5:30pm» K.N. Raghavan, The Institute of Mathematical Sciences- Time:
- 5:30pm
- Description:
- Date and Time: 2 October 2020, 5:30pm IST/ 12:00GMT / 08:00am EDT
(joining time: 5:15 pm IST - 5:30 pm IST)
Google meet link: meet.google.com/vog-pdxx-fdt
Speaker: K.N. Raghavan, The Institute of Mathematical Sciences
Title: Multiplicities of points on Schubert varieties in the Grassmannian
- Part 2
Given an arbitrary point on a Schubert (sub)variety in a Grassmannian, how
to compute the Hilbert function (and, in particular, the multiplicity) of
the local ring at that point? A solution to this problem based on
"standard monomial theory" was conjectured by Kreiman-Lakshmibai circa
2000 and the conjecture was proved about a year or two later by them and
independently also by Kodiyalam and the speaker. The two talks will be an
exposition of this material aimed at non-experts in the sense that we will
not presume familiarity with Grassmannians (let alone flag varieties) or
Schubert varieties.
There are two steps to the solution. The first translates the problem from
geometry to algebra and in turn to combinatorics. The second is a solution
of the resulting combinatorial problem, which involves establishing a
bijection between two combinatorially defined sets. The two talks will
roughly deal with these two steps respectively.
Three aspects of the combinatorial formulation of the problem (and its
solution) are noteworthy: (A) it shows that the natural determinantal
generators of the tangent cone (at the given point) form a Groebner basis
(in any "anti-diagonal" term order); (B) it leads to an interpretation of
the multiplicity as counting certain non-intersecting lattice paths; and
(C) as was observed by Kreiman some years later, the combinatorial
bijection is a kind of Robinson-Schensted-Knuth correspondence, which he
calls the "bounded RSK".
Determinantal varieties arise as tangent cones at points on Schubert
varieties (in the Grassmannian), and thus one recovers multiplicity
formulas for these obtained earlier by Abhyankar and Herzog-Trung. (The
multiplicity part of the Kreiman-Lakshmibai conjecture was also proved by
Krattenthaler, but by very different methods.)
What about Schubert varieties in other (full or partial) flag varieties
(G/Q with Q being a parabolic subgroup of a reductive algebraic group G)?
The problem remains open in general, even for the case of the full flag
variety GL(n)/B, although there are several papers over the last two
decades by various authors using various methods that solve the problem in
various special cases. Time permitting, we will give some indication of
these results, without however any attempt at comprehensiveness.
- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: Mainak Poddar, IISER Pune
Time: Monday 5th October 4 to 5pm (joining time 3.50pm IST)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: Equivariant splitting of toric principal bundles over projective
spaces.
Abstract: I will describe a classification of torus equivariant principal
G-bundles over a complex nonsingular toric variety where G is a complex
linear algebraic group. I will discuss a connection between their
equivariant automorphisms and equivariant reduction of structure group.
Using this we will show the existence of a torus equivariant splitting of
such a bundle over the projective space of dimension n when G is a
reductive subgroup of GL(r) for r < n. This generalizes a theorem of
Kaneyama on the existence of equivariant splitting of any torus
equivariant vector bundle of rank r < n over a projective space of
dimension n. The talk is based on joint works with Indranil Biswas, Jyoti
Dasgupta, Arijit Dey and Bivas Khan.
- Time:
- 5:30pm
- Description:
- Date/Time: 6 October 2020, 5:30pm IST/ 12:00GMT / 08:00am EDT (joining
time: 5:15 pm IST - 5:30 pm IST)
Google meet link: meet.google.com/wif-eiof-jvd
Speaker: Mrinal Das, ISI Kolkata
Title: Some open problems in projective modules and complete intersections
Abstract: Consider a surjective $k$-algebra ($k$ field) morphism from a
polynomial ring of $n$ variables to a polynomial ring of $m$ variables
over $k.$ Is the kernel generated by $n - m$ elements? Our discussion will
primarily be around this question and its variants.
- Time:
- 7:00pm
- Description:
- Title: Matrix orbit closures and their Hilbert functions.
Speaker: Alex Fink, Queen Mary University of London.
Time: 7pm IST (gate opens 6:45 pm IST).
Google Meet Link: meet.google.com/upg-tmyo-ekw.
Phone: (US) +1 929-266-1977 PIN: 832 926 004#.
