Algebraic Geometry Seminar

Wednesday, 1 March 2023,  4.00pm
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Venue: Room No 215

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (retd)

Title: Local Criterion of Flatness-II

Commutative algebra seminar

Thursday, 2 March 2023, 4 pm

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Venue: Room 215

Host: Tony Puthenpurakal

Speaker: Sudeshna Roy

Affiliation: TIFR, Mumbai

Title: Asymptotic behavior of the Castelnuovo-Mumford regularity of (saturations of) products of powers of homogeneous ideals.

Abstract: In this talk, we will give a brief survey on results regarding the asymptotic linear bounds of the Castelnuovo-Mumford regularity of (saturations of) products of powers of ideals. We also aim to discuss a finer description of the asymptotic behavior of this invariant under some extra assumptions.

Virtual Commutative Algebra Seminar

Friday,  3 March 2023, 6:30pm

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Venue: meet.google.com/mjq-ahwy-oxo

Speaker: Vaibhav Pandey

Affiliation: Purdue University, West Lafayette, IN, USA

Title: Linkage and F-regularity of generic determinantal rings

Abstract: We prove that the generic link of a generic determinantal ring of maximal minors is strongly F-regular, hence it has rational singularities. In the process, we strengthen the result of Chardin and Ulrich. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that they are, in fact, strongly F-regular. 

In the mid-1990s, Hochster and Huneke showed that generic determinantal rings are strongly F-regular; however, their proof is quite involved. The techniques that we discuss will allow us to give a new and simple proof of the strong F-regularity of generic determinantal rings defined by maximal minors. Time permitting, we will also share new proof of the strong F-regularity of determinantal rings defined by minors of any size. This is joint work with Yevgeniya Tarasova.

Lecture series on Lie groups

Monday, 6 March at 4 pm

Tea: 3.50 pm

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

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

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.

Annual Progress seminar

Tuesday,14 March 2023, 11.30 am

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

Venue: Room 105

Host: Shripad M. Garge

Speaker: Deepkumar Makadiya

Affiliation: IIT Bombay

Title: Twisted Chevalley groups

Abstract: The talk aims at discussing the twisted Chevalley group and its automorphisms. The first part of the talk deals with the existing literature regarding the twisted Chevalley group and we present some results in this direction. The main aim of the second half is to characterize the class of automorphisms of the twisted Chevalley group. En route, we present some useful results to build the setting of the desired characterization.

Number Theory seminar

Tuesday, 14 March 2023, 2.30 pm

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Venue: Google meet: Link: https://meet.google.com/jtv-udzf-bbx

Host: Kummari Mallesham

Speaker: Keshav Aggarwal

Affliation: Alfréd Rényi Institute of Mathematics

Title:  Analytic study of L-functions: Subconvexity bound problem

Abstract: Much of modern number theory revolves around the study of L-functions. The L-functions associated with arithmetic objects called automorphic forms are of great interest and many conjectures have been made about their properties. The best known is the Riemann hypothesis (RH) which has many well-known consequences. One such is the Lindelof hypothesis (LH) which is a conjecture about the growth rate of automorphic L-functions on the critical line. Results achieving partial progress towards LH are known as Subconvexity Bound results and have rich arithmetic applications, like equidistribution of points on certain surfaces, and Quantum Unique Ergodicity. Recent years have seen much activity in this area by combining powerful tools like the Delta method, Motohashi-type formulas, and estimates on moments of L-functions. In this talk, we give an introduction to the subconvexity bound problem and a recent result on bounding a short second moment of GL(3) L-functions. This is joint work with Joseph Leung and Ritabrata Munshi and yields the yet best-known subconvexity bound for GL(3) L-functions in the t-aspect.

Algebraic Groups Seminar

Tuesday, 14 march 2023, 4 pm

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Venue: Room 114

Host: Shripad M. Garge

Speaker: Deepkumar Makadiya

Affiliation: IIT Bombay

Title: Linear algebraic groups, basic themes. IV

Abstract: We complete the discussion around the Jordan decomposition in this lecture.

