ESF Mathematics Conference in Partnership with EMS and ERCOM
Teichmüller Theory and its Interactions in Mathematics and Physics
28 June - 3 July 2010
Programme |
This conference is organised in partnership with
| Tuesday 29 June | |
| 08:45 - 09:00 | Welcome from the Chairs and the ESF Rapporteur |
| 09:15 - 10:15 | Norbert A'Campo, Universitaet Basel, CH Minimal mapping classes and monodromy of singularities Positive rational Dehn twist along the boundary and positive rational multi twists of a compact oriented surface extend to special relative mapping classes that can appear as monodromies of singularities if minimal. |
| 10-15 - 10:30 | Coffee break |
| 10:30 - 11:30 | Nathalie Wahl, Copenhagen University, DK Hochschild homology of structured algebras and complexes of graphs Generalizing the work of Costello, we consider the spaces of operations existing on the Hochschild complex of various types of algebras. For "A_\infty-Frobenius" algebras, Costello finds an action of a chain complex computing the homology of the mapping class group of surfaces. We show that this action factors through a smaller complex when the Frobenius algebra is strict, a complex closely related to compactified Sullivan diagrams. The machinery can also be applied for example to finite dimensional algebras, non-compact Frobenius algebras, og associative algebras with a commuting action of some operad. |
11:30 - 12:30 | Ursula Hamenstädt, Bonn University, DE Bowen's construction for the Teichmueller flow We show that Bowen's construction applied to a component of a stratum in the moduli space of quadratic differentials gives rise to the Lebesgue measure. We also discuss several global properties of the flowon strata. |
| 13:00 - 15:30 | Lunch and break |
| 15:30 - 16:00 | Coffee break |
| 16:00 - 17:00 | William Goldman, Maryland University, US 3-dimensional affine space forms and hyperbolic geometry Flat Riemannian manifolds arse quotients of Euclidean space by discrete groups of isometries, and correspond to classical crystallographic groups. Such structures can equivalently be defined as systems of local coordinates into affine space where the coordinate changes are locally isometries. The theorems of Bieberbach provide an effective classification of such structures. Analogous questions for manifolds with flat connections, or equivalently, quotients by groups of affine transformations are considerably more difficult, and presently unsolved. In this talk I will describe how the classification in dimension three, reduces to a question on hyperbolic geometry on open 2-manifolds. This represents joint work with Fried, Drumm, Margulis, Labourie, Charette and Minsky. |
| 17:00 - 18:00 | Kirill Krasnov, University of Nottingham, UK Renormalized volume of hyperbolic 3-manifolds (joint work with J. M. Schlenker and a recent review arXiv:0907.2590) The notion of renormalized volume of, in general, asymptotically hyperbolic Einstein manifolds has origins in the AdS/CFT correspondence of string theory. We review its applications to the case of 3 dimensional Einstein manifolds. These are hyperbolic and completely characterized by the conformal structures of their boundary components. The renormalized volume becomes a function on several copies of Teichmuller space. This very interesting function is related to what in physics is known as the Liouville theory, and can be shown to be a Kahler potential for the Weil-Petersson metric on Teichmuller space. The arising relation between 2-dimensional (boundary) structures and 3-dimensional geometrical ones turns out to be quite useful, for it allows to prove some non-trivial 2-dimensional statements by rather elementary computations done in the 3-dimensional setting. Examples include McMullen's quasifuchsian reciprocity and the fact that Thurston's grafting map is symplectic. |
| 18:00 - 18:30 | Break |
| 18:30 - 19:30 | Vladimir Markovic, Warwick University, UK Random skew-pants in hyperbolic 3-manifolds I will discuss my recent work with Jeremy Kahn on constructing closed essential surfaces in hyperbolic 3-manifold and. I will sketch the main ideas and explain the role that random "long" pants are playing in Riemann surface theory and low dimensional topology. |
| Wednesday 30 June | |
| 09:00 - 10:00 | Jorgen Ellegaard Andersen, Aarhus University, DK TQFT, Hitchin's connection and Toeplitz operators In the talk we will review the geometric gauge theory construction of the vector spaces the Reshetikhin-Turaev TQFT associates to a closed oriented surface. Hence we will build the Hitchin connection in the vector bundles over Teichmüller space, which is obtained by applying geometric quantization to the moduli space of flat connections on the surface. This will be followed by a discussion of the relation between the Toeplitz operator construction and the Hitchin connection. The talk will end with a discussion of various results about the RT-TQFT's which we have proved using these geometric constructions. |
