Analysis Seminar
Thursday,  1 February 2024, 3.30 pm
========================

Venue: meet.google.com/fxo-yvfh-avr
Host: Santanu Dey

Speaker: Repana Devendra
Affiliation: IIT Madras

Title: C*-extreme points of unital entanglement breaking maps on matrix algebras.

Abstract: In this talk, we discuss various characterizations of C*-extreme points of the C*-convex set of unital entanglement breaking (EB-) maps on matrix algebras.  A key contribution is made through the establishment of a Radon-Nikodym type theorem for a specific class of EB-maps, offering a comprehensive understanding of these points. Additionally, a noncommutative counterpart to the Krein-Milman theorem is derived for
C*-convexity within the set of unital EB- maps. Lastly, we highlight the significant connection between C*-extreme points and the notion of degradability in the context of quantum information theory.

Commutative algebra seminar

IPDF talk
Friday, 2nd Feb, 3:30-4:30 pm

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

Host: Manoj Keshari

Venue: https://meet.google.com/gyb-jfbn-oiu?authuser=0

Speaker: Parnashree Ghosh

Affiliation: ISI Kolkata

Title: Applications of exponential maps to the epimorphism and Zariski
cancellation problem

Abstract: In the first part, we will discuss the Epimorphism Problem and also discuss the famous Abhyankar-Sathaye Epimorphism Conjecture. We will introduce ``Generalised Asanuma varieties" (GAV) of higher dimensions \geq 3 and see some necessary and sufficient conditions for these varieties to be isomorphic to the affine space. We see that this characterization immediately yields a family of higher dimensional hyperplanes satisfying the Abhyankar-Sathaye Conjecture.

In the second part, we see some necessary conditions for two GAVs to be isomorphic and also describe automorphisms of a certain subfamily of GAV. These results show that for each d \geq 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. We further give a complete description of two important invariants called Makar-Limanov and Derksen invariants of a certain subfamily of GAV.

This talk is based on a joint work with Neena Gupta.

Algebraic Groups seminar

Friday, 2 February, 4.00-5.15

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

Venue: Room 105

Host: Dipendra Prasad

Speakers: Dibyendu Biswas, Chayan Karmakar, Mohammed Saad and Deep Makadiya

Affiliation: IIT Bombay

Title: Regular elements in semi-simple algebraic groups

Abstract: We will discuss a variety of topics in Algebraic groups through reading some of the papers that have become classics in the subject. The first few seminars will be on the paper by ROBERT STEINBERG, Regular elements of semi-simple algebraic groups Publications mathématiques de l'I.H.É.S., tome 25 (1965), p. 49-80

Analysis seminar

IPDF talk

Friday, 2nd Feb. 4 pm - 5 pm 

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

Venue: Online, link TBA

Host: Sanjay Pusti

Speaker: Jitendra Kumar Senapati 

Affiliation : BITS Pilani, Goa

Title: Restriction theorem for the Fourier-Dunkl transform and its applications to Strichartz inequalities

Abstract: We define the Fourier-Dunkl transform, which generalizes the Fourier transform. We prove Strichartz’s restriction theorem for the Fourier-Dunkl transform for certain surfaces, namely, cone, paraboloid, sphere, and hyperboloid, and its generalization to the family of orthonormal functions. Finally, as an application of these restriction theorems, we establish versions of Strichartz estimates for orthonormal families of initial data associated with the Schrodinger propagator, wave propagator, and Klein-Gordon propagator for the case of the Dunkl Laplacian. This restriction theorem generalizes Stein-Tomas and Strichartz’s restrictions theorems in special cases. This is a joint work with B. Pradeep, S. S. Mondal, and H. Mejjaoli.

