- Time:
- 11:30am
- Description:
**Algebraic geometry seminar****Date**Tuesday, 2 April, 11.30 am

**Venue**Room 215

**Host**Sudarshan Gurjar

**speaker**Nitin Nitsure

**Affiliation**Bhaskaracharya Pratishthana, Pune

**Title**Overview and summary of my 50 lectures on Algebraic Stacks and

Moduli spaces, 2022-24.**Abstract**I gave a series of about 50 lectures on Algebraic Stacks and Moduli spaces in the Department of Mathematics, IIT-B, spread across four semesters from 2022 to 2024. This is the final lecture of this series, in which I will summarise the main themes, and make suggestions for further study.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Algebraic groups Seminar****Date**Tuesday, April 2, 2024, 4 pm

**Venue**Ramanujan Hall

**Host**Shripad M. Garge

**speaker**Akash Yadav

**Affiliation**IIT Bombay

**Title**Connected solvable groups

**Abstract**We begin studying connected, solvable, linear algebraic groups starting with the Lie-Kolchin theorem.

- Time:
- 4:00pm - 5:00pm
- Location:
- Room 215, Mathematics Department
- Description:
**Commutative Algebra Seminar****Date**Tuesday, 2 April, 2024, 4-5 pm

**Venue**Room 215

**Host**Tony J. Puthenpurakal

**speaker**Om Prakash

**Affiliation**IIT Bombay

**Title**Numerical Semigroups and associated Semigroup Rings-I

**Abstract**In this series of two lectures, we will study numerical semigroups and their associated semigroup rings. Initially, we will define numerical semigroups, state their fundamental properties, and introduce relevant invariants. Subsequently, we aim to prove the following fundamental results: (i) The Frobenius number of a numerical semigroup S equals the degree, viewed as a rational function, of the Hilbert series of the numerical semigroup ring k[S]. (ii) The Cohen-Macaulay type of the numerical semigroup ring $k[S]$ corresponds to the number of pseudo-Frobenius elements of $S$. Consequently, we derive a well-known result concerning Gorenstein numerical semigroup rings (credited to Kunz) asserting that k[S] is Gorenstein if and only if S is symmetric.

- Time:
- 4:00pm
- Location:
- Room No 105, Department of Mathematics
- Description:
**Algebraic groups Seminar****Date**Thursday, 4 April, 4 pm

**Venue**Room 105

**Host**Dipendra Prasad

**speaker**Deep Makadiya

**Affiliation**IIT Bombay

**Title**Regular elements of semi-simple algebraic groups

**Abstract**We will continue with the seminar on Algebraic groups reading the paper of Robert Steinberg, Regular elements of semi-simple algebraic groups Publications mathématiques de l’I.H.É.S., tome 25 (1965), p. 49-80

- Time:
- 10:15am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Topology and Related Topics Seminar****Date**Friday, 5 April 2024, 10.15 am

**Venue**Ramanujan Hall

**Host**Rekha Santhanam

**speaker**Sudarshan Gurjar

**Affiliation**IIT Bombay

**Title**Vector bundles and Characteristic Classes

**Abstract**This is the third talk in the series of three talks. We will give an introduction to the characteristic classes of a vector bundle. Characteristic classes are invariants of a vector bundle taking values in the singular cohomology of the base and satisfying the obvious functoriality property concerning pullback. They are the measure of the non-triviality of the vector bundle.

