Commutative algebra seminar

Tuesdays from 3.30 pm-4.45 pm

Venue: Ramanujan Hall

Speaker: H. Ananthnarayan

Title: Boij-Soderberg Conjectures and the Multiplicity Conjecture

Abstract: In an article published in 2008, Boij and Soderberg introduced
the notion of a cone related to the graded Betti numbers of a graded
module over the polynomial ring over a field, and stated a couple of
conjectures related to the extremal rays of this cone. They also showed
that a positive answer to these conjectures resolves the Multiplicity
conjecture. Eisenbud-Schreyer (2009) show that the Boij-Soderberg
conjectures are true.

In these talks, we will introduce the multiplicity conjectures, indicate
their connection to the Boij-Soderberg conjectures, and give an idea of 
how
Eisenbud-Schreyer resolve the latter conjectures. We explore similar
results over other standard graded rings.

 Prof. Nitin Nitsure will give the second talk of his lecture series on
'Algebraic Stacks and Moduli Theory' tomorrow (4th Oct) at 5:00 pm. The
talk will be in Ramanujan Hall and of 75 minutes duration.

Speaker: Daniel Erman, University of Wisconsin, Madison
Date/Time: 7 October 2022, 6:30pm IST/ 1:00pm GMT /9:00am ET (joining 
time  6:20 pm IST)

Gmeet link: meet.google.com/jeh-ucaw-ddk [1]

Title:  Matrix factorizations of generic polynomials
Abstract:  I'll discuss the Buchweitz-Greuel-Schreyer Conjecture on the 
minimal size of a matrix factorization, and my recent proof that the 
conjecture holds for generic polynomials.

For more information and links to previous seminars, visit the website 
of VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar [2]



Links:
------
[1] http://meet.google.com/jeh-ucaw-ddk
[2] https://sites.google.com/view/virtual-comm-algebra-seminar

Speaker: Prof. Umesh Dubey: HRI

Title: A functorial construction of moduli of parabolic sheaves.

Abstract:
The moduli construction for vector bundle over smooth projective curves due
to Mumford and Seshadri was extended to moduli of torsion-free sheaves over
higher dimensional varieties by Gieseker and Maruyama. Simpson later
generalized it to the moduli of pure sheaves on higher dimension projective
schemes and Langer extended it to mixed characteristics.
   Alvarez-Consul and King used embedding of the category of regular
sheaves to the category of Kronecker representations to get a functorial
moduli construction of pure sheaves.
 In this talk, we will briefly describe the construction of the Consul and
King. If time permits, we will also mention the related results obtained
jointly with Sanjay Amrutiya for parabolic sheaves using the moduli of
filtered Kronecker representations.

Commutative algebra seminar

Tuesday, 11 October 2022

Time: 3.30 pm-4.45 pm

Venue: Ramanujan Hall

Speaker: H. Ananthnarayan

Title: Boij-Soderberg Conjectures and the Multiplicity Conjecture-II

Abstract:
In an article published in 2008, Boij and Soderberg introduced the 
notion of a cone related to the graded Betti numbers of a graded
module over the polynomial ring over a field, and stated a couple of 
conjectures related to the extremal rays of this cone. They also showed
that a positive answer to these conjectures resolves the Multiplicity 
conjecture. Eisenbud-Schreyer (2009) show that the Boij-Soderberg
conjectures are true.

In these talks, we will introduce the multiplicity conjectures, indicate 
their connection to the Boij-Soderberg conjectures, and give an idea of
how Eisenbud-Schreyer resolve the latter conjectures. We explore similar 
results over other standard graded rings.

This is a reminder for the upcoming talk on Thursday 13 October.
https://sites.google.com/math.iitb.ac.in/geometric-analysis/home

Speaker: Luca Martinazzi (Sapienza University of Rome)

Time: October 13, Thursday, 4 pm (Indian Standard Time)

Title: Critical points of the Moser-Trudinger functional on closed surfaces

Abstract: Given a 2-dimensional closed surface, we will show that the
Moser-Trudinger functional has critical points of arbitrarily high energy.
Since the functional is too critical to directly apply to it the known
variational methods (in particular the Struwe monotonicity trick), we will
approximate it by subcritical ones, which in fact interpolate it to a
Liouville-type functional from conformal geometry. Hence our result will
also unify and give common results for these two apparently unrelated
problems. This is a joint work with F. De Marchis, A. Malchiodi and P-D.
Thizy.

Google Meet joining info
Video call link: https://meet.google.com/cpu-tchr-nvu
Or dial: ‪(US) +1 240-812-1225‬ PIN: ‪262 484 324‬#

Prof. Nitin Nitsure will continue his lecture series on
'Algebraic Stacks and Moduli Theory' tomorrow (13th Oct) at 5:00 pm. The
talk will be in Ramanujan Hall and of 75 minutes duration.

Speaker: Ramya Dutta (TIFR-CAM)

Time: October 14, Friday, 11:30 am
Venue: Room 216

Title:  Apriori decay estimates for Hardy-Sobolev-Maz'ya equations and
application to a Brezis-Nirenberg problem.

Abstract: In this talk we will discuss some qualitative properties and
sharp decay estimates of solutions to the Euler-Lagrange equation
corresponding to Hardy-Sobolev-Mazya inequality with cylindrical weight.
Using these sharp asymptotics we will establish a Brezis-Nirenberg type
existence result for class of $C^1$ sublinear perturbations of the
p-Hardy-Sobolev equation with cylindrical weight in a bounded domain in
dimensions $n > p^2$ and an appropriate notion of positivity for these
perturbations.

Virtual Commutative Algebra Seminar

Speaker: Parnashree Ghosh, Indian Statistical Institute Kolkata, India
Date/Time: 14 October 2022, 5:30pm IST/ 12:00pm GMT /8:00am ET (joining 
time 5:20 pm IST)

Gmeet link: meet.google.com/eap-qswg-xvg [1]

Title: On the triviality of a family of linear hyperplanes
Abstract:  Let k be a field, m a positive integer, V an affine 
subvariety of $A^{m+3}$ defined by a linear relation of the form $x_1^{ 
r_1} · · · x_r^{r_m} y = F(x_1, . . . , x_m, z, t),$ A the coordinate 
ring of V and $G = X_1^{ r_1} · · · X_r^{r_m} Y - F(X_1, . . . , X_m, Z, 
T).$ We exhibit several necessary and sufficient conditions for V to be 
isomorphic $A^{m+2}$ and G to be a coordinate in $k[X_1, . . . , X_m, Y, 
Z, T],$ under a certain hypothesis on F. Our main result immediately 
yields a family of higher-dimensional linear hyperplanes for which the 
Abhyankar-Sathaye Conjecture holds.

We also describe the isomorphism classes and automorphisms of integral 
domains of type A under certain conditions. These results show that for 
each integer d ⩾ 3, there is a family of infinitely many pairwise 
non-isomorphic rings which are counterexamples to the Zariski 
Cancellation Problem for dimension d in positive characteristic.

This is joint work with Neena Gupta.

For more information and links to previous seminars,

visit the website of VCAS: 
https://sites.google.com/view/virtual-comm-algebra-seminar [2]



Links:
------
[1] http://meet.google.com/eap-qswg-xvg
[2] https://sites.google.com/view/virtual-comm-algebra-seminar

Prof MS Raghunathan will give a course of lectures aimed at beginning PhD students

on a topic of basic importance to all of mathematics. Title and abstract are given below.

The course will begin on Monday 17th October at 4pm in the room A1-A2 of CDEEP on the ground floor of the Math building. Each lecture will be of 90 minutes. The course will run

roughly through the middle of December, so about 8 lectures.  Since Monday 24th October is  Deepawali, the lecture will be organized on 26th afternoon for which I will separately announce the precise timing.

It will be a lecture course in hybrid mode so that others not in IIT can also benifit from this course. Feel free to tell your friends in case it may interest them. Here is the zoom link in case you cannot attend in person:

Hope to see you there!

Best wishes, Dipendra

-------------------------------------------------------------------

Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G,G]=G, π1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is a compact. Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially  the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  
I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology. 

Commutative algebra seminar

3.30 pm on Tuesday, 18 October 2022

Venue: Ramanujan Hall

Speaker: H. Ananthnarayan

Title: Boij-Soderberg Theory and multiplicity conjecture-III

Abstract: 
In an article published in 2008, Boij and Soderberg introduced the notion of a cone related to the graded Betti numbers of a graded
module over the polynomial ring over a field, and stated a couple of conjectures related to the extremal rays of this cone. They also showed
that a positive answer to these conjectures resolves the Multiplicity conjecture. Eisenbud-Schreyer (2009) show that the Boij-Soderberg
conjectures are true.

In these talks, we will introduce the multiplicity conjectures, indicate their connection to the Boij-Soderberg conjectures, and give an idea of 
how Eisenbud-Schreyer resolve the latter conjectures. We explore similar results over other standard graded rings.

Department Colloquium: Prof. Chandrashekhar Khare:  University of California Los Angeles

Title: Modularity of elliptic curves over number fields

Abstract: I will give an account of developments arising from Wiles’s  work on modularity of elliptic curves over the rational numbers  and Fermat’s Last Theorem.  I will focus on recently announced results of Ana Cariani and James Newton which prove modularity of all elliptic curves over Gaussian numbers. Their result uses as a key step a result of Patrick Allen, Jack Thorne and myself which proves the modularity of mod 3 representations arising from such elliptic curves. This provides a starting point for Cariani and Newton in the same way as a result of Langlands-Tunnel was a starting point for Wiles.

This lecture will be a guided tour of Wiles’s breakthrough in 1994 and the numerous developments since in this very active area of number theory.