Algebraic Geometry seminar

Wednesday, 15 March 2023, 11.30 am

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

Venue: Ramanujan Hall

Host: Saurav Bhaumik

Speaker: Pinakinath Saha

Affilaition: IIT Bombay

Title: On (weak) Fano $G$-Bott-Samelson-Demazure-Hansen varieties

Abstract: Let $G$ be a semi-simple simply connected algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $W$ be the Weyl group of $G$ with respect to $T$. For an arbitrary sequence $w$ of simple reflections in $W,$ let $Z_{w}$ be the Bott-Samelson-Demazure-Hansen variety (BSDH-variety for short) corresponding to $w.$ Bott-Samelson-Demazure-Hansen-varieties are an important tool in geometric representation theory. They were originally defined as desingularizations of Schubert varieties and were used to describe the geometry of Schubert varieties. There is a natural action of $B$ on $Z_{w}$ given by the left multiplication. Let $\widetilde{Z_{w}}:=G\times^{B}Z_{w}$ be the fibre bundle associated to the principal $B$-bundle $G\to G/B.$ We call it $G$-Bott-Samelson-Demazure-Hansen variety ($G$-BSDH-variety for short). In the first part of the talk, we will describe a basis of the Picard group of $G$-BSDH variety, which we will refer as the $\mathcal{O}(1)$-basis. Then we will characterize the nef, globally generated, ample and very ample line bundles on $G$-BSDH variety in terms of the $\mathcal{O}(1)$-basis. Finally, we will provide a characterization of (weak) Fano $G$-BSDH varieties. We introduce a few more notations for the second part of my talk. Let $G=SO(8n,\mathbb{C})\big/SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the end simple root of its Dynkin diagram. In the second part of the talk, we will discuss the projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle. The first part of the talk will be based on joint work with Saurav Bhaumik. Here is the link of the preprint: \url{https://arxiv.org/abs/2212.10366}. The second part of the talk will be based on joint work with Arpita Nayek. Here are the links of the preprints: \url{https://arxiv.org/abs/2207.01477} and \url{https://arxiv.org/abs/2302.00555}.

Geometry and Topology Seminar

Wednesday, 15 March 2023, 2.30 pm

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

Venue: Ramanujan Hall

Host: Sandip Singh

Speaker: Arghya Mondal

Affiliation: CMI, Chennai

Title: A higher dimensional generalization of Margulis' construction of expander graphs

Abstract: Expanders are a family of finite graphs whose vertex sizes go to infinity but edge sizes grow at most linearly in vertex sizes, while still remaining highly connected. The first explicit construction of such graphs was by Margulis using discrete groups having Property (T), a rigidity property defined in terms of unitary representations. In recent years various higher dimensional generalizations of expanders, replacing graphs by simplicial complexes of a fixed dimension, have been considered. We will discuss a group theoretic construction of one such generalization, which is an extension of Margulis' construction to higher dimensions.

Mathematics Colloquium

15 March 2023, 4 pm

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

Venue: Ramanujan Hall

Host: Mayukh Mukherjee

Speaker: Arunima Ray

Affiliation: MPI, Bonn

Title: Embedding surfaces in 4-manifolds

Abstract: Manifolds are fundamental objects in topology since they locally model Euclidean space. Within a given ambient manifold, we are often interested in finding embedded submanifolds, which would then enable cutting and pasting operations, such as surgery. The study of surfaces in 4-dimensional manifolds has led to breakthroughs such as Freedman's proof of the 4-dimensional Poincare conjecture. Important open questions on 4-manifolds can also be reduced to the question of finding certain embedded surfaces.

I will consider the following question: When is a given map of a surface to a 4-manifold homotopic to an embedding? I will give a survey of related results, including the celebrated work of Freedman and Quinn, and culminating in a general surface embedding theorem, arising in joint work with Daniel Kasprowski, Mark Powell, and Peter Teichner.

Speaker: Bikram Bir, IIT Bombay

Date and Time (of the first lecture): Thursday 16 March 2023 at 02.30PM

Venue: Ramanujan Hall, Department of Mathematics.