**The background assumed will depend on the audience present.**

- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Algebraic geometry seminar****Date**Friday, 5 April, 11.30 am

**Venue**Ramanujan Hall

**Host**Sudarshan Gurjar

**speaker**Nitin Nitsure

**Affiliation**Bhaskaracharya Pratishthana, Pune

**Title**Introduction to the Tohoku (1957) paper of Grothendieck-part 1

**Abstract**In 1957, Alexander Grothendieck published a long paper in

two parts in the Tohoku Mathematical Journal, Japan,

whose title can be translated as `On some points of homological

algebra'. This paper revolutionized the subject of homological

algebra and it sheds unifying light on topics in diverse areas such as group theory, Lie theory, number theory, algebraic topology, etc. It had profound applications to Algebraic Geometry, which were made apparent by Grothendieck and his school in the next few decades. The ideas of abelian categories and additive functors between them, and their derived functors, come from this paper. The famous `Grothendieck spectral sequence' first appeared here. I will give a series of two talks, introducing this paper's ideas and reporting some subsequent developments.

The first talk will assume no prior knowledge except of the basics of groups, rings, and modules at an undergraduate level. In it,

I will explain the historical context, and introduce the paper's themes in simple terms. The second talk will assume some

acquaintance with Algebraic Geometry.

- Time:
- 4:00pm - 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Analysis seminar****Date**Friday, April 5, 4 pm - 5 pm

**Venue**Ramanujan Hall

**Host**Chandan Biswas

**speaker**Prachi Mahajan

**Affiliation**IIT Bombay

**Title**The Squeezing function & the Fridman invariant

**Abstract**The squeezing function and its dual, the Fridman invariant, are biholomorphic invariants, both of which capture the coarse metric geometry of the given domain. I will describe some results on the squeezing function and Fridman invariant such as their boundary behavior, their utility in classifying the unit ball under various hypotheses, and estimates near the boundary of the given domain. In the second part, I will compare this pair of invariants by showing that they are both equally capable of determining the boundary geometry of a bounded domain if their boundary behavior is apriori known.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Date**Monday, April 8, 2024, 4 pm

**Venue**Ramanujan Hall

**Host**Shripad M. Garge

**speaker**Akash Yadav

**Affiliation**IIT Bombay

**Title**Connected solvable groups-II

**Abstract**We prove the structure theorem for connected solvable linear algebraic groups over an algebraically closed field, that any such group G is a semi-direct product of its unipotent radical, [G, G], and any one of the maximal tori of G.

- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Algebraic geometry seminar****Date**Wednesday, 10 April 11.30 am

**Venue**Ramanujan Hall

**Host**Sudarshan Gurjar

**speaker**Nitin Nitsure

**Affiliation**Bhaskaracharya Pratishthana, Pune

**Title**Introduction to the Tohoku (1957) paper of Grothendieck-part 1

**Abstract**In 1957, Alexander Grothendieck published a long paper in

two parts in the Tohoku Mathematical Journal, Japan,

whose title can be translated as `On some points of homological

algebra'. This paper revolutionized the subject of homological

algebra shed unifying light on topics in diverse areas such as group theory, Lie theory, number theory, algebraic topology, etc. It had profound applications to Algebraic Geometry, which were made apparent by Grothendieck and his school in the next few decades. The ideas of abelian categories and additive functors between them, and their derived functors, come from this paper. The famous `Grothendieck spectral sequence' first appeared here. I will give a series of two talks, introducing the ideas of this paper, and also reporting some subsequent developments.

The first talk will assume no prior knowledge except of the basics of groups, rings and modules at an undergraduate level. In it,

I will explain the historical context, and introduce the themes of the paper in simple terms. The second talk will assume some

acquaintance with Algebraic Geometry.

- Time:
- 4:00pm
- Location:
- Room No 105, Department of Mathematics
- Description:
**Algebraic groups Seminar****Date**Friday, 12 April, 4 pm

**Venue**Room 105

**Host**Dipendra Prasad

**speaker**Dibyendu Biswas

**Affiliation**IIT Bombay

**Title**Regular elements of semi-simple algebraic groups

**Abstract**We will continue with the seminar on Algebraic groups by reading the paper of Robert Steinberg, Regular elements of semi-simple algebraic Groups Publications mathématiques de l’I.H.É.S., tome 25 (1965), p. 49-80

- Time:
- 11:30am - 12:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Topology and Related Topics Seminar

