Lecture series on Lie groups

Monday, 6 March at 4 pm

Tea: 3.50 pm

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Venue: A1-A2, CDEEP, Mathematics Department

Host: Dipendra Prasad

Speaker: M. S. Raghunathan

Affiliation: CEBS, Mumbai

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, π_1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is  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.

Algebraic Groups Seminar

 Wednesday, 08/03/2023, 10.30 am

Venue: https://meet.google.com/jcn-nwpx-nmq

Host:  Shripad M. Garge

Speaker: Arpita Nayek

Affiliation: IIT Bombay

Title: On the torus quotients of Schubert varieties in Grassmannians

Abstract: Let G=SO(8n, C) or SO(8n+4, C) and T be a maximal torus of G. Let P be the maximal parabolic subgroup of G corresponding to the simple root \alpha_{4n} (respectively, \alpha_{4n+2}). In the first part of the talk, we will discuss the projective normality of the GIT quotients of certain Schubert varieties in G/P with respect to a T-linearized very ample line bundle on G/P. Let G_{r,n} be the Grassmannian of all r-dimensional subspaces of C^n. For r and n coprime, let X(w_{r,n}) be the unique minimal dimensional Schubert variety in G_{r,n} admitting semi-stable points. Let X^v_{w_{r,n}} be the Richardson variety in G_{r,n} corresponding to the Weyl group elements v and w_{r,n}. In the second part of the talk, we will discuss the sufficient conditions on v such that the GIT quotient of X^v_{w_{r,n}} is the product of projective spaces. The first part of my talk is based on a joint work with Pinakinath Saha and the second part of my talk is based on a joint work with Somnath Dake and Shripad Garge.

Algebraic Geometry Seminar

Wednesday, 8 March 2023,  12.00pm
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Venue: Ramanujan Hall, Department of Mathematics

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (retd)

Title: Local Criterion of Flatness-II

Analysis Seminar

Wednesday, 08/03/2023, 4 pm

Host: B. K. Das

Venue: Ramanujan Hall

Speaker: Haripada Sau

Affiliation:  IISER Pune

Title: Multivariable Operator Theory: Rational Dilation,

         Realization Formula, and Distinguished Varieties.

Abstract: In 1951, John von Neumann proved a pathbreaking inequality involving contractive linear transformations acting on Hilbert spaces and bounded analytic functions on the unit disk. The inequality -- now referred to as the von Neumann inequality -- has an extraordinary influence on the development of operator theory. A couple of years later, a simpler proof of the von Neumann inequality emerged when Sz.-Nagy proved his dilation theorem - the genesis of the dilation theory. Sz.-Nagy's discovery prompted several mathematicians to consider a multivariable generalization of the classical dilation theory. This is called the rational dilation problem -- the Holy Grail of spectral theory. In this talk, we shall discuss in brief the rational dilation problem corresponding to the unit disk, the bidisk, the symmetrized bidisk, and the tetrablock. We shall also discuss in brief a realization formula for bounded analytic functions on the disk, the bidisk, and the symmetrized bidisk. We shall see how a special class of algebraic varieties - the so-called distinguished varieties - becomes a natural object of study in the context of multivariable von Neumann inequality. A new characterization of such varieties will be presented. The last part of the talk will be about an ongoing project concerning a constrained two-variable dilation problem.

Analysis Seminar

Thursday, 09/03/2023, 11.30 am

Venue: Ramanujan Hall

Host: Prachi Mahajan

Speaker: Ratna Pal

Affiliation: IISER Mohali

Title: Rigidity properties of Henon maps in $\mathbb{C}^2$ and Short $\mathbb{C}^2$.