Title: Oldroyd Model of Order One: Theory and Numerics

Abstract:  In these lectures, we discuss about a few finite element methods
for the equations of motion arising in the two-dimensional Oldroyd model
of order one; a model that represents linear viscoelastic fluid flows and
that can be viewed as an integral perturbation of the Navier-Stokes
equations.
First, we study existence and uniqueness of the continuous and weak
solution. Then, we discretize the space variable based on standard
Galerkin finite element method and the temporal variable based on
different time-discrete schemes. We next employ different finite element
methods like the two-grid method, the penalty method, the grad-div
stabilization method, the projection method and the nonconforming finite
element method. In all these cases, our main aim will be to obtain an
optimal error estimate for the velocity and the pressure. Finally, we
present some
numerical experiments to validate our theoretical findings.
 

Statistics and Probability  seminar

Thursday, 16 March 2023, 3 pm

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

Venue: Online, link will be sent later

Host: Debraj das

Speaker: Sandipan Roy

Affiliation: University of Bath

Title: Statistical Inference in Complex Data with Network Structure

Abstract: New technological advancements have allowed the collection of datasets of large volume and different levels of complexity. Many of these datasets have an underlying network structure. Networks are capable of capturing dependence relationships among a group of entities and hence analyzing these datasets unearth the underlying structural dependence among the individuals. Examples include gene regulatory networks, understanding stock markets, protein-protein interaction within the cell, online social networks etc. We present two important aspects of large high-dimensional data with network structure. The first one focuses on a data with a network structure that evolves over time. Examples of such data sets include time course gene expression data, voting records of legislative bodies etc. Traditionally, the estimation of Gaussian graphical models (GGM) is performed in an i.i.d setting. More recently, such models have been extended to allow for changes in the distribution, but primarily where changepoints are known a priori. In this work, we study the Group-Fused Graphical Lasso (GFGL) which penalizes partial correlations with an L1 penalty while simultaneously inducing block-wise smoothness over time to detect multiple changepoints. The other aspect that we examine is heterogeneity in a network structure and how we can use such heterogenous features in a predictive model. We use a linear latent variable model viz. PCA and its extensions to learn an underlying network structure from data varying over time. We then employ the learned network as a feature in a predictive model to perform the downstream task in the test data. Neuroimaging-driven prediction of brain age, defined as the predicted biological age of a subject using only brain imaging data, is an exciting avenue of research. In this work, we seek to build models of brain age based on functional connectivity while prioritizing model interpretability and understanding. This way, the models serve to both provide accurate estimates of brain age as well as allow us to investigate changes in functional connectivity that occur during the aging process.

Commutative algebra seminar

Thursday, 16 March 2023, 4 pm

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

Venue: Ramanujan Hall

Host: Tony Puthenpurakal

Speaker: Prof. R. V. Gurjar

Affiliation: Former Professor, IIT Bombay

Title: Positively Graded domains

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 conjectured around 1990 and proved by O.Mathieu In 2002 will be discussed. It has some new consequences for rings of invariants of reductive algebraic group action on an affine space.  4. Divisor Class Groups of positively graded domains. Works of Brieskon Flenner, Samuel, Scheja-Storch, Anurag Singh, etc, will be mentioned. The connection with the Topology of these results will be discussed.

Geometry seminar

Thursday, 16th March 2023, 5 pm

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

Venue: Ramanujan Hall

Host: Mayukh Mukherjee

Speaker: Mitul Islam

Affiliation: Heidelberg University

Title: Relatively hyperbolic groups and convex projective structures

Abstract: Studying discrete subgroups of linear groups using a preserved geometric structure has a long tradition. For instance, using real hyperbolic geometry to study discrete subgroups of SO(n,1). Convex projective structures, a generalization of real hyperbolic structures, has recently received much attention in the context of studying discrete subgroups of PGL(n). In this talk, I will discuss convex projective structures and discuss results (joint with A. Zimmer) on relatively hyperbolic groups that preserve convex projective structures. In particular, I will discuss a complete characterization of relative hyperbolicity in terms of the geometry of the projective structure. 

