July 27, 2020 - July 31, 2020
Public Access Category: All |
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- Time:
- 6:30pm-7:30pm
- Description:
- Date and Time: Tuesday 28 July, 6:30 pm IST - 7:30 pm IST (joining time :
6:15 pm IST - 6:30 pm IST)
Google Meet Link: https://meet.google.com/oku-xudb-imy
Speaker: Melvin Hochster, University of Michigan.
Title: Tight Closure, lim Cohen-Maculay sequences, content of local
cohomology, and related open questions - Part 2.
Abstract: The talks will give multiple characterizations of tight closure,
discuss some of its applications, indicate connections with the existence
of big and small Cohen-Macaulay algebras and modules, as well as variant
notions, and also explain connections with the theory of content. There
will be some discussion of the many open questions in the area, including
the very long standing problem of proving that Serre intersection
multiplicities have the behavior one expects.
- Time:
- 5:30pm
- Description:
- Date and Time: Friday 31 July 2020, 5:30 pm IST / 12:00 GMT / 08:00am EDT
(joining time : 5:15 pm IST - 5:30 pm IST)
Google Meet link: https://meet.google.com/xxm-cidr-yqa
Speaker: Neena Gupta, ISI Kolkata.
Title: On the triviality of the affine threefold $x^my = F(x, z, t)$ -
Part 2.
Abstract: In this talk we will discuss a theory for affine threefolds of
the form $x^my = F(x, z, t)$ which will yield several necessary and
sufficient conditions for the coordinate ring of such a threefold to be a
polynomial ring. For instance, we will see that this problem of four
variables reduces to the equivalent but simpler two-variable question as
to whether F(0, z, t) defines an embedded line in the affine plane. As one
immediate consequence, one readily sees the non-triviality of the famous
Russell-Koras threefold x^2y+x+z^2+t^3=0 (which was an exciting open
problem till the mid 1990s) from the obvious fact that z^2+t^3 is not a
coordinate. The theory on the above threefolds connects several central
problems on Affine Algebraic Geometry. It links the study of these
threefolds with the famous Abhyankar-Moh “Epimorphism Theorem” in
characteristic zero and the Segre-Nagata lines in positive characteristic.
We will also see a simplified proof of the triviality of most of the
Asanuma threefolds (to be defined in the talk) and an affirmative solution
to a special case of the Abhyankar-Sathaye Conjecture. Using the theory,
we will also give a recipe for constructing infinitely many counterexample
to the Zariski Cancellation Problem (ZCP) in positive characteristic. This
will give a simplified proof of the speaker's earlier result on the
negative solution for the ZCP.