BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Amartya Datta (ISI\, Kolkata) DTSTART:20201110T120000Z DTEND:20201110T130000Z DTSTAMP:20260423T021007Z UID:VCAS/33 DESCRIPTION:Title: G_ a-actions on Affine Varieties: Some Applications - Part 1\nby Amartya Datta (ISI\, Kolkata) as part of IIT Bombay Virtual Commutative Algebra Se minar\n\n\nAbstract\nOne of the hardest problems that come up in affine al gebraic geometry is to decide whether a certain d-dimensional factorial a ffine domain is ``trivial''\, i.e.\, isomorphic to the polynomial ring in d variables. There are instances when the ring of invariants of a suitabl y 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 faile d to make the distinction. It was using one such invariant that Makar-Lim anov 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 l arge class of Asanuma threefolds leading to her solution of the Zariski Ca ncellation Problem in positive characteristic.\n\nG_a actions are also inv olved in the algebraic characterisation of the affine plane by M. Miyanish i and the algebraic characterisation of the affine 3-space.by Nikhilesh D asgupta and Neena Gupta. Miyanishi's characterisation had led to the solut ion of Zariski's Cancellation Problem for the affine plane. Using G_a-act ions\, a simple algebraic proof for this cancellation theorem was obtaine d three decades later by Makar-Limanov.\n\nIn this talk (in two parts)\, w e will discuss the concept of G_a-actions along with the above application s\, 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- acti on corresponds to the kernel of the corresponding locally nilpotent deriva tion (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. \n\nWe will also discuss the close alignment of the kernel of a locally n ilpotent derivation on a polynomial ring over a field of characteristic ze ro 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 l ow-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 ri ng of invariants of a G_a action on the polynomial ring over a Noetherian normal domain\, obtained by Bhatwadekar-Dutta and Chakrabarty-Dasgupta-Dut ta-Gupta.\n LOCATION:https://researchseminars.org/talk/VCAS/33/ END:VEVENT END:VCALENDAR