Seminar on Complex Analysis and Complex Geometry - Fall 2015

On Wednesday when no extramural speaker is listed, the members of the complex geometry group may meet for a working seminar. For more information, send an email to Gordon Heier.

Wednesday, September 16, 2015, 11am-12noon, PGH 646

Title: Informal introductory lecture on stability and the Yau-Tian-Donaldson conjecture

Speaker: Xiaowei Wang, Rutgers University-Newark

Abstract: See title.

Wednesday, September 16, 2015, 3pm-4pm, PGH 646 (COLLOQUIUM)

Title: Moduli space of Fano Kahler-Einstein manifolds

Speaker: Xiaowei Wang, Rutgers University-Newark

Abstract: In this talk, we will discuss our construction of compact Hausdorff Moishezon moduli spaces parametrizing smoothable K-stable Fano varieties. The solution relies on the recent solution of Yau-Tian-Donaldson conjecture by Chen-Donaldson-Sun and Tian. In particular, we prove the uniqueness of the degeneration of Fano Kahler-Einstein manifolds and more algebraic properties that are needed to construct an algebraic moduli space. (This is a joint work with Chi Li and Chenyang Xu).

Friday, September 18, 2015, 10am-11am, PGH 646 (NOTE THE SPECIAL TIME)

Title: Informal introductory lecture on stability and the Yau-Tian-Donaldson conjecture, part II

Speaker: Xiaowei Wang, Rutgers University-Newark

Abstract: See title.

Wednesday, October 21, 2015, 3pm-4pm, PGH 646 (COLLOQUIUM)

Title: Totally geodesic subvarieties in the moduli space of Riemann surfaces

Speaker: Ronen Mukamel, Rice University

Abstract: The moduli space of Riemann surfaces carries a natural Teichmuller metric. Amazingly, there is an explicit description of the geodesics for this metric. In this talk, we will describe some subvarieties of moduli space which are totally geodesic for the Teichmuller metric, and we will discuss some of the remarkable properties enjoyed these subvarieties.

Wednesday, October 28, 2015, time: 10-11am, room: PGH 646

Title: Dynamical Approach in CR-geometry and Applications

Speaker: Ilya Kossovskiy, University of Vienna

Abstract: Study of equivalences and symmetries of real submanifolds in complex space goes back to the classical work of Poincar\'e and Cartan and was further developed in later work of Tanaka and Chern and Moser. This work initiated far going research in the area (since 1970's till present), which is dedicated to questions of regularity of mappings between real submanifolds in complex space, unique jet determination of mappings, solution of the equivalence problem, and study of automorphism groups of real submanifolds.

Current state of the art and methods involved provide satisfactory (and sometimes complete) solution for the above mentioned problems in nondegenerate settings. However, very little is known for more degenerate situations, i.e., when real submanifolds under consideration admit certain degeneracies of the CR-structure.

The recent CR (Cauchy-Riemann Manifolds) - DS (Dynamical Systems) technique, developed in our joint work with Shafikov and Lamel, suggests to replace a real submanifold with a CR-singularity by an appropriate dynamical systems. This technique has recently enabled us to solve a number of long-standing problems in CR-geometry, in particular, related to a Conjecture by Treves and that by Ebenfelt and Huang. The technique also has applications to Dynamics. In this talk, we give an overview of the technique and the results obtained recently by using it.