Tuesday, 16 April 2024, 11:30 am-12:30 pm

=========================

Venue: Ramanujan hall

Host: Rekha Santhanam

Speaker: Bittu Singh

Affiliation: IIT Bombay

Title: Topological Hochschild homology

Abstract: This is the first of a series of two talks. We will discuss cyclic homology, Symmetric monoidal category of spectra and S^1 action on a cyclic set.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Algebraic Groups Seminar

Tuesday, April 16, 2024, 4 pm

====================Venue: Ramanujan Hall

Host: Shripad M. Garge

Speaker: Akash Yadav

Affiliation: IIT Bombay, Mumbai

Title: Borel and parabolic subgroups

Abstract: We complete the 6th chapter of Springer's book with some important properties of Borel and parabolic subgroups of linear algebraic groups.

- Time:
- 4:00pm - 5:00pm
- Location:
- Room 215, Mathematics Department
- Description:
Commutative Algebra Seminar

Speaker: Om Prakash

Affiliation: IIT Bombay

Host: Tony J. Puthenpurakal

Time: Tuesday, 16 April 2024, 4:00-5:00 pm

Venue: Room # 215

Title: Commutative Algebra SeminarTitle: Numerical Semigroups and associated Semigroup Rings-II.

Abstract: In this series of two lectures, we will study numerical

semigroups and their associated semigroup rings. Initially, we will define

numerical

semigroups, state their fundamental properties, and introduce relevant

invariants. Subsequently, we aim to prove the following fundamental

results: (i) The Frobenius number of a numerical semigroup S equals the

degree, viewed as a rational function, of the Hilbert series of the

numerical semigroup ring k[S]. (ii) The Cohen-Macaulay type of the

numerical semigroup ring k[S] corresponds to the number of

pseudo-Frobenius elements of S. Consequently, we derive a well-known

result concerning Gorenstein numerical semigroup rings (credited to

Kunz) asserting that k[S] is Gorenstein if and only if S is symmetric.

- Time:
- 10:00am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Speaker: Anamay Tengse

Affiliation: Reichman University, Herzliya

Date/Venue: 18 April (Thursday), 10 AM.

Title: Equations for efficiently computable polynomials

Abstract: Algebraic circuits are a natural model for computing

polynomials, as they essentially capture the minimum number of

sums and products required to evaluate a polynomial at any given

input. In algebraic complexity theory, circuits are used to study

how the cost of computing a polynomial varies as a function of its

number of variables. For instance, the nxn symbolic determinant has

a cost that is a polynomial in n, and is therefore 'efficiently

computable'. A central object of interest here, is the class

of (sequences of) polynomials that are efficiently computable.The 'algebraic P vs NP' question, asks whether there are 'explicit' polynomials

that are not efficiently computable. This question was posed in a work of

Valiant's in 1979, and remains open to this date. In fact, the best 'lower

bound' against circuits is Omega(n log n), which is a result from 1983.

A lot of works have therefore focussed on more structured forms of

circuits, hoping that the techniques there could eventually be generalized.

Super-polynomial lower bounds are now known against many of these

structured models. However, since these results have not yet lead to any

better lower bounds against general circuits, some recent works have

studied a 'meta-question': are these techniques fundamentally incapable of

leading to circuit lower bounds?In particular, the works of Forbes, Shpilka and Volk (2018), and Grochow,

Kumar, Saks and Saraf (2017), observed that most of the current techniques

in fact yield 'efficiently computable equations' for the sets of

polynomials that are computable by the corresponding models. A nautral

question therefore, is whether there are such equations for circuits. In

joint works with Chatterjee, Kumar, Saptharishi and Ramya, we shine some

light on the classes that have (or do not have) such efficiently computable

equations. In the talk, we will first briefly introduce the relevant

concepts from algebraic complexity, and then go over some of our findings

- Time:
- 11:30am - 12:30pm
- Location:
- Room 215, Mathematics Department
- Description:
Topology and Related Topics Seminar

Friday, 19 April 2024, 11:30 am-12:30 pm

=========================

Venue: 215

Host: Rekha Santhanam

Speaker: Bittu Singh

Affiliation: IIT Bombay

Title: Topological Hochschild homology

Abstract: This is the second of a series of two talks. We will discuss cyclic homology, Symmetric monoidal category of spectra and S^1 action on a cyclic set.

