10:00 to 11:00 
Dror Varolin (Stony Brook University, USA) 
Hilbert Space Techniques in Complex Analysis and Geometry (Lecture 10) This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry. After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds. We shall then develop the twisted estimates of DonnellyFeffernanOhsawa, following an approach that we first learned from Siu. The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi. Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles. An application is a proof of a special case of the L^2 extension theorem but with optimal constant. We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.



11:00 to 12:30 
Problem/Discussion Session 
Problem/Discussion Session 


12:30 to 14:30 
Break 
Lunch 


14:30 to 15:00 
Vikramjeet Singh Chandel (IITB, India) 
The 3point spectral Pick interpolation problem unit ball. The spectral unit ball is the set of all those n n complex matrices that have spectral radius less than 1. The interest in the interpolation problem arises from problems in Control Theory. The spectral unit ball is a nonconvex, unbounded domain for which the group of biholomorphic transformations does not act transitively. Because of this latter fact, the well known Schuralgorithm could not be applied to address the interpolation problem. Bercovici?Foias?Tannenbaum, from an operatortheoretic viewpoint, gave a (somewhat hardtocheck) characterization for the existence of an interpolant under a very mild restriction. Later, Agler?Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc, leading to a necessary condition for the existence of an interpolant. This condition is difficult to check since it involves checking the positive definiteness of infinitely many matrices. In this talk, I shall present a couple of results that are based on one of my recent articles on the interpolation problem. The principal result is a necessary condition for the existence of a 3point holomorphic interpolant F : D →?n, n ≥ 2. This condition is inequivalent to the necessary conditions hitherto known for this problem. The condition generically involves a single inequality and is reminiscent of the Schwarz lemma.



15:00 to 15:30 
Sayani Bera (RKMVERI, Howrah, India) 
Dynamics of polynomial shiftlike maps Existence of nonwandering Fatou component for rational maps in one complex variable is a celebrated result, known due to Sullivan. In this talk, I will discuss the analog questions in higher dimensions, i.e., in $C^2$ for Henon maps and polynomial maps. Next, I will discuss on a nonwandering phenomenon of certain Henon maps, that can be extended to higher dimensions, i.e., on $C^3$ and higher dimensions for the class of shiftlike maps.



15:30 to 16:00 
Break 
Tea/coffee 


16:00 to 17:00 
Dror Varolin (Stony Brook University, USA) 
Hilbert Space Techniques in Complex Analysis and Geometry (Lecture 11) This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry. After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds. We shall then develop the twisted estimates of DonnellyFeffernanOhsawa, following an approach that we first learned from Siu. The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi. Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles. An application is a proof of a special case of the L^2 extension theorem but with optimal constant. We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.


