October 26, 2020 - October 30, 2020
Public Access Category: All |
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- Time:
- 11:30am
- Description:
- Date and Time: Monday 26 October, 11.30am
Speaker: Omkar Javadekar
Google Meet link: https://meet.google.com/afe-nzqz-sgt
Title: Serre’s conjecture for projective modules
Abstract: Also known as Quillen-Suslin theorem, Serre's conjecture is a result concerning the
relationship between free and projective modules over polynomial rings. It states that every finitely
generated projective module over a polynomial ring over a field is free. The statement was conjectured
by Serre in 1955, and the first proofs were given independently by Quillen and Suslin in 1976. In this
talk we will see a proof of Serre's conjecture.
We begin by defining unimodular extension property. We then show that polynomial rings have unimodular extension property. Finally, appealing to the result that finitely generated projective modules over polynomial rings are stably free, we conclude the proof of Serre's conjecture by showing that stably free modules over a ring having unimodular extension property are free.
- Time:
- 4:00pm-5:00pm
- Description:
- Speaker: Praveen Roy, TIFR
Time: Monday 26th October 4 to 5pm (joining time 3.50pm IST)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: Seshadri Constant on Surfaces.
Abstract: Seshadri constant is a tool to study/quantify the positivity of
a line bundle on a projective variety. It was defined by Demailly in late
80s to study the Fujita conjecture, but afterwards it arose as an
independent area or research with computing and bounding the constant as
some of the main topics of research. In this talk we will see some of such
results obtained on Hyperelliptic surfaces and on surfaces of general
type.
- Time:
- 6:30pm
- Description:
- Speaker: Satya Mandal, The University of Kansas, KS, USA
Date/Time: 27 October 2020, 6:30pm IST/ 1:00pm GMT/ 9:00am EDT (joining
time: 6:15 pm IST - 6:30 pm IST)
Google meet link: meet.google.com/zcj-xnpb-ffo
Title: Quillen $K$-Theory: A reclamation in Commutative Algebra - Part 1
Abstract: In these two talks I take a pedagogic approach to Quillen
$K$-theory. What it takes to teach (and learn) Quillen $K$-theory? I am at
the tail end of completing a book on this, which would eventually be
available through some outlet. This is based on a course I taught. Current
version has nearly 400 pages, in eleven chapters. I finish with Swan’s
paper on quadrics. I tried to do it in a reader friendly way, and tried to
avoid expressions like “left to the readers”. I would give an overview and
a road map.
To justify the title, let me remind you that $K$-theory used to be part of
Commutative algebra. In this endeavor, I consolidate the background
needed, in about 100 pages, for a commutative algebraist to pick up the
book and give a course, or learn. There is a huge research potential in
this direction. This is because, with it, topologists have done what they
are good at. However, these higher $K$-groups have not been described in a
tangible manner. That would be the job of commutative algebraist, and
would require such expertise.
- Time:
- 4:00pm
- Description:
- Date and Time: Wednesday 28 October, 04.00pm
Speaker: Uttam Ojha
Google Meet link: https://meet.google.com/tbg-fghh-nmg
Title: Hensel's Lemma
Abstract: We begin with the notions of completion of a module and completeness. Then we prove Hensel's Lemma for a complete ring and deduce the Implicit function theorem as a Corollary.
- Time:
- 6:00pm
- Description:
- Speaker name: Professor Yves Benoist of University of Paris - Saclay.
The title of the talk is: Harish-Chandra tempered representations and homogeneous spaces
The video talk premieres on Wednesday, 28th October, 2020 at 6:00 PM, Indian Standard Time (IST), at: https://youtu.be/Y5dbZLZkqLQ
The duration of the talk is about 45 minutes and it will continue to remain available at the above link after its initial release tomorrow.
A live interaction with Prof. Yves Benoist is scheduled on Wednesday, 11th November, 2020 at 6:00 PM (IST).
After viewing the video talk, you are welcome to send questions or comments for the speaker by filling out the Google form at the link below:
https://forms.gle/HbYDntDKXwaboNsR9
Please be sure to send the questions/comments before November 9, 2020, 06:00 PM IST
- Time:
- 5:30pm
- Description:
- Speaker: N. V. Trung, Institute of Mathematics, Hanoi, Vietnam
Date/Time: 29 October 2020, 5:30pm IST/ 12:00 GMT/ 8:00am EDT (joining
time: 5:15 pm IST - 5:30 pm IST)
Google meet link: meet.google.com/zcj-xnpb-ffo
Title: Multiplicity sequence and integral dependence
Abstract: The first numerical criterion for integral dependence was proved
by Rees in 1961 which states that two m-primary ideals $I \subset J$ in an
equidimensional and universally catenary local ring $(R, m)$ have the same
integral closure if and only if they have the same Hilbert-Samuel
multiplicity. This result plays an important role in Teissier's work on
the equisingularity of families of hypersurfaces with isolated
singularities. For hypersurfaces with non-isolated singularities, one
needs a similar numerical criterion for integral dependence of
non-$m$-primary ideals. Since the Hilbert-Samuel multiplicity is no longer
defined for non-$m$-primary ideals, one has to use other notions of
multiplicities that can be used to check for integral dependence. A
possibility is the multiplicity sequence which was introduced by Achilles
and Manaresi in 1997 and has its origin in the intersection numbers of the
Stuckrad-Vogel algorithm. It was conjectured that two arbitrary ideals $I
\subset J$ in an equidimensional and universally catenary local ring have
the same integral closure if and only if they have the same multiplicity
sequence. This talk will present a recent solution of this conjecture by
Polini, Trung, Ulrich and Validashti.
- Time:
- 6:30pm
- Description:
- Speaker: Satya Mandal, The University of Kansas, KS, USA
Date/Time: 30 October 2020, 6:30pm IST/ 1:00pm GMT/ 9:00am EDT (joining
time: 6:15 pm IST - 6:30 pm IST)
Google meet link: meet.google.com/zcj-xnpb-ffo
Title: Quillen $K$-Theory: A reclamation in Commutative Algebra - Part 2
Abstract: In these two talks I take a pedagogic approach to Quillen
$K$-theory. What it takes to teach (and learn) Quillen $K$-theory? I am at
the tail end of completing a book on this, which would eventually be
available through some outlet. This is based on a course I taught. Current
version has nearly 400 pages, in eleven chapters. I finish with Swan’s
paper on quadrics. I tried to do it in a reader friendly way, and tried to
avoid expressions like “left to the readers”. I would give an overview and
a road map.
To justify the title, let me remind you that $K$-theory used to be part of
Commutative algebra. In this endeavor, I consolidate the background
needed, in about 100 pages, for a commutative algebraist to pick up the
book and give a course, or learn. There is a huge research potential in
this direction. This is because, with it, topologists have done what they
are good at. However, these higher $K$-groups have not been described in a
tangible manner. That would be the job of commutative algebraist, and
would require such expertise.