CACAAG seminar

Friday, 17 March, 2023, 10 am

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

Venue: Ramanujan Hall

Host: Madhusudan Manjunath

Speaker: Madhusudan Manjunath

Affiliation: IIT Bombay

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

Abstract:  We will start with a recap and take a more conceptual approach to this topic (with the goal of touching upon recent developments).  We will not assume any particular background and encourage students and those who missed out last week to attend.

Date and time:  Friday, March 17th, 2023

Venue: 11:00 a.m. - 11:50 a.m.

Host: Krishnan Sivasubramanian

Speaker:  Rakesh Jana

Affiliation: IIT Bombay

Title: Distance Matrices of Trees Inspired by Buneman’s Four-Point Condition

Date and time:  March 17, 2023, 12:30 to 1.30 pm

Venue: Ramanujan hall

Host: Prof. P. Vellaisamy

Speaker:  Mostafizar

Affiliation: IIT Bombay

Title: Generalized counting process: its non-homogeneous and          time-changed versions

Virtual Commutative algebra seminar

Friday, 17 March 2023, 6.30 pm

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

Host: J. K. Verma

Venue: meet.google.com/oes-jruv-qup

Speaker: Christine Berkesh

Affiliation: University of Minnesota, USA

Title: Differential operators, retracts, and toric face rings 

Abstract: Toric face rings, introduced by Stanley, are simultaneous generalizations of Stanley–Reisner rings and affine semigroup rings, among others. We use the combinatorics of the fan underlying these rings to inductively compute their rings of differential operators. Along the way, we discover a new differential characterization of the Gorenstein property for affine semigroup rings. Our approach applies to a more general class of rings, which we call algebras realized by retracts. This is joint work with C-Y. Chan, P. Klein, L. Matusevich, J. Page, and J. Vassilev.

Lecture series on Lie groups.

Monday, 20 March at 4 pm

Tea: 3.50 pm

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

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 center and its commutator subgroup (which is a 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 abelianization is trivial is compact.

Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch 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.  

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

PDE Seminar

Tuesday, 21 March at 4 pm

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

Venue: 216, Mathematics Department

Host: Saikat Mazumdar

Speaker: Dr. Souptik Chakraborty

Affiliation: IIT Bombay

Title: Quantitative stability results of some functional inequalities

Abstract: In 1992, Bianchi-Egnell proved that the "Sobolev deficit" is bounded below by the distance of the function to the manifold of optimizers which gives a quantitative version of Lions result. In recent years, Figalli and other mathematicians obtained quantitative stability results for the critical exponent problem which is the Euler-Lagrange equation associated to Sobolev inequality. In this talk, I'm going to present recent developments in this direction. In particular, I will talk about some progress in the quantitative stability aspects of Hardy-Sobolev inequality. This is an ongoing joint project with Saikat Mazumdar.

Algebraic groups Seminar

Tuesday, 21 March at 4 pm

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

Venue: 114, Mathematics Department

Host: Shripad Garge

Speaker: Shripad M. Garge

Affiliation: IIT Bombay

Title: Linear algebraic groups, basic themes. V

Abstract: We discuss the last section in Springer's second chapter. It is on determining a group by its representations.

Commutative Algebra Seminar

Thursday, 23 March 2023, 4 pm

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

Venue: Ramanujan Hall

Host: Tony Puthenpurakal

 Speaker: Tony Puthenpurakal

Affiliation: IIT Bombay

Title: Asymptotic primes and analytic spread

Abstract: We discuss two results of Brodmann. The first result shows that over a Noetherian ring, Ass M/I^nM stabilizes for n large. The second result shows that depth M/I^nM is constant for n large.

Algebraic Geometry Seminar

Friday 24 March 2023,  11.00am
===========================

Venue: Ramanujan Hall, Department of Mathematics

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (retd)

Title: Local Criterion of Flatness-II

GGT Seminar 

Date and time: Friday 24 March, 2023 at 12:35 pm

Venue: Room No 114, Department of Mathematics

Host: Rekha Santhanam

Speaker:  Ankit Rai 

Affiliation: IIT Bombay

Title: The Ping-Pong Lemma.

Abstract : The talk will be based on the chapter titled The Ping-Pong Lemma from the book Office hours with a geometric group theorist. The  chapter consists of ping-pong lemma and it's application(s), a construction of Farey tree and a gentle introduction to hyperbolic upper half plane.