Combinatorics Seminar

Monday, 05th Feb. 4 pm

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

Venue: meet.google.com/ite-hvpv-oqy

Host: S. Krishnan

Speaker: Pranshu Gupta

Affiliation: University of Passau

Title: Minimal Ramsey graphs and the Erdős Rothschild problem

Abstract: In this talk, I will provide an overview of my research thus far. My main area of focus during my time as a doctoral candidate had

been the minimal Ramsey graphs. A graph G is said to be Ramsey for a graph H if, for any two colouring of the edges of G, there exists a monochromatic

copy of H. Determination of the smallest such clique for a graph H has been

a topic of intense research. A graph G is said to be minimal Ramsey if G

itself is Ramsey for H but no proper subgraph of it is Ramsey for H. In

1976, in their seminal paper, Burr, Erdős, and Lovász initiated a

systematic study of said graphs. One such popular problem has been the

determination of the minimum of the minimum degree among all minimal Ramsey graphs, for a given H. I will introduce this notion and summarize my

contributions to this field. I will follow this up with a discussion on one of my more recent works, with my colleagues Pehova, Powierski, and Staden,

on the Erdős Rothschild problem. The Erdős Rothschild problem from 1974 asks, given positive integers n,s, and k, what the maximum number of edge colourings, with s colours, an n vertex graph can have which avoids a monochromatic copy of clique of size k. We initiate a systematic study of the generalisation of this problem to a given forbidden family of colourings of cliques of size k.I will elaborate on the problem of forbidden triangles with exactly two colours. We solve this problem for all integers s ≥ 2 and large n and show in particular that every extremal graph is a Turán graph on an even number of parts; r is a solution to a certain optimisation 

problem which grows with s. This extends the work of Hoppen, Lefmann and Schmidt who solved the cases s ≤ 26, in which case r is 2 and proves their conjecture for s=27.

Algebraic Stacks lecture series

Tuesday, 6 February 2024, 11:30 AM

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

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: Bhaskaracharya Pratishthana

Title: Going up, liftings and valuative criteria.

Abstract: This lecture is in the series on separated and proper morphisms of topological spaces, schemes, algebraic spaces and algebraic stacks, We will begin with some commutative algebraic results which have the broad theme of `going up', `extensions' and `lifts'. These are of three kinds: (1) Results about lifting primes, (2) Properties of valuation rings and extensions of local domains to maximal such in their quotient fields, and (3) the formulations of valuative criteria as problems of lifting morphisms. This will be followed by a close look at specialization and generalization of points. (This will be re-visited when we go to algebraic stacks, as surprising new behaviour can occur.) With the above preliminaries done, we will give an exposition of the proof of the valuative criteria for separatedness and universal closedness of morphisms of schemes. Preparatory reading for students: Hartshorne `Algebraic Geometry' Chapter 2 section 4, and the portion on valuation rings in Atiyah-Macdonald, `Commutative Algebra' chapter 5.

Commutative Algebra Seminar

Tuesday 6 Feb, 4.00-5:30 pm

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

Venue: Room 215

Host: Tony J. Puthenpurakal

Speaker : Samarendra Sahoo

Affiliation: IIT Bombay

Title: Eisenbud conjecture on bounded Betti number

Abstract: Eisenbud conjecture: Let Q be an NLR and I be generated by Q-regular sequence. Set A=Q/I. Let F.\to M be a minimal-free resolution of M such that the ranks of the free modules F¡ are bounded, then F is eventually periodic of period 2. He proved that it is true for the Complete intersection ring. I will continue my lecture and discuss proof of this.

Algebraic Groups seminar

Tuesday, February 6, 2024, 4 pm

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

Venue: Ramanujan Hall

Host: Shripad M. Garge

Speaker: Shripad M. Garge

Affiliation: IIT Bombay

Title: Homogeneous Spaces - I

Abstract: We introduce the notion of homogeneous spaces for linear algebraic groups. 

Probability and statistics seminar

Wednesday, 7th February at 3 pm

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

Venue: Ramanujan Hall

Host: Debraj Das

Speaker: Dr. Samriddha Lahiry

Affiliation: Harvard University

Title: Quantum Statistical Inference

Abstract: Recent advancements in quantum technology, such as quantum

computing, communication, and metrology, have given rise to questions

related to quantum measurements, which can be elegantly formulated in the

language of mathematical statistics. However, quantum mechanics,

inherently noncommutative, yields inferential results that are distinctly

non-trivial, compared to their counterparts in classical statistics.