Abstract: The broad research area of my talk is Complex Dynamics in Several Variables. Classically complex dynamics was studied for rational endomorphisms of the Riemann sphere. In the past three decades, this field of research has flourished to a great extent and the holomorphic dynamics in higher dimensions has attracted a lot of attention. In particular, the dynamics of the polynomial automorphisms in higher dimensions mushroomed as one of the central themes of study. In $\mathbb{C}^2$, the most important polynomial automorphisms are the Henon maps and in this talk they will play the role of the protagonist. In the first part of the talk, we shall see a couple of rigidity properties of Henon maps. Loosely speaking, by rigidity properties we mean those properties of Henon maps which determine the underlying Henon maps almost uniquely. In the latter part of the talk, we shall survey a few recent results obtained for Short $\mathbb{C}^2$'s. A Short $\mathbb{C}^2$ is a proper domain of $\mathbb{C}^2$ that can be expressed as an increasing union of unit balls (up to biholomorphism) such that the Kobayashi metric vanishes identically, but allows a bounded above pluri-subharmonic function. The sub-level sets of the Green's functions of Henon maps are classical examples of Short $\mathbb{C}^2$'s. Note that the Green's function of a Henon map $H$ is the global pluri-subharmonic functions on $\mathbb{C}^2$ which is obtained by measuring the normalized logarithmic growth rate of the orbits of points in $\mathbb{C}^2$ under the iterations of the Henon map $H$. In this part of the talk, we shall first see a few interesting natural properties of Short $\mathbb{C}^2$'s. Then we give an effective description of the automorphism groups of the sublevel sets of Green's functions of Henon maps (recall that the sublevel sets of the Green's functions of Henon maps are Short $\mathbb{C}^2$'s). It turns out that the automorphism groups of this class of Short $\mathbb{C}^2$'s are not very large. Thus it shows that, unlike in a bounded set-up, although the Euclidean balls have large automorphism groups, the automorphism group of an increasing union of balls (up to biholomorphism) might flatten out when the final union is unbounded. A part of the results which will be presented in this talk is obtained in several joint works with Sayani Bera, John Erik Fornaess and Kaushal Verma.

Commutative algebra Seminar

Thursday, 09/03/2023, 4 pm

 Venue: Ramanujan Hall

Host: Tony J. Puthenpurakal

Speaker: Prof. R. V. Gurjar

Affiliation: Former Professor, IIT Bombay

Title: Positively Graded domain

Abstract:  I will continue my lectures on this topic. Following results will be discussed.  1. Demazure's construction of normal affine positively graded domains. Some applications of this will be discussed.  2. Flenner and Keiichi Watanabe's rationality of singularities criterion for positively graded affine domains.  3. A very general result I conjectures around 1990 and proved by O.Mathieu In 2002 will be discussed. It has some new consequence for rings of invariants of reductive algebraic group action on an affine space.  4. Divisor Class Groups of positively draded domains. Works of Brieskon Flenner, Samuel, Scheja-Storch, Anurag Singh etc, will be mentioned. Connection with Topology of these results will be discussed.

CACAAG seminar

10 AM Friday, 10 March, 2023.

Venue: Ramanujan Hall

Host: Madhusudan Manjunath

Speaker: Madhusudan Manjunath

Affiliation: IIT Bombay

Title: Unimodality and Log concavity in Algebra, Geometry and Combinatorics.

Abstract:  We will start with a gentle introduction to this topic with the goal of touching upon recent developments. 

This is intended as the first of a series of talks on this topic. We will not assume any particular background and

encourage students to attend. 

GGT seminar

Date and time: 10/03/23 at 12:30 PM

Venue: Ramanujan Hall, Department of Mathematics

Host: Rekha Santhanam

Speaker:  Samyak Jha

Affiliation:  IIT Bombay

Title: Free Groups and Objects

Abstract:  In this seminar, I will discuss about Free Objects in a category , the topology of Free Groups and the combinatorial way of describing groups

GGT seminar

Date and time: 17th March 2023  at 12:35 PM

Venue: Room no 114, Department of Mathematics

Host: Rekha Santhanam

Shantanu Nene (BS Math 3rd year) will be speaking on;

Title: Free Groups and Folding
Abstract: In this seminar, I will discuss folding of graphs, graph immersions, and a few applications of folding.

Analysis Seminar

Friday, 10/03/2023, 4 pm

Venue:  Ramanujan Hall

Host: Sanjoy Pusti

Speaker: Chandan Biswas

Affiliation: IISc, Bangalore

Title: A gentle introduction to Fourier restriction inequalities

Abstract: Initiated by Elias Stein in late 1960’s the Fourier restriction conjecture has played a central part in the development of modern harmonic analysis. Despite continuous progress over the last five decades, currently this remains out of reach in dimensions bigger than two. To get a better sense of restriction inequalities, we consider Fourier restriction estimates onto curves $\gamma : \R \to \Rd$. Even in this well-explored setting, there are many basic questions that remain open such as the question of existence of maximizers for such inequalities. This talk will be a gentle introduction to such questions and some recent progress on these. This is based on our recent works with Betsy Stovall (at University of Wisconsin Madison).

Virtual commutative algebra seminar

Friday, 10/03/23, 6.30 pm

Venue: meet.google.com/ibs-zwea-xqr

Host: J. K. Verma

Speaker: Cheng Meng

Affiliation: Purdue University, West Lafayette, IN, USA

Title:  Multiplicities in flat local extensions

Abstract:  We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if (R,m) and (S,n) are Noetherian local rings of the same dimension, S is a flat local extension of R,and up to completion S is standard graded over a field and I=mS is homogeneous, then the multiplicity of R is no greater than that of S.