Geometry and Topology seminar
Speaker: Sudarshan Gurjar, IIT Bombay
Host: Rekha Santhanam
Title: Kodaira Embedding Theorem
Time, day and date: 11:30:00 AM - 12:30:00 PM, Monday, October 13
Venue: Ramanujan Hall
Abstract: This is a continuation of my earlier talk. After recalling the theorem, I will outline a proof of it.

Ph.D thesis defense
Speaker: Samarendra Sahoo
Date: 13 October 2025
Day: Monday
Venue: Ramanujam Hall
Time: 4-5 pm

Title: Minimal free resolutions, $I$-stable filtrations and some lower
bounds of Hilbert coefficients.

Abstract: Let $(A, \mathfrak{m})$ be a Cohen-Macaulay local ring and $