In classical statistics, a fundamental paradigm involves approximating

complex models with simpler ones. One commonly establishes asymptotic

equivalence between i.i.d models, characterized by a local parameter, and

a Gaussian shift model. This approximation, known as local asymptotic normality (LAN),

facilitates the construction of an estimator based on a procedure in the

Gaussian model, offering comparable risk bounds.

Notably, local asymptotic equivalence can be extended to quantum scenarios,

linking quantum i.i.d. models with quantum Gaussian models. In this context,

we obtain optimal estimators in the complex former models based on optimal

estimators in the simpler latter models.

Analysis Seminar

Thursday, 8 February 2024, 3.30 pm

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

Venue: Link: meet.google.com/kxj-tmsk-zou

Host: Santanu Dey

Speaker: Shanola Sequeira

Affiliation: IIT Hyderabad

Title: Absolutely norm-attaining and absolutely minimum attaining operators

Abstract: A bounded linear operator $T$ between the Hilbert spaces $H_1$ and $H_2$ is called absolutely norm attaining if the restriction of $T$ to any non-zero closed subspace of $H_1$ attains its norm and absolutely minimum attaining if every restriction to a non-zero closed subspace of $H_1$ attains its minimum modulus. In this talk, we discuss some properties and spectral representations of these two classes of operators. Further, we also characterize absolutely norm attaining and absolutely minimum attaining Toeplitz and Hankel operators on the Hardy Hilbert space.

Topology and Related Topics seminar

Friday, 9 Feb, 11.30-12.30

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

Venue: Room 215

Host: Rekha Santhanam

Speaker: Priyanka Magar

Affiliation: IIT Bombay

Title: Classifying Spaces of Categories

Abstract: The classifying space $BC$ of a category $C$ is a way to turn a category into a topological space. If the category is equipped with additional structure, the associated classifying space reflects this structure. We discuss this mainly for a topological category.

Analysis seminar

Friday, Feb 9, 4 pm – 5 pm

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

Venue: Ramanujan Hall

Host: Chandan Biswas

Speaker: Debraj Chakrabarti

Affiliation: Central Michigan University, Mt. Pleasant, MI

Title: Projection operators on Bergman spaces of Reinhardt domains.

Abstract: It is a famous result of M. Riesz that the Szegö projection operator, initially defined as the orthogonal projection from the space of square integrable functions on the circle to the Hardy space of the disc extends continuously as a projection operator from onto. There is a long history of similar results in the setting of Bergman spaces, and a long list of domains where an analogous statement does not hold in the Bergman setting. We try to understand the geometric distinction between the Hardy and the Bergman situations on Lebesgue spaces, and propose a new projection operator on Reinhardt domains which is expected to have better mapping properties. We verify that the new operator satisfies Lebsegue space estimates in some situations where the Bergman projection operator does not satisfy such estimates. This is joint work with Luke Edholm of the University of Vienna.

Algebraic Groups Seminar

Friday, February 9, 4-5:15 PM

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

Venue: Room No. 105, Department of Mathematics

Host: Dipendra Prasad

Speaker: Dibyendu Biswas

Affiliation: IIT Bombay

Title: Regular elements in semi-simple algebraic groups

Abstract: We will discuss the paper by Robert Steinberg, Regular elements

of semi-simple algebraic groups Publications mathématiques de l'I.H.É.S.,

tome 25 (1965

), p. 49-80.

Algebraic Stacks lecture series
Tuesday, 13 February, 11:30 AM

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

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: Bhaskaracharya Pratishthana

Title: Separated and proper morphisms for algebraic stacks.

Abstract: We will begin by revisiting the valuative criteria for universal closedness and separateness for morphisms between schemes. This will be followed by interesting elementary examples that illustrate why these criteria do not directly apply to algebraic spaces and algebraic stacks. We will end with the modified criteria that apply to algebraic spaces and algebraic stacks.

Topology and Related Topics Seminar

Tuesday, 13 February, 12:00-1:00

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

Venue: Ramanujan Hall (Hybrid Lecture) (Speaker will be online).

Host: Rekha Santhanam

Speaker: Suraj Krishna

Affiliation: Technion - Israel Institute of Technology

Title: Mapping tori in geometric group theory.

Abstract: In this talk, I will provide a gentle introduction to 
geometric group theory using the concept of mapping tori. Using pictures 
and examples, I will introduce the relevant topics and give a glimpse of 
some of the spectacular progress made by stalwarts like Bill Thurston. 
Additionally, I will mention some recent progress in the field, along 
with my forays.

This talk is a background lecture especially aimed at graduate students 
for the talk  Suraj Krishna will give the following Tuesday.
 

Commutative Algebra Seminar

Tuesday, 13 Feb, 3 pm—4 pm

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

Venue:   Room 215

Host: Tony J. Puthenpurakal

Speaker: Sudeshna Roy

Affiliation: TIFR Bombay

Title:  Epsilon multiplicity in two-dimensional standard  graded algebras

Abstract: The notion of epsilon multiplicity, a generalization of the Hilbert-Samuel multiplicity, was introduced by B. Ulrich and J. Validashti to detect the integral dependence of arbitrary ideals. This invariant is difficult to handle as there are examples where it can be irrational and the epsilon function is very far from being polynomial-like. Let $A$ be a standard graded normal domain of dimension two over a field with the unique homogeneous maximal ideal $m$. Let $I$ be a homogeneous ideal in $A$. Our objective is to show that the epsilon multiplicity of $I$ is a rational number.

Algebraic Groups Seminar
Tuesday, February 13, 4 PM--5 PM
=========================

Venue: Ramanujan Hall, Department of Mathematics
Host: Dipendra Prasad

Speaker: Mohammed Saad Qadri
Affiliation: IIT Bombay

Title: Regular elements in semi-simple algebraic groups

Abstract: We will discuss the paper by Robert Steinberg, Regular elements of semi-simple algebraic groups Publications mathématiques de l'I.H.É.S., tome 25 (1965), p. 49-80.

Commutative Algebra seminar

Tuesday, 13 February 4 pm-5 pm

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

Venue: Ramanujan Hall

Host: J. K. Verma

Speaker: S. K. Khanduja

Affiliation: IISER Mohali

Title: Some well-known Irreducibility Criteria for polynomials over rationals and their generalizations

Abstract: We will review the classical irreducibility criteria of Eisenstein, Schoenemann, and Dumas, and then discuss their extensions using Newton polygons and the theory of valuations. We shall also present recently proved generalizations of the well-known

results of Schur as well as of Filaseta and Trifonov regarding the irreducibility of the truncated exponential polynomial and the Bessel polynomial. This talk is partly based on joint work with Ankita Jindal, Bablesh Jhorar, and Sanjeev Kumar.

Mathematics Colloquium:

Wednesday, 14  February  2023, 4:00 pm

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

Venue: Ramanujan hall

Host: Ravi Raghunathan

Speaker: Manish Patnaik

Affiliation: University of Alberta

Title: Eisenstein series and some variants on Loop Groups

Abstract: We will first review some aspects of the reduction theory for loop groups studied by Howard Garland in the 1970s. Following this, we explain how features of the Siegel domains in this setting essentially force certain cuspidal Eisenstein series on loop groups to be entire functions, in stark contrast to the finite-dimensional situation. 

Switching to the case of function fields, we then introduce, following ideas of Braverman and Kazhdan, new “regularized” cuspidal Eisenstein series for loop groups. These series are no longer entire and their Fourier coefficients encode (finite-dimensional) automorphic L-functions that have previously remained inaccessible by the standard Langlands-Shahidi method.

Probability and statistics seminar

Wednesday, 14th Feb, 4 pm--5pm

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

Venue: Room 113, Maths Dept

Host: Siuli Mukhopadhyay

Speaker: Subhra Sankar Dhar

Affiliation:   IIT Kanpur

Title: Inspecting discrepancy between multivariate distributions using half-space depth

 Abstract: In this talk, we inspect whether a multivariate distribution is different from a specified distribution or not AND two multivariate distributions are equal or not. In the course of this study, a graphical toolkit using well-known half-spaced depth is proposed, which is a two-dimensional plot, regardless of the dimension of the data, and it is even useful in comparing high-dimensional distributions. The simple interpretability of the proposed graphical toolkit motivates us to formulate test statistics to carry out the corresponding testing of hypothesis problems. It is established that the proposed tests are consistent, and the asymptotic distributions of the test statistics under contiguous/local alternatives are derived, which enables us to compute the asymptotic power of these tests. Furthermore, it is observed that the computations associated with the proposed tests are unburdensome. Besides, these tests perform better than many other tests available in the literature when data are generated from various distributions such as heavy-tailed distributions, which indicates that the proposed methodology is robust as well. Finally, the usefulness of the proposed graphical toolkit and tests is shown on two benchmark real data sets. This is a joint work with Pratim Guha Niyogi (Johns Hopkins University, USA). 

Analysis Seminar: Annual Progress Seminar
Thursday,  15 February, 3.30 pm
===============================

Venue: Room 215

Host: Sourav Pal

Speaker: Priyanka Aroda
Affiliation: IIT Bombay

Title: Geometric and function theoretic aspects of the tetrablock
Abstract: We study the tetrablock, a domain in \mathbb{C}^3 given by 

\mathbb{E}=\left\{\left(x_1,x_2,x_3\right) \in \mathbb{C}^3 : \ 1-x_1 z-x_2 w+x_3 zw \neq 0 \ \ \text{for} \ \ |z| \leq 1,|w| \leq 1\right\}.\]The tetrablock is related to a certain function-theoretic problem that arises in control engineering. We discuss several characterizations for the elements in$\mathbb{E}$ and in its closure. We present a set of necessary and sufficient conditions that provide a Schwarz lemma for $\mathbb{E}$. We also describe the automorphism group of $\mathbb{E}$. 

Geometry and Number Theory seminar
Thursday, 15th February 2024, 4 pm-5 pm
===========================

Venue: Ramanujan Hall
Host: K. Mallesham.

Speaker: Dr. Rahul Kumar Singh
Affiliation: IIT Patna

Title: Zero Mean Curvature Surfaces and Ramanujan's Identities

Abstract: In this talk, we discuss an interesting connection between examples of zero mean curvature surfaces in Euclidean and Lorentz-Minkowski 3-space and some special Ramanujan identities.

Topology and Related Topics Seminar

Thursday, 16 February  2023, 11:30-12:30

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

Venue: Ramanujan hall

Host: Rekha Santhanam

Speaker: Sahin Mandal

Affiliation: IIT Bombay

Title: Quillen Q-Construction

Abstract: In this series of two talks we will define higher algebraic k 
theory using Quillen's Q construction.
 

Algebraic Groups Seminar

Friday, February 23, 2024, 4 pm --5:15 pm

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

Venue: Ramanujan Hall, Department of Mathematics

Host: Dipendra Prasad

Speaker: Chayan Karmakar

Affiliation: IIT Bombay

Title: Regular elements in semi-simple algebraic groups

Abstract: We will discuss the paper by Robert Steinberg, Regular elements of semi-simple algebraic groups Publications mathématiques de l'I.H.É.S., tome 25 (1965), p. 49-80.

Pre-synopsis Seminar

 Monday, 19 Feb, 9.45 am

 =================

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Priyanka Magar

Affiliation: IIT Bombay

Title: Smooth structures on PL-manifolds of dimensions between 8 and 10

Annual Progress Seminar

Monday, 19 Feb. 11 am

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

Venue: Ramanujan Hall

Host: Tony Puthenpurakal

Speaker: Samarendra Sahoo

Affiliation: IIT Bombay

Title: Quasi-pure resolutions and some lower bounds of Hilbert coefficients of Cohen-Macaulay modules.

Pre-synopsis Seminar

 Monday, February 19, 2024, 4 PM

 ======================

 Venue: Ramanujan Hall, Department of Mathematics

 Host: Dipendra Prasad

 Speaker: Dibyendu Biswas

 Affiliation: IIT Bombay

 Title: Stability of Tensor Products and Branching Laws for Classical Groups

Abstract: We consider the decomposition of tensor products of an irreducible representation of GL(n, C) and of the corresponding representations of GL(n + 1, C) and prove a 'stability result' about such tensor products. We go on to discuss similar questions for classical groups. Then we also discuss the branching laws for the classical pairs (GL(m,C), GL(n, C)), (Sp(2m, C), Sp(2n, C)), (SO(q, C), SO(p, C)) for all m<n, and all q<p.

Algebraic Stacks lecture series

Tuesday, 20 February, 11:30 AM

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

Venue: Room 113

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: Bhaskaracharya Pratishthana

Title: Separated and proper morphisms for algebraic stacks

Abstract: This talk is devoted to the generalization of algebraic stacks of the theory of separated and proper morphisms for schemes. We will begin by recalling the relevant basics of algebraic spaces and of algebraic stacks, and their morphisms, illustrated with elementary examples. This will be followed by a description of specializations of points on algebraic stacks, which is the background needed for the valuative criteria for universal closedness and separateness.

Lecture series on Bayesian analytics

20 and 22 Feb 11:35 am to 1:00 pm

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

Venues: 

Ramanujan  Hall on 22nd Feb.

Room 216 on 20 Feb.

Host: Radhendushka Srivastava

Speaker: Prof. Sujit Ghosh, NCSU.

Title: A short course on "Bayesian Analytics in Practice".

Abstract: The Bayesian paradigm provides a natural and practical way for building complex analytical models by expressing the joint model through a sequence of simpler conditional models, making it useful for various hierarchical data structures. This series of lectures will first introduce the general notions of Bayesian methods via hierarchical models, and then expand the topic with the more realistic and complex models that have recently emerged as a result of current Machine Learning (ML) literature. These models will be illustrated through practical applications to various real case studies avoiding much of the theoretical underpinnings. However, pointers to relevant theory will be provided as supplements with additional resources. Participants with basic knowledge of probability theory and statistical inferential framework will find the lectures useful in expanding their toolkit with the advanced use of Bayesian analytical methods. Popular topics such as prior sensitivity analysis, model comparisons, and uncertainty quantification for machine learning methods will be covered. In particular, the lectures will provide the necessary theory and practice for handling missing and censored data, a topic largely ignored in traditional ML methods. The concepts and methods discussed will be demonstrated primarily using R software illustrations, but the methodologies presented can also be carried out by other software (e.g., Python). Group activities during lab will be encouraged, allowing participants to have a hands-on experience. 

Topology and Related Topics Seminar

Tuesday, 20 February 2024, 12:00-1:00

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

Venue: Ramanujan Hall (Hybrid Lecture) (Speaker will be online). 

Host: Rekha Santhanam

Speaker: Suraj Krishna

Affiliation: Technion - Israel Institute of Technology

Title: Cubulating hyperbolic mapping tori.

Abstract: A group is cubulated if it acts properly and cocompactly on a CAT(0) cube complex, which is a generalisation of a product of trees. Cubulated groups possess remarkable algebraic and geometric features and are an important topic of study in geometric group theory. In this talk, I will show that semidirect products of hyperbolic groups with $\mathbb{Z}$, which are themselves hyperbolic, are cubulated. 

Two prominent examples of our setup are: (1) Mapping tori of fundamental groups of closed hyperbolic surfaces over pseudo-Anosov automorphisms, and (2) Mapping tori of free groups over atoroidal automorphisms.

Both of these classes of groups are known to be cubulated due to outstanding works. Our proof builds upon these two noteworthy results and places them in a unified framework. This is based on joint work with François Dahmani and Jean Pierre Mutanguha.

Algebraic Groups Seminar

Tuesday, February 20, 4 pm

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

Venue: Ramanujan Hall

Host: Shripad M. Garge

Speaker: Shripad M. Garge

Affiliation: IIT Bombay, Mumbai

Title: Homogeneous Spaces - II

Abstract: We study homogeneous spaces for linear algebraic groups.

Commutative Algebra Seminar

Tuesday, 20 Feb, 4 pm—5 pm

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

Venue: Room 215

Host: Tony J. Puthenpurakal

Speaker: Sudeshna Roy

Affiliation: TIFR Bombay

Title: Epsilon multiplicity in two-dimensional standard graded algebras

Abstract: The notion of epsilon multiplicity, a generalization of the Hilbert-Samuel multiplicity, was introduced by B. Ulrich and J. Validashti to detect the integral dependence of arbitrary ideals. This invariant is difficult to handle as there are examples where it can be irrational and the epsilon function is very far from being polynomial-like. Let $A$ be a standard graded normal domain of dimension two over a field with the unique homogeneous maximal ideal $m$. Let $I$ be a homogeneous ideal in $A$. Our objective is to show that the epsilon multiplicity of $I$ is a rational number.

Lecture series on Bayesian analytics

22 Feb 11:35 am to 1:00 pm

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

Venues:  Ramanujan  Hall 

Host: Radhendushka Srivastava

Speaker: Prof. Sujit Ghosh, NCSU.

Title: A short course on "Bayesian Analytics in Practice".

Abstract: The Bayesian paradigm provides a natural and practical way for building complex analytical models by expressing the joint model through a sequence of simpler conditional models, making it useful for various hierarchical data structures. This series of lectures will first introduce the general notions of Bayesian methods via hierarchical models, and then expand the topic with the more realistic and complex models that have recently emerged as a result of current Machine Learning (ML) literature. These models will be illustrated through practical applications to various real case studies avoiding much of the theoretical underpinnings. However, pointers to relevant theory will be provided as supplements with additional resources. Participants with basic knowledge of probability theory and statistical inferential framework will find the lectures useful in expanding their toolkit with the advanced use of Bayesian analytical methods. Popular topics such as prior sensitivity analysis, model comparisons, and uncertainty quantification for machine learning methods will be covered. In particular, the lectures will provide the necessary theory and practice for handling missing and censored data, a topic largely ignored in traditional ML methods. The concepts and methods discussed will be demonstrated primarily using R software illustrations, but the methodologies presented can also be carried out by other software (e.g., Python). Group activities during lab will be encouraged, allowing participants to have a hands-on experience. 

Algebra and Number Theory Seminar

Thursday, 22 February 2024, 4:00 pm

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

Venue: Ramanujan hall

Host: Sandip Singh

Speaker: Sumit Chandra Mishra

Affiliation: IISER Mohali

Title: Local-global principles for norms and products of norms over semi-global fields

Abstract: A well-known result of Hasse states that the local-global principle holds for norms over number fields for cyclic extensions. In other words, if L/F is a cyclic extension of number fields then an element \lambda \in F^{\times} is in the image of norm map N_{L/F}: L^{\times} \rightarrow F^{\times} if and only if \lambda is in the image of the norm map locally everywhere,  i.e., for  completions associated to all archimedean and non-archimedean places of F.  In this talk, we would discuss local-global principles for norms and product of norms over fields which are function fields of curves over complete discretely valued fields, for example, \mathbb{C}((t))(x).

Annual Progress seminar 

Friday, 23 February, 11:30 am -12:30 pm 

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

Venue: Ramanujan Hall

Host: Rekha Santhanam 

Speaker: Lekha Das (22D0790)

Title: Whitehead's Theorem

Abstract: We aim to show that if f is a continuous mapping between CW complexes X and Y inducing isomorphisms on all homotopy groups, then f is a homotopy equivalence.

Abstract: The lowest-order nonstandard finite element schemes namely Morley, discontinuous Galerkin, and 𝐶0-interior penalty methods are presented for numerical discretization of the biharmonic wave equation with clamped boundary conditions in a unified framework. The regularity results we presented ensure the boundedness of various norms for the different derivatives of the continuous solution required in the error analysis. Stability results are obtained for both semidiscrete and fully discrete (implicit and explicit) schemes. A novel Ritz projection operator is used to obtain the error estimates. The expected convergence rates are confirmed by the numerical results, highlighting the versatility of the methods.

Abstract:  The limiting spectral distribution (LSD) of high-dimensional Kendall's correlation matrix and tests of independence based on this matrix have been studied in the literature when observations are absolutely continuous with respect to the Lebesgue measure, and are independent and identically distributed. We investigate the LSD of this matrix under much weaker assumptions which accommodate discrete and/or non-identical observations, and also identify the limit distribution by using free probability. We also propose a graphical test of independence under these assumptions.

Abstract: Statistical test for high-dimensional means under missing observations seems to be very rare in the literature. We propose a new two-sample test for high-dimensional means based on independent observations with missing values. The critical region of the proposed test is based on a bootstrap estimate of the sample quantiles of the proposed test statistic. Unlike the existing tests, this test does not require any distributional assumptions or any particular correlation structure of the covariance matrices.  We establish the Gaussian approximation result for the proposed test statistic which is a non-trivial extension of the two-sample Gaussian approximation result with no missing values.  The rate of accuracy of the bootstrap approximation of the sample quantile of the proposed test statistic is also derived. This Gaussian approximation result and the accuracy of the bootstrap estimators together provide the theoretical guarantees on the size and power of the proposed test.

Annual Progress seminar

Wed, 28 Feb 11:00 am -12:00 pm

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

Venue: Ramanujan Hall

Host: Suresh Kumar

Speaker: Vikrant Desai (174093013)

Title: Mean field games with degenerate diffusions

Abstract: We study mean field games with degenerate diffusions. We prove

the existence of a mean field equilibrium for class of mean field games with state dynamics given by controlled degenerate diffusions.

Abstract: In this seminar, we discuss three types of Kirchhoff-type PDE with memory involving usual time derivative, fractional time derivative, and fractional Laplacian. In the first half, we focus on well-posedness and regularity results. Then, we derive a semi-discrete error analysis of Galerkin FEM by defining a modified Ritz-Volterra projection  operator.  Finally, we develop fully discrete formulations and implement some examples to validate the proven theoretical results.

Abstract: In the first half of the talk, I will introduce key ideas and methodologies of causal inference in the scientific foundations of evidence-based medicine. I will trace the history (20th century) of modern drug development through the evolution of the law known as The Food, Drugs and Cosmetic Act. The law was instituted and further evolved in response to some major global health events/crises. The US Congress gave sweeping powers to the FDA to interpret and execute the law to develop a drug approval process based on empirical data and science. We will also trace the law’s impact on global drug policies. In that context, we will discuss ‘adequate and well-controlled studies and the role of statistics in the process. We will discuss some of the important examples on the way. 

Bio of the speaker:  As a statistician, I worked in academia, the federal government, and the Biotechnology/Pharma industry. I was in leadership and supervisory positions in the government (Department of Biostatistics at Center for Drug Development and Research, Food and Drug Administration), and in the pharma industry at Johnson & Johnson  (large pharma), Amgen (mid-size pharma when I began there) and Esperion (small biotech). The early years of my career (about 10 years) were in academic positions starting as a lecturer at Bombay University followed by several years at the University of California, Santa Barbara with a brief sojourn to IIT, Mumbai. I continued methodological research and publishing during my time in the government and industry as well. Currently, I offer statistical and regulatory consulting services to pharmaceutical companies specializing in strategic and secondary review of submission modules, DMC statistical membership, strategic reviews of protocols, and statistical analysis plans mostly in dermatology, immunology, cardiovascular, and antiviral therapeutic areas.  I am an elected Fellow of the American Statistical Association.

Abstract: The classical theorem of Beckman and Quarles states the following: A function from the Euclidean plane to itself that preserves unit distances must preserve all distances. We will discuss some key steps of the proof. We will also discuss our recent result generalizing this theorem. This is joint work with Ben Schmidt. We will try to make the talk accessible to a wide audience.