CACAAG seminar

4 PM Friday, 24 March 2023

Please note the deviation in the time from the previous seminars.

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

Venue: Ramanujan Hall

Host: Madhusudan Manjunath

Speaker: Madhusudan Manjunath

Affiliation: IIT Bombay

Title: Unimodality and Log concavity in Algebra, Geometry, and Combinatorics: h-vectors of Polytopes and Toric Geometry.

Abstract:  We discuss unimodality questions on h-vectors of polytopes and sketch an approach via toric geometry.

Virtual Commutative Algbera Seminar

Friday, 24 March 2023, 6.30 p.m.

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Venue: meet.google.com/upz-rfxh-uob

Host: J. K. Verma

Speaker: Takumi Maruyama

Affiliation: Purdue University, USA

Title: Uniform bounds on symbolic powers in regular rings via closure theory

Abstract: The containment problem asks: For a fixed ideal I, which symbolic powers of I are contained in an ordinary power of I? We present a closure-theoretic proof of the theorem which says that for ideals I in regular rings R, there is a uniform containment of symbolic powers of I in ordinary powers of I.

Algebraic Geometry seminar

11.30 am on 28 March 2023

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Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Former Affiliation: TIFR, Mumbai

Title: Smooth morphisms

Algebraic groups Seminar

Tuesday, 28 March at 4 pm

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Venue: 114, Mathematics Department

Host: Shripad Garge

Speaker: Shripad M. Garge

Affiliation: IIT Bombay

Title: Linear algebraic groups, basic themes. VI

Abstract: We complete the second chapter in Springer's book with a 
discussion on Tannaka's theorem.

Topology Seminar

Date and time: Thursday 30 March 2023 at 10 AM

Venue: Video call link: https://meet.google.com/faa-xzqt-xyn

Host: Rekha Santhanam

Speaker:  Bikramjit Kundu 

Affiliation: ISI Bangalore

Title: The index of some Stiefel and flag manifolds

Abstract:  In this talk I will do the Fadell-Husseini index calculations for certain
group actions on Stiefel manifolds and flag manifolds. The calculations for the
Stiefel manifold are related to generalizations of the Kakutani’s theorem in geom-
etry. I will also briefly discuss index of flag manifolds of equal dimensional p orthogonal
subspaces on which C_p acts by the cyclic permutation. The result bears geometric consequences about
measures of orthogonal shadows of convex subsets. This is an odd primary ana-
logue of a calculation of Blagozevic et al for the Grassmannian.

Commutative algebra seminar

4 pm on Thursday, 30 March, 2023

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Venue: Ramanujan Hall

Host: Tony Puthenpurakal

Speaker: Manoj Keshari

Affiliation: IIT Bombay

Title: On complete intersection ideals

Abstract: I will give a survey on the problem of complete intersection 
ideals in a polynomial ring

Geometric Group Theory (GGT)  Seminar

12.30 pm on 31 March 2023

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Venue: Room 114

Host: Rekha Santhanam

Speaker: Soumyadeb Samanta

Affiliation: IIT Bombay

Title: Quasi-isometries

Abstract: In this talk, I will give an introduction to quasi-isometries on metric spaces and groups, and discuss some basic results concerning them. Then we will see the statement and proof of the celebrated Milnor-Schwarz lemma.

CCAAG Seminar

4 pm on Friday, 31 March 2023

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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 review the relevant tools from Hodge's theory.

Virtual commutative algebra seminar

6.30 pm on Friday, 31 March 2023

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Venue: meet.google.com/rhf-jzen-fyf

Host: J. K. Verma

Speaker: Alexandra  Seceleanu

Affiliation: University of Nebraska-Lincoln

 Title: Principal symmetric ideals

Abstract: Consider a homogeneous polynomial f in variables x_1,...,x_n. The set of polynomials obtained from f by permuting the variables in all possible ways generates an ideal, which we call a principal symmetric ideal. What can we say about the Betti numbers of a principal symmetric ideal? I will give a general answer in this talk. This is joint work with Megumi Harada and Liana Sega.