- 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