show that the volume of the convex core is roughly equivalent to the L^1 distance between the left and right metrics and we will then compare it to more classical distances (Teichmuller, Thurston and Weil-Petersson) defined on Teichmuller space. This is a joint work with F. Bonsante and A. Seppi.

• Thursday 29 November, 12h30 to 14h00, Venue: Room G002
  Andrew Yarmola (University of Luxembourg).
  Quasiisometric harmonic maps between rank one symmetric spaces - VI
• Thursday 22 November, 12h30 to 14h00, Venue: Room G002
  Andrew Yarmola (University of Luxembourg).
  Quasiisometric harmonic maps between rank one symmetric spaces - V
• Tuesday 15 November, 12h30 to 14h00, Venue: Room G002
  Clara Aldana (University of Luxembourg).
  Heat kernel and determinant of the Laplacian on finite area Euclidean sectors.

  Abstract: I will talk about a variational Polyakov formula for the determinant of the Laplace operator on finite area Euclidean sectors. This formula shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle of the sector. I will not concentrate on the differentiation of the determinant (or the spectral zeta function) but in the actual computation of the variation in terms of the trace of the heat operator multiplied by the operator multiplication by the conformal factor. I will explain how we compute explicitly all the contributions to this formula coming from the different parts of the sector. Time permitting, I will mention how to compute the zeta-regularized determinant of rectangular domains of unit area and how this is uniquely maximized by the square. The results presented in the talk are in collaboration with Julie Rowlett.

• Thursday 3 November, 14h00 to 15h30, Venue: Room G002
  Andrew Yarmola (University of Luxembourg).
  Quasiisometric harmonic maps between rank one symmetric spaces - IV
• Tuesday 18 October, 12h30 to 14h00, Venue: Room G002
  Qiyu Chen (University of Luxembourg).
  Quasiisometric harmonic maps between rank one symmetric spaces - III
• Tuesday 11 October, 11h30 to 13h00, Venue: Room G002
  Qiyu Chen (University of Luxembourg).
  Quasiisometric harmonic maps between rank one symmetric spaces - II
• Tuesday 4 October, 11h30 to 13h00, Venue: Room G002
  Andrea Tamburelli (University of Luxembourg).
  Quasiisometric harmonic maps between rank one symmetric spaces - I
• Tuesday 27 September, 13:45 to 15:15, Venue: Room G002
  Nicolas Tholozan (University of Luxembourg).
  The Pressure Metric on Teichmüller Space - II
• Tuesday 20 September, 11:30 to 13:00, Venue: Room G004
  Nicolas Tholozan (University of Luxembourg).
  The Pressure Metric on Teichmüller Space - I

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Spring Semester sessions of the Academic year 2015-2016

Organizer: Jean-Marc Schlenker, Daniel Monclair and Hengnan Hu

Unless otherwise specified, the reading group will meet on Wednesday, 3:30 p.m - 4:30 p.m. at Room B14 (Campus Kirchberg)
Contact: Hengnan Hu (hengnan.hu -at- uni.lu)

References

• Martin Bridgeman, Richard Canary and Andrés Sambarino, An introduction to pressure metrics on higher Teichmüller spaces, 2015, (ArXiv).
• Martin Bridgeman, Richard Canary, Francois Labourie and Andrés Sambarino, The pressure metric for Anosov representations, 2013, (ArXiv).
• Francois Labourie, Anosov Flows, Surface Groups and Curves in Projective Space, 2004, (ArXiv).
• Rufus Bowen, The Equidistribution of Closed Geodesics, 1972, (American Journal of Mathematics).
• Rufus Bowen, Periodic Orbits for Hyperbolic Flows, 1972, (American Journal of Mathematics).

Upcoming sessions

• 22 June (Wednesday), 3:30 p.m - 4:30 p.m, Venue: Room B14
  Nicolas Tholozan (University of Luxembourg).
  Introduction to Anosov representation - V

Past sessions

• 11 May (Wednesday), 3:30 p.m - 4:30 p.m, Venue: Room B14
  Daniel Monclair (University of Luxembourg).
  Introduction to Anosov representation - IV
• 4 May (Wednesday), 3:30 p.m - 4:30 p.m, Venue: Room B14
  Daniel Monclair (University of Luxembourg).
  Introduction to Anosov representation - III
• 27 Apr (Wednesday), 3:30 p.m - 4:30 p.m, Venue: Room B14
  Daniel Monclair (University of Luxembourg).
  Introduction to Anosov representation - II
• 20 Apr (Wednesday), 3:30 p.m - 4:30 p.m, Venue: Room B14
  Daniel Monclair (University of Luxembourg).
  Introduction to Anosov representation - I
• 11 Apr (Monday), 2 p.m - 3 p.m, Venue: Room A02
  Daniel Matei ( IMAR, Bucharest).
  An Introduction to character varieties - II
• 6 Apr (Wednesday), 3:30 p.m - 4:30 p.m, Venue: Room B14
  Daniel Matei ( IMAR, Bucharest).
  An Introduction to character varieties - I

