11:30am» Nitin Nitsure, Bhaskaracharya Pratishthana- 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.