| 10:00 - 10:30 | Coffee break |
| 10:30 - 11:30 | Vladimir Fock, Strasbourg University, FR |
| 11:30 - 13:00 | Two short talks |
| 13:00 | Lunch |
| 16:00 | Excursion in Barcelona - Barrio Gotico |
| 19:30 | Conference dinner |
| Thursday 1 July | |
| 09:00 - 10:00 | Andrei Zelevinsky, Northeastern University, US Cluster algebra formalism: F-polynomials and g-vectors (joint work with H.Derksen and J.Weyman) The formalism of cluster algebras (as well as that of Fock-Goncharov cluster X- and A-varieties) is based on several discrete integrable systems on a n-regular tree. Among these systems a fundamental role is played by the dynamics of F-polynomials (certain integer polynomials in n indeterminates) and g-vectors (certain integer n-vectors) introduced jointly with S.Fomin in the "Cluster Algebras IV" paper. Both F-polynomials and g-vectors have several interesting categorifications. If time allows, we discuss one of them based on quivers with potentials and their representations. |
| 10:00 - 10:30 | Coffee break |
| 10:30 - 11:30 | Anna Wienhard, Princeton University, US Deformation spaces of geometric structures and higher Teichmueller spaces Higher Teichmueller spaces arise as connected components of representation varieties of fundamental groups of surfaces into Lie groups. They share many properties with classical Teichmueller space regarded as subset of representations of the fundamental group of the surface into PSL(2,R). In joint work with Olivier Guichard we interpret higher Teichmueller spaces as deformation spaces of geometric structures on compact bundles over surfaces. |
| 11:30 - 13:00 | Two short talks |
| 13:00 - 15:30 | Lunch and break |
| 15:30 - 16:00 | Coffee break |
| 16:00 - 17:00 | John Smillie, Cornell University, US Polygonal billiards, renormalization and moduli space I will discuss some connections between the dynamics of polygonal billiards and the dynamics of flows on certain moduli spaces. |
| 17:00 - 18:00 | Gabriele Mondello, Università La Sapienza, IT SL(2,R)-flows on the Teichmueller space We construct a family of SL(2,R) local flows on the cotangent space of the Teichmueller space that specialize to the well-known action of SL(2,R) on the space of quadratic differentials. |
| 18:00 - 19:30 | Two short talks |
| Friday 2 July | |
| 09:00 - 10:00 | Benson Farb, University of Chicago, US Representation theory and homoloigcal stability Homological stability is a remarkable phenomenon in the study of groups and spaces. For certain sequences G_n of groups, for example SL(n,Z) or the braid group B_n, it states that the homology group H_i(G_n) does not depend on n once n is large enough. However, there are many natural sequences, from pure braid groups to congruence groups to Torelli groups, for which homological stability fails horribly. In these cases the rank of H_i(G_n) blows up to infinity, and in the latter two cases almost nothing is known about H_i(G_n); indeed it is possible that there is no nice "closed form" for the answers. While doing some homology computations for the Torelli group, Tom Church and I found what looked like the shadow of an overarching pattern. In order to explain it and to formulate a specific conjecture, we came up with a notion of "stability of a sequence of representations of groups". We proved that some sequences, including the homology of pure braid groups and certain Malcev Lie algebras, are representation-stable in this sense. For other cases, including Torelli groups and congruence subgroups, this notion provides natural conjectures (some of which have since been proved by other people). In this talk I will explain our broad picture via some of its many instances, including a connection between representation stability for pure braid groups and the analytic number theory of polynomials over finite fields. No knowledge of representation theory or group homology will be assumed. |
| 10:00 - 10:30 | Coffee break |
| 10:30 - 11:30 | Jean-Marc Schlenker, Université Paul Sabatier, FR |
| 11:30 - 13:00 | Two short talks |
| 13:00 - 15:00 | Lunch and break |
| 15:00 - 15:30 | Coffee break |
| 15:30 - 17:00 | Two short talks |
| 17:00 - 18:00 | Scott Wolpert, University of Maryland, US Understanding Weil-Petersson geometry and curvature A report is presented on the program to describe the intrinsic geometry of the Weil-Petersson metric and geodesic-length functions. Formulas for the metric, covariant derivative and the curvature tensor are presented. The distant sum estimate is included. Recent and new applications are sketched, including results of Liu-Sun-Yau, the Burns-Masur-Wilkinson ergodicity result, an examination of the Yamada model metric and a description of Jacobi fields along geodesics to the boundary. The written report is available at arXiv:0809.3699 |
| Saturday 3 July | |
| Breakfast and Bus Departure at 8:00 and 10:00 | |