- Time:
- 4:00pm
- Location:
- Room No 105, Department of Mathematics
- Description:
Algebraic groups Seminar

Date : Friday, 19 April, 4 pm

Venue: Room 105

Host: Dipendra Prasad

speaker: Mohammed Saad Qadri

Affiliation: IIT BombayTitle: Regular elements of semi-simple algebraic groups

Abstract: We will continue with the seminar on Algebraic groups by reading the paper of Robert Steinberg, Regular elements of semi-simple algebraic Groups Publications mathématiques de l’I.H.É.S., tome 25 (1965), p. 49-80

- Time:
- 11:00am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Algebraic Groups Seminar (Shripad)

Tuesday, April 23, 2024, 11 am

====================Venue: Ramanujan Hall

Host: Shripad M. Garge

Speaker: Akash Yadav

Affiliation: IIT Bombay, Mumbai

Title: Borel and parabolic subgroups II

Abstract: We complete the 6th chapter of Springer's book with some important properties of Borel and parabolic subgroups of linear algebraic groups.

- Time:
- 11:00am
- Location:
- Room No 105, Department of Mathematics
- Description:
Algebraic groups Seminar (Dipendra)

Date : Wednesday, 11 AM.

Venue: Room 105

Host: Dipendra Prasad

speaker: Chayan Karmakar

Affiliation: IIT BombayTitle: Regular elements of semi-simple algebraic groups

Abstract: We will continue with the seminar on Algebraic groups by reading the paper of Robert Steinberg, Regular elements of semi-simple algebraic Groups Publications mathématiques de l’I.H.É.S., tome 25 (1965), p. 49-80.

- Time:
- 4:30pm - 5:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
PDE-Seminar

Wednesday, 24 th April 2024, 16:30 am-17:30 pm

=========================

Venue: Ramanujan hall, Department of Mathematics, IIT Bombay

Host: Debanjana Mitra

Speaker: Dr. Dharmatti Sheetal

Affiliation: Department of Mathematics, IISER Thiruvananthapuram

Title: Cahn-Hilliard-Navier-Stokes equations with Nonhomogeneous Boundary: Existence, Uniqueness, Regularity and Optimal Control

Abstract: The evolution of two isothermal, incompressible, immiscible fluids in a bounded domain is governed by Cahn-Hilliard-Navier-Stokes equations (CHNS System). In this work we study the well-posedness results for CHNS systems with nonhomogeneous boundary conditions for the velocity equation. We obtain the existence of global weak solutions in the two dimensional bounded domain using semi Galerkin approximation. We further prove the continuous dependence of the solution on initial conditions and boundary data that will provide the uniqueness of the weak solution. The existence of strong solutions is also established in this work. Furthermore, we study optimal boundary control using the continuous dependance of strong solution. Using Pontryagin's maximum principle we show that the optimal control is characterised as a unique solution of the appropriate adjoint system.

- Time:
- 4:00pm - 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Date and Time: April 30, 4pm - 5pm

Venue: Ramanujan HallSpeaker: Mayukh Mukherjee

Affiliation: IITB

Title: Spectra of negatively curved Riemannian manifolds

Abstract: We discuss various issues surrounding the spectra of complete Riemannian manifolds (and sometimes orbifolds) of non-positive curvature. These include, among other topics, absolute and (absence of) singularly continuous spectra, small eigenvalues and eigenfunction decay. This describes previous (and ongoing) joint work with Ballmann and Polymerakis.