


Analysis Seminar
Thursday, 09/03/2023, 11.30 am
Venue: Ramanujan Hall
Host: Prachi Mahajan
Speaker: Ratna Pal
Affiliation: IISER Mohali
Title: Rigidity properties of Henon maps in $\mathbb{C}^2$ and Short $\mathbb{C}^2$.
Abstract: The broad research area of my talk is Complex Dynamics in Several Variables. Classically complex dynamics was studied for rational endomorphisms of the Riemann sphere. In the past three decades, this field of research has flourished to a great extent and the holomorphic dynamics in higher dimensions has attracted a lot of attention. In particular, the dynamics of the polynomial automorphisms in higher dimensions mushroomed as one of the central themes of study. In $\mathbb{C}^2$, the most important polynomial automorphisms are the Henon maps and in this talk they will play the role of the protagonist. In the first part of the talk, we shall see a couple of rigidity properties of Henon maps. Loosely speaking, by rigidity properties we mean those properties of Henon maps which determine the underlying Henon maps almost uniquely. In the latter part of the talk, we shall survey a few recent results obtained for Short $\mathbb{C}^2$'s. A Short $\mathbb{C}^2$ is a proper domain of $\mathbb{C}^2$ that can be expressed as an increasing union of unit balls (up to biholomorphism) such that the Kobayashi metric vanishes identically, but allows a bounded above plurisubharmonic function. The sublevel sets of the Green's functions of Henon maps are classical examples of Short $\mathbb{C}^2$'s. Note that the Green's function of a Henon map $H$ is the global plurisubharmonic functions on $\mathbb{C}^2$ which is obtained by measuring the normalized logarithmic growth rate of the orbits of points in $\mathbb{C}^2$ under the iterations of the Henon map $H$. In this part of the talk, we shall first see a few interesting natural properties of Short $\mathbb{C}^2$'s. Then we give an effective description of the automorphism groups of the sublevel sets of Green's functions of Henon maps (recall that the sublevel sets of the Green's functions of Henon maps are Short $\mathbb{C}^2$'s). It turns out that the automorphism groups of this class of Short $\mathbb{C}^2$'s are not very large. Thus it shows that, unlike in a bounded setup, although the Euclidean balls have large automorphism groups, the automorphism group of an increasing union of balls (up to biholomorphism) might flatten out when the final union is unbounded. A part of the results which will be presented in this talk is obtained in several joint works with Sayani Bera, John Erik Fornaess and Kaushal Verma.
Commutative algebra Seminar
Thursday, 09/03/2023, 4 pm
Venue: Ramanujan Hall
Host: Tony J. Puthenpurakal
Speaker: Prof. R. V. Gurjar
Affiliation: Former Professor, IIT Bombay
Title: Positively Graded domain
Abstract: I will continue my lectures on this topic. Following results will be discussed. 1. Demazure's construction of normal affine positively graded domains. Some applications of this will be discussed. 2. Flenner and Keiichi Watanabe's rationality of singularities criterion for positively graded affine domains. 3. A very general result I conjectures around 1990 and proved by O.Mathieu In 2002 will be discussed. It has some new consequence for rings of invariants of reductive algebraic group action on an affine space. 4. Divisor Class Groups of positively draded domains. Works of Brieskon Flenner, Samuel, SchejaStorch, Anurag Singh etc, will be mentioned. Connection with Topology of these results will be discussed.