Abstract: The Gauss-Bonnet equality and inequality and Thurton's lesson: " Use geometry, especially hyperbolic geometry," provide a more elementary proof.
Abstract: Smooth mapping class groups of compact surfaces punctured along Cantor sets have
strong finiteness properties and they are closely related to Thompson groups.
Similar constructions in higher dimensions can be used to recover Brin's finitely
Abstract : By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere
with prescribed n cone singularities of positive curvature is a complex hyperbolic orbifold of dimension n-3. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra,
we observe that this space has a natural decomposition into real hyperbolic convex polyhedra
of dimensions n-3 and \leq 1/2(n-1).
By a result of W.~Veech, the moduli space of flat metrics on a compact surface with prescribed cone singularities of negative curvature has a foliation whose leaves have a local structure of complex
pseudo-spheres, coming again from the area of the metric. The form can be degenerate; its signature depends on the collection of angles.
Using polyhedral surfaces in Minkowski space, we show that this moduli space has a natural decomposition into spherical convex polyhedra.
Abstract: After making a quick survey of the "classical" mirror symmetry
in 90's, I will discuss two interesting examples of complete intersection
Calabi-Yau manifolds (CICYs) which have birational automorphisms
of infinite order. I will describe the mirror symmetry (mirror family) of these
Calabi-Yau manifolds, and observe that the birational automorphisms
correspond nicely to certain monodromy transformations of the family.
If time permit, I will show "Picard-Lefschetz monodromy" which corresponds
to flopping curves. This talk is based on collaborations with Hiromichi Takagi.
Abstract: Teichmüller space is studied and applied from several points of view.
In this talk, I will give my recent progress on an attempt to unify
Topological aspects (Thurston theory) and Complex analytical aspect
(Ahlfors-Bers, Kodaira-Spencer theory).
Abstract: In the last years there arose an interest in definitions and calculations of sectional and Ricci
curvatures of sub-Riemannian manifolds. For examples, one can mention
recent papers of Agrachev-Barilari-Rizzi, Baudoin-Garofalo, Sturm, and others.
I will discuss briefly some classical work papers connected with this subject by Schouten,
Wagner, and especially a later paper by A.F.Solov'ev (1984) for rigged metrized distributions
on Riemannian manifolds. To apply results of these papers, the author suggests to use
in some cases special riggings of corresponding invariant bracket generated
distributions on homogeneous sub-Riemannian manifolds. In particular, this method works
for contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, for horizontal
distribution of natural submersion of full connected semisimple isometry group of Riemannian
symmetric space onto this space.
Abstract: I will describe some manifolds admitting parabolic geometric
structures whose holonomy is a Hitchin or a Quasi-Hitchin
representation. This generalizes the Thurston's theories of Fuchsian
and Quasi-Fuchsian representations to higher rank Lie groups. The
results come from a joint work with Qiongling Li and a joint work with
Sara Maloni and Anna Wienhard.
The study of the Poisson geometry of the Teichmuller space and the moduli space of local systems gave rise to the discovery of the Goldman bracket of curves on an oriented surface which in turn led Chas and Sullivan to discover string topology operations on chains on the free loop space of an arbitrary oriented manifold. Their string topology operations also generalized the Turaev cobracket which did not come from a Poisson geometric origin, and the search for the geometric meaning of all string topology operations continues. In this direction, I will discuss some Poisson geometric aspects of the moduli stack of Z-graded Chen connections and how in the large N-limit an additional relevant structure should appear (N=dimension of the fibre). Unlike the Z-graded case, the somewhat conceptually different Z/2 graded case, studied several years ago with Hossein Abbaspour, did not require the use of derived geometry. The simple reason is that there are very few maps (e.g. traces) from a Z-graded object to a ground ring concentrated in degree zero, whereas in the Z/2 graded setting, viable maps exist. In the derived setting the single ground ring is replaced by the class of all non-positively graded differential graded algebras, with the differential going up towards the origin, as the test objects (i.e. a deformation functor). I plan to review the necessary background material before discussing recent work. This is part of a joint work in progress with Gregory Ginot and Owen Gwilliam.
Globally hyperbolic AdS space-times are Lorentzian manifolds of constant curvature -1
with topological support the product of a surface and the real line.
Once the genus of the surface is fixed, Mess showed that the relevant moduli space is the product of two copies
of the Teichmüller space of the corresponding surface.
In analogy with the quasi-Fuchsian case those Lorentzian manifolds contain a convex core.
In the talk, after briefly revising the theory of GH AdS space-times, I will determine the coarse behavior
of the volume of the convex core in terms of the L^1-energy between the two hyperbolic metrics associated to
the space-time by Mess.
This is a joint work with A. Seppi and A. Tamburelli.
Abstract: The Teichmüller TQFT is a combinatorial model of a three-dimensional TQFT of infinite type where the underlying vector spaces associated with surfaces are infinite dimensional. It is expected to be part of exact quantum Chern-Simons theory with non-compact gauge groups PSL(2,R) and PSL(2,C), the orientation preserving isometry groups in hyperbolic geometries in dimensions 2 and 3 respectively. There exist two versions of the Teichmüller TQFT called « old » and « new » formulations which are not equivalent in general but coincide when restricted to integer homology spheres. The talk is based on the works done in collaboration with Joergen Ellegaard Andersen.
Abstract: The goal of this talk is to describe two different approaches of using harmonic maps into metric spaces of non-positive curvature in the sense of Alexandrov to prove rigidity.
Abstract: In this talk I will revisit the asymptotic structure of the SL(2,C) character variety of a closed surface group. Recent work of Taubes and Mazzeo, et.al. describes the large scale behavior of solutions to the Hitchin equations in terms of certain limiting configurations. I will show how these correspond, via harmonic maps, exactly to Bonahon's parametrization of pleated surfaces in hyperbolic 3-space by transverse and bending cocycles for a geodesic lamination. This is joint work with Andreas Ott, Jan Swoboda, and Mike Wolf.
Abstract: In 1976, W. Thurston published two forms of the h-principle for
foliations of codimension 1, on closed manifolds of all dimensions. I
shall recall his methods and give some more recent improvements in ambiant
dimension at least 4, avoiding the use of Mather's homology equivalence in
the proof, and obtaining foliations with all leaves dense.
Abstract: Based on a collaborative project with Marcus Khuri and Gilbert Weinstein, we construct solutions to the 4+1 dimensional vacuum Einstein equation. We impose stationarity and two axisymmetries of the spacetime which would reduce the Einstein equation to a semi-linear elliptic system, which in turn is identified with the harmonic map equation into a symmetric space. In this construction, we obtain a whole new set of the solutions to the Einstein equation whose blackhole horizons have the entire range of topological types appearing as the irreducible elements in the prime decomposition theorem of three dimensional manifolds.