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9:00am 


10:00am 


11:00am 
[11:00am] Hiranya Kishore Dey
 Description:
 Presynopsis seminar
Student: Hiranya Kishore Dey
Date and Time: Tuesday, 10 November 2020 at 11.00am
Title: Descents, Excedances and Alternatingruns in Positive elements of
Coxeter Groups
Google Meet Link: https://meet.google.com/cpjfbhoapd
All interested are cordially invited.


12:00pm 


1:00pm 


2:00pm 


3:00pm 


4:00pm 


5:00pm 
[5:30pm] Amartya Kumar Datta, ISI Kolkata
 Description:
 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/jometrzbdd
Title: G_aactions 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 ddimensional 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_aaction has been able to distinguish between two rings (i.e., to prove they are nonisomorphic), when all other
known invariants failed to make the distinction. It was using one such
invariant that MakarLimanov proved the nontriviality of the
RussellKoras threefold, leading to the solution of the Linearization
Problem; and again, it was using an invariant of G_aactions 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 3space.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_aactions, a simple algebraic proof for
this cancellation theorem was obtained three decades later by
MakarLimanov.
In this talk (in two parts), we will discuss the concept of G_aactions
along with the above applications, and the closely related theme of
Invariant Theory. The concept of G_aaction can be reformulated in the
convenient ringtheoretic 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 lowdimensional 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 DaigleFreudenburg, BhatwadekarDaigle,
BhatwadekarGuptaLokhande and DasguptaGupta. 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
BhatwadekarDutta and ChakrabartyDasguptaDuttaGupta.


6:00pm 

