Commutative algebra and algebraic Geometry seminar Speaker: R. V. Gurjar Dates: Tuesday, 8 and 15 November 2022 Time: 3.30-5.00 pm Venue: Ramanujan Hall Title : Positively Graded Domains Abstract : We will discuss positively graded affine domains over complex field from algebraic, geometric, and topological viewpoints. Important results by M. Demazure, I. Dolgachev, H. Flenner, S. Goto, A. Grothendieck, S. Mori, W. Neumann, P. Orlik- P. Wagreich, H. Pinkham, Keiichi Watanabe will be mentioned. A very general result "conjectured" by me in 1990 and proved by O. Mathieu around 2003 will be discussed. It has important consequences for rings of invariants of reductive algebraic groups. Many naturally occurring examples of positively graded domains will be discussed. If time permits, I will mention closely related results proved recently by A. Pramanik-S. Thandar-R.V. Gurjar about affine surfaces with finite fundamental group at infinity.