Tue, November 10, 2020
Public Access

Category: All

November 2020
Mon Tue Wed Thu Fri Sat Sun
2 3 4 5 6 7 8
9 10 11 12 13 14 15
16 17 18 19 20 21 22
23 24 25 26 27 28 29
11:00am [11:00am] Hiranya Kishore Dey
Pre-synopsis seminar Student: Hiranya Kishore Dey Date and Time: Tuesday, 10 November 2020 at 11.00am Title: Descents, Excedances and Alternating-runs in Positive elements of Coxeter Groups Google Meet Link: https://meet.google.com/cpj-fbho-apd All interested are cordially invited.

5:00pm [5:30pm] Amartya Kumar Datta, ISI Kolkata
Date and Time: 10 November 2020, 5:30pm IST/ 12:00GMT/07:00am EDT (joining time 5:15pm - 5:30pm IST) Speaker: Amartya Kumar Datta, ISI Kolkata Google meet link: https://meet.google.com/jom-etrz-bdd Title: G_a-actions on Affine Varieties: Some Applications - Part 1 Abstract: One of the hardest problems that come up in affine algebraic geometry is to decide whether a certain d-dimensional factorial affine domain is ``trivial'', i.e., isomorphic to the polynomial ring in d variables. There are instances when the ring of invariants of a suitably chosen G_a-action has been able to distinguish between two rings (i.e., to prove they are non-isomorphic), when all other known invariants failed to make the distinction. It was using one such invariant that Makar-Limanov proved the non-triviality of the Russell-Koras threefold, leading to the solution of the Linearization Problem; and again, it was using an invariant of G_a-actions that Neena Gupta proved the nontriviality of a large class of Asanuma threefolds leading to her solution of the Zariski Cancellation Problem in positive characteristic. G_a actions are also involved in the algebraic characterisation of the affine plane by M. Miyanishi and the algebraic characterisation of the affine 3-space.by Nikhilesh Dasgupta and Neena Gupta. Miyanishi's characterisation had led to the solution of Zariski's Cancellation Problem for the affine plane. Using G_a-actions, a simple algebraic proof for this cancellation theorem was obtained three decades later by Makar-Limanov. In this talk (in two parts), we will discuss the concept of G_a-actions along with the above applications, and the closely related theme of Invariant Theory. The concept of G_a-action can be reformulated in the convenient ring-theoretic language of ``locally nilpotent derivation'' (in characteristic zero) and ``exponential map'' (in arbitrary characteristic). The ring of invariants of a G_a- action corresponds to the kernel of the corresponding locally nilpotent derivation (in characteristic zero) and the ring of invariants of an exponential map. We will recall these concepts. We will also mention a theorem on G_a actions on affine spaces (or polynomial rings) due to C.S. Seshadri. We will also discuss the close alignment of the kernel of a locally nilpotent derivation on a polynomial ring over a field of characteristic zero with Hilbert's fourteenth problem. While Hilbert Basis Theorem had its genesis in a problem on Invariant Theory, Hilbert's fourteenth problem seeks a further generalisation: Zariski generalises it still further. The connection with locally nilpotent derivations has helped construct some low-dimensional counterexamples to Hilbert's problem. We will also mention an open problem about the kernel of a locally nilpotent derivation on the polynomial ring in four variables; and some partial results on it due to Daigle-Freudenburg, Bhatwadekar-Daigle, Bhatwadekar-Gupta-Lokhande and Dasgupta-Gupta. Finally, we will state a few technical results on the ring of invariants of a G_a action on the polynomial ring over a Noetherian normal domain, obtained by Bhatwadekar-Dutta and Chakrabarty-Dasgupta-Dutta-Gupta.