- Time:
- 3:30pm - 4:45pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
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.

- Time:
- 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
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.

- Time:
- 6:30pm
- Location:
- Online
- Description:
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

- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
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.

- Time:
- 3:30pm - 4:45pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
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.

- Time:
- 4:00pm
- Location:
- Video call link: https://meet.google.com/cpu-tchr-nvu
- Description:
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#

- Time:
- 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
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.

- Time:
- 11:30am
- Location:
- Room 216, Mathematics Department
- Description:
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.

- Time:
- 5:30pm
- Location:
- Gmeet link: meet.google.com/eap-qswg-xvg
- Description:
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

- Time:
- 4:00pm - 4:45pm
- Location:
- Room No. A1A2, CDEEP, Department of Mathematics
- Description:
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.

- Time:
- 3:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**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.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
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.

- Time:
- 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
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

- Time:
- 11:00am - 12:30pm
- Location:
- Room No 215, Department of Mathematics
- Description:
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/

- Time:
- 4:00pm
- Location:
- Room No. A1A2, CDEEP, Department of Mathematics
- Description:
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: 156831Hope 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.

- Time:
- 5:00pm
- Location:
- Video call link: https://meet.google.com/nir-pyzj-hxf
- Description:
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#

- Time:
- 6:30pm
- Location:
- Gmeet link: meet.google.com/ngr-tekr-vfb
- Description:
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]

- Time:
- 4:00pm - 4:45pm
- Location:
- Room No. A1A2, CDEEP, Department of Mathematics
- Description:
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.