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Autumn Semester sessions of the Academic year 2015-2016

Organizer: Jean-Marc Schlenker, Daniel Monclair and Hengnan Hu

Unless otherwise specified, the reading group will meet on Wednesday, 1:30 p.m - 2:30 p.m. at Room B21
Contact: Hengnan Hu (hengnan.hu -at- uni.lu)

References

• Martin Bridgeman, Richard Canary and Andrés Sambarino, An introduction to pressure metrics on higher Teichmüller spaces, 2015, (ArXiv).
• Martin Bridgeman, Richard Canary, Francois Labourie and Andrés Sambarino, The pressure metric for Anosov representations, 2013, (ArXiv).
• Francois Labourie, Anosov Flows, Surface Groups and Curves in Projective Space, 2004, (ArXiv).
• Rufus Bowen, The Equidistribution of Closed Geodesics, 1972, (American Journal of Mathematics).
• Rufus Bowen, Periodic Orbits for Hyperbolic Flows, 1972, (American Journal of Mathematics).

Past sessions

• 2 Dec (Wednesday), 1:30 p.m - 2:30 p.m, Venue: Room B21
  Nicolas Tholozan (University of Luxembourg).
  Introduction to Anosov representation - III
• 25 Nov (Wednesday), 1:30 p.m - 2:30 p.m, Venue: Room B21
  Nicolas Tholozan (University of Luxembourg).
  Introduction to Anosov representation - II
• 11 Nov (Wednesday), 1:30 p.m - 2:30 p.m, Venue: Room B21
  Nicolas Tholozan (University of Luxembourg).
  Introduction to Anosov representation - I
• 4 Nov (Wednesday), 1:30 p.m - 2:30 p.m, Venue: Room B21
  Andrea Tamburelli (University of Luxembourg).
  Geodesic flows on hyperbolic surfaces - II
• 28 Oct (Wednesday), 1:30 p.m - 2:30 p.m, Venue: Room B21
  Andrea Tamburelli (University of Luxembourg).
  Geodesic flows on hyperbolic surfaces - I
• 21 Oct (Wednesday), 1:30 p.m - 2:30 p.m, Venue: Room B21
  Jean-Marc Schlenker (University of Luxembourg).
  Introduction to pressure metric - II
• 07 Oct (Wednesday), 1:30 p.m - 2:30 p.m, Venue: Room B21
  Jean-Marc Schlenker (University of Luxembourg).
  Introduction to pressure metric - I

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Spring Semester sessions of the Academic year 2014-2015

Organizer: Jean-Marc Schlenker, Jérémy Toulisse and Nicolas Tholozan

Unless otherwise specified, the reading group will meet on on Monday, 2:00 p.m to 3:30 p.m Room B14.
Contact: Jérémy Toulisse

Past sessions

• 28 May (Thursday) 2 p.m at Venue: Room ?
  Clara Aldana (University of Luxembourg).
  Markovich's proof of Schoen's conjecture.
• 18 May (Monday) at Venue: Room B21
  Jérémy Toulisse (University of Luxembourg).
  Geometrization of the Hitchin component in PSL(4,R) - II.
• 14 May (Thursday) 2 p.m at Venue: Room B24A
  Jérémy Toulisse (University of Luxembourg).
  Geometrization of the Hitchin component in PSL(4,R) - I.
• 30 April (Thursday) 2 p.m at Venue: Room B24A
  Jean-Marc Schlenker (University of Luxembourg).
  Anti-de Sitter and Margulis spacetimes (after Danciger-Guéritaud-Kassel).
• 30 March (Monday)
  Nicolas Tholozan (University of Luxembourg).
  Closed anti-de Sitter spacetimes (after Guéritaud-Kassel).
• 23 March (Monday)
  Jean-Marc Schlenker (University of Luxembourg).
  Introduction to anti-de Sitter geometry in dimension 3

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Other Seminars

• Geometry and Topology seminar.
• Informal lunch seminar Computer Science meets Mathematics.
• Colloquium.
• General Mathematics Seminar.
• Seminar for PhD candidates.