Abstract: If an ordered point configuration in projective space is
represented
by a matrix of coordinates, the resulting matrix is determined up to
the action of the general linear group on one side and the torus of
diagonal matrices on the other. We study orbits of matrices under the
action of the product of these groups. The main question is what
properties of closures of these orbits, or quotients in other ambient
spaces, are determined by the matroid of the point configuration. The
main result is that the finely-graded Hilbert function is so
determined in characteristic 0 (we think also in general).
The results of mine in this talk are mostly joint with Andy Berget.
- Time:
- 5:30pm
- Description:
- Date/Time: 9 October 2020, 5:30pm IST/ 12:00GMT / 08:00am EDT (joining
time: 5:15 pm IST - 5:30 pm IST)
Google meet link: meet.google.com/wif-eiof-jvd
Speaker: Sarang Sane, IIT Madras
Title: $K_0$ and ideals
Abstract: We begin by discussing $K_0$ and defining $K_1$ for a ring $R$
and the exact sequence connecting them on localization with respect to a
multiplicative set $S$. More generally, there is a similar localization
exact sequence for an open set $V(I)^c$ of Spec(R) connecting $K_0$ and
$K_1$, and we relate the properties of the ideal $I$ with the intermediate
term in the sequence.
- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: V Balaji, CMI, Chennai
Time: Monday 12th October 4 to 5pm (joining time 3.45 pm IST)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: Torsors on semistable curves and the problem of degenerations.
Abstract: Let G be an almost simple, simply connected algebraic group over
the field of complex numbers. In this talk I will discuss a basic question
in the classification of G-torsors on curves, which is to construct a flat
degeneration of the moduli stack G-torsors on a smooth projective curve
when the curve degenerates to an irreducible nodal curve.The question has
a long history and I will discuss the classical results and discuss the
new difficulties and solutions.
- Time:
- 5:30pm
- Description:
- Speaker: Kamran Divaani Aazar, IPM Tehran.
Date/Time: 13 October 2020, 5:30pm IST/ 12:00GMT / 08:00am EDT (joining
time: 5:15 pm IST - 5:30 pm IST)
Google meet link: https://meet.google.com/fwn-qimp-rxk
Title: A survey on the finiteness properties of local cohomology modules -
Part 1
Abstract:
The theory of local cohomology has been developed significantly during the
six decades of research after its introduction by Grothendieck. The study
of the finiteness properties of local cohomology modules initiated, in
1962, with a question asked by Grothendieck in his algebraic geometry
seminar.
In this survey talk, first, we will recall the basic definitions and
properties in the theory of local cohomology. Then, we shall list some
major problems on the finiteness properties of local cohomology modules.
Next, we will focus on the notion of cofiniteness. We shall continue by
examining some generalizations of local cohomology modules, and the
reformulation of finiteness properties for them.
To study finiteness properties for the widest generalization of local
cohomology modules, if time allows, at the end of the second session, we
will review the derived category approach to local cohomology.
- Time:
- 5:30pm
- Description:
- Speaker: Kamran Divaani Aazar, IPM Tehran.
Date/Time: 16 October 2020, 5:30pm IST/ 12:00GMT / 08:00am EDT (joining
time: 5:15 pm IST - 5:30 pm IST)
Google meet link: https://meet.google.com/fwn-qimp-rxk
Title: A survey on the finiteness properties of local cohomology modules -
Part 2
-------------------------------------------
Abstract:
The theory of local cohomology has been developed significantly during the
six decades of research after its introduction by Grothendieck. The study
of the finiteness properties of local cohomology modules initiated, in
1962, with a question asked by Grothendieck in his algebraic geometry
seminar.
In this survey talk, first, we will recall the basic definitions and
properties in the theory of local cohomology. Then, we shall list some
major problems on the finiteness properties of local cohomology modules.
Next, we will focus on the notion of cofiniteness. We shall continue by
examining some generalizations of local cohomology modules, and the
reformulation of finiteness properties for them.
To study finiteness properties for the widest generalization of local
cohomology modules, if time allows, at the end of the second session, we
will review the derived category approach to local cohomology.
- Time:
- 11:30am
- Description:
- Date and Time: Monday 19 October, 11.30am
Speaker: Nitin Tomar
Google Meet Link: https://meet.google.com/afe-nzqz-sgt
Title: Pro C* algebras
Abstract: Pro C*-algebras are the generalisation of C*-algebras in which the C*-norm is replaced by a family of C*-seminorms. In this talk, we discuss the analogy between pro C*-algebra and C*-algebra. Also, we discuss that any pro C* algebra arises as the projective limit of C*-algebras (hence, the name). With this observation, we write the spectrum of a pro C*-algebra and see how it differs from its C*-algebra counterpart. We will also see necessary and sufficient conditions under which a pro C*-algebra isomorphic to a C*-algebra.
- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: Joachim Jelisiejew, University of Warsaw, Poland
Time: Monday 19th October 4 to 5pm (joining time 3.45pm IST)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: Questions and recent results on the Hilbert scheme of points.
Abstract: In the talk I will present the open questions and state of the
art on Hilbert schemes of points, focusing on the most accessible
questions. In one sentence: the Hilbert scheme parameterizes deformations
of finite rank algebras, which accounts for its usefulness, while it is
highly singular which accounts for the difficulties, jointly these
features make its investigation a very active area.
- Time:
- 4:00pm
- Description:
- Date and Time: Wednesday 21 October, 04.00pm
Speaker: Saad Qadri
Google Meet Link: https://meet.google.com/tbg-fghh-nmg
Title: Lindemann-Weierstrass theorem
Abstract: A (complex) number is said to be algebraic (over rationals) if it satisfies a nonzero polynomial
equation with integer coefficients. A number that is not algebraic is said to be transcendental. Our goal
in this talk will be to prove the Lindemann Weierstrass theorem which states that if b_j's are distinct algebraic numbers then exp(b_j)'s are linearly independent over the field of algebraic numbers (over Q). This gives as its corollary the fact that pi and e are transcendental.
- Time:
- 6:30pm
- Description:
- Date and Time: 23 October 2020, 6:30pm IST/ 1:00pm GMT / 09:00am EDT
(joining time: 6:15 pm IST - 6:30 pm IST)
Speaker: Jack Jefferies, University of Nebraska-Lincoln, NE, USA
Google meet link: meet.google.com/hzo-wzpe-tht
Title: Faithfulness of top local cohomology modules in domains.
Abstract: Inspired by a question of Lynch, we consider the following
question: under what conditions is the highest non-vanishing local
cohomology module of a domain R with support in an ideal I, faithful as an
R-module? We will review some of what is known about this question, and
provide an affirmative answer in positive characteristic when the
cohomological dimension is equal to the number of generators of the ideal.
This is based on joint work with Mel Hochster.
- Time:
- 4:00pm
- Description:
- Date and Time: Saturday 24 October, 04.00pm
Speaker: Ashish Shukla
Google Meet Link: https://meet.google.com/tbg-fghh-nmg
Title: Representation theory of the symmetric group
Abstract: We give a glimpse into the representation theory of the symmetric group (S_n). Here we
begin by establishing the basics of representation theory by addressing questions such as: what is a
representation? what is a module? how many irreducible representations are there? etc. We then
answer these questions for the symmetric group. We define certain terminologies, building the basics
of representation theory from introductory knowledge of linear algebra and group theory. We explore
an intimate connection between Young tableaux and representations of the symmetric group. We
describe the construction of Specht modules which are irreducible representations of S_n
- Time:
- 11:30pm
- Description:
- Date and Time: Saturday 24 October, 11.30am
Speaker: Deep Makadiya
Google Meet Link: https://meet.google.com/afe-nzqz-sgt
Title: Schreier’s theorem
Abstract: For any group G, Schreier theorem states that any two subnormal series of G have isomorphic refinements. This is one of the fundamental results in group theory. The proof involves another interesting lemma called Butterfly Lemma (also known as Zassenhaus Lemma). As a consequence of Schreier's theorem, we shall also outline a proof of Jordan-Hölder theorem for composition series.
- 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.
- Time:
- 11:30pm
- Description:
- Date and Time: Saturday 31 October, 11.30am
Speaker: Sai Krishna
Google Meet link: https://meet.google.com/afe-nzqz-sgt
Title: Krull's Generalized Principal Ideal Theorem
Abstract: In this talk, we will discuss the proof of Krull's Generalized Principal Ideal Theorem without
using the the Dimension theorem. We will briefly introduce the notion of Krull dimension and prove the
Principal ideal theorem and the generalized version. We will look at a few consequences of this result