Wednesday September 20, 2017 2:30 PM  3:30 PM Math Tower P131
 Tali Pinsky, TIFR
On the volumes of modular geodesicsA closed geodesic gamma on the modular surface is a knot in the complement of a trefoil knot in S^3. It is always a hyperbolic knot, and thus has a well defined volume. I will show how to construct the modular template, due to Ghys, and use it to obtain an upper bound for this volume.

Wednesday September 27, 2017 2:30 PM  3:30 PM Math Tower P131
 Liviana Palmisano, Institut MittagLeffler
The rigidity conjectureA central question in dynamics is whether the topology of a system determines its geometry, whether the system is rigid. Under mild topological conditions rigidity holds in many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. We will discuss the case of circle maps with a flat interval. The class of maps with Fibonacci rotation numbers is a C1 manifold which is foliated with co dimension three rigidity classes. Finally, we summarize the known nonrigidity phenomena in a conjecture which describes how topological classes are organized into rigidity classes.

Wednesday October 04, 2017 2:30 PM  3:30 PM Math Tower P131
 Adi Glücksam, Tel Aviv University
Measurably entire functions and their growthLet (X,B,P) be a standard probability space. Let T:C> PPT(X) be a free action of the complex plane on the space (X,B,P). We say that the function F:X> C is measurably entire if it is measurable and for Pa.e x the function F_x(z):=F(T_zx) is entire. B. Weiss showed in '97 that for every free C action there exists a nonconstant measurably entire function. In this talk I will present upper and lower bounds for the growth of such functions. The talk is partly based on a joint work with L. Buhovsky, A.Logunov, and M. Sodin.

Wednesday October 11, 2017 2:30 PM  3:30 PM Math Tower P131
 Carlos Cabrera, Instituto de Matemáticas UNAM, Unidad Cuernavaca
On the Hurwitz class of rational mapsIn this talk we discuss both geometrical and algebraic aspects of Hurwitz classes of rational maps. The first aspect is related to cobordisms of rational maps while the other is given in terms of isomorphisms of sandwich semigroups. In particular, sandwich semigroups characterize also the conjugacy relation on rational maps.

Wednesday November 08, 2017 2:30 PM  3:30 PM Math Tower P131
 Leon Takhtajan, Stony Brook University
Schwarz equation and symplectic geometry of the space of complex projective structuresThe space of complex projective connections on Riemann surfaces is an affine bundle, modeled on the holomorphic cotangent bundle of the moduli space, and carries a holomorphic symplectic form  a differential of the Liouville 1form on the cotangent bundle. The monodromy map is a holomorphic mapping of this space into the PSL(2,C)character variety of a Riemann surface. It has a natural symplectic form, introduced by Bill Goldman for compact surface. The PSL(2,C)character is nonabelian analog of a C*character variety, the set of onedimensional representations of the fundamental group of a surface. The symplectic nature of the latter variety was studied by Riemann.
I will explain how variational theory of Schwarz equation allows to determine the differential of the monodromy map and to prove the nonabelian analog of the Riemann bilinear relations, the relation between symplectic structures on the holomorphic tangent bundle of the moduli space of Riemann surfaces and on the PSL(2,C)character variety of a given surface. This relation holds for the general case of orbifold Riemann surface.

Wednesday November 15, 2017 2:30 PM  3:30 PM Math Tower P131
 Leon Takhtajan, Stony Brook University
Schwarz equation and symplectic geometry of the space of complex projective structuresThe space of complex projective connections on Riemann surfaces is an affine bundle, modeled on the holomorphic cotangent bundle of the moduli space, and carries a holomorphic symplectic form  a differential of the Liouville 1form on the cotangent bundle. The monodromy map is a holomorphic mapping of this space into the PSL(2,C)character variety of a Riemann surface. It has a natural symplectic form, introduced by Bill Goldman for compact surface. The PSL(2,C)character is nonabelian analog of a C*character variety, the set of onedimensional representations of the fundamental group of a surface. The symplectic nature of the latter variety was studied by Riemann.
I will explain how variational theory of Schwarz equation allows to determine the differential of the monodromy map and to prove the nonabelian analog of the Riemann bilinear relations, the relation between symplectic structures on the holomorphic tangent bundle of the moduli space of Riemann surfaces and on the PSL(2,C)character variety of a given surface. This relation holds for the general case of orbifold Riemann surface.