 Prof. Nitin Nitsure will deliver his next talk on Thursday, 20th Oct at
5:00 pm in Ramanujan Hall. Meanwhile he has shared the following link to
the recent notes by Jarod Alper on 'Moduli and Stacks', which has a vast
overlap with what Prof. Nitsure plans to cover. Jarod Alper is one of the
leading experts in this area.

https://sites.math.washington.edu/~jarod/moduli.pdf

We will have *Basudev Pattanayak* speaking in the RTAG seminar from *11AM
to 12:30PM* on *Friday(tomorrow)*.


Here are the necessary details for his talk:

Time: Friday, 21 October, 11:00AM – 12:30 PM.

Venue : Room 215, Department of mathematics.

Title: A Visit to the Local Langlands Conjecture - 4

Abstract: In this series of talks, we first recall some important results
of class field theory. Then we will discuss the representation theory of
p-adic groups. Here we will discuss the Hecke algebra attached to
Bushnell-Kutzko types. With little basic setup, later we will state the
local Langlands Conjecture and its enhancement. For some special cases, we
will discuss their proofs.



We have now a dedicated website where one can find the notes and resources
from the past meets and announcements of the upcoming meetings:
https://sites.google.com/view/rtag/
 

Prof MS Raghunathan will give a course of lectures aimed at beginning PhD students

on a topic of basic importance to all of mathematics. Title and abstract are given below.

The course will begin on Monday 17th October at 4pm in the room A1-A2 of CDEEP on the ground floor of the Math building. Each lecture will be of 90 minutes. The course will run

roughly through the middle of December, so about 8 lectures.  Since Monday 24th October is  Deepawali, the lecture will be organized on 26th afternoon for which I will separately announce the precise timing.

It will be a lecture course in hybrid mode so that others not in IIT can also benifit from this course. Feel free to tell your friends in case it may interest them. Here is the zoom link in case you cannot attend in person:

https://us06web.zoom.us/j/86048537440?pwd=c2FUTXYzOU84dkpPc1NQTGpDSUYvQT09

Meeting ID: 860 4853 7440
Passcode: 156831

Hope to see you there!

Best wishes, Dipendra

-------------------------------------------------------------------

Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G,G]=G, π1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is a compact. Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially  the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  
I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology. 

This is a reminder for the upcoming talk on Thursday 27 October.
https://sites.google.com/math.iitb.ac.in/geometric-analysis/home

Speaker: Panchugopal Bikram  (NISER-Bhubaneswar)

Time: October 27, Thursday, 5 pm (Indian Standard Time)

Title:  On the non-commutative Neveu decomposition and stochastic ergodic
theorems

Abstract: In this talk we discuss the non-commutative analogue of Neveu
decomposition for actions of locally compact amenable groups on finite von
Neumann algebras. In addition, we assume $G = \Z_+$ or $G$ is a locally
compact group of polynomial growth with a symmetric compact generating set
$V$, then for a state preserving action $\alpha$ of $G$ on a finite von
Neumann algebra $M$, discuss the convergence in bilateral almost uniformly
of the ergodic averages associated with the predual action on $M_{*}$
corresponding to the F\o lner sequence $\{K_n\}_{n \in \N}$ (where $K_n =
\{ 0, 1, \ldots n-1 \}$ for $G= \Z_+$ and $K_n = V^n$ otherwise) . At the
end, using these results, we establish the stochastic ergodic theorem.


Google Meet joining info:
Video call link: https://meet.google.com/nir-pyzj-hxf
Or dial: ‪(US) +1 978-615-9747‬ PIN: ‪439 474 149‬#

Virtual commutative algebra seminar

Speaker: Xianglong Ni, University of California, Berkeley, CA, USA

Date/Time: 28 October 2022, 6:30pm IST/ 1:00pm GMT /9:00am ET (joining 
time 6:20 pm IST)

Gmeet link: meet.google.com/ngr-tekr-vfb [1]

Title: Linkage in codimension three
Abstract:  All perfect ideals of codimension two are in the linkage 
class of a complete intersection (licci), but in codimension three and 
beyond this is no longer the case. I will share some ongoing work, joint 
with Lorenzo Guerrieri and Jerzy Weyman, which illustrates how the 
theory of "higher structure maps" originating from Weyman's generic ring 
may be used to distinguish licci ideals within the broader class of 
perfect ideals of codimension three.

For more information and links to previous seminars, visit the website 
of VCAS:

  https://sites.google.com/view/virtual-comm-algebra-seminar [2]

Prof MS Raghunathan will give a course of lectures aimed at beginning PhD students

on a topic of basic importance to all of mathematics. Title and abstract are given below.

The course will begin on Monday 17th October at 4pm in the room A1-A2 of CDEEP on the ground floor of the Math building. Each lecture will be of 90 minutes. The course will run

roughly through the middle of December, so about 8 lectures.  Since Monday 24th October is  Deepawali, the lecture will be organized on 26th afternoon for which I will separately announce the precise timing.

It will be a lecture course in hybrid mode so that others not in IIT can also benifit from this course. Feel free to tell your friends in case it may interest them. Here is the zoom link in case you cannot attend in person:

Hope to see you there!

Best wishes, Dipendra

-------------------------------------------------------------------

Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G,G]=G, π1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is a compact. Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially  the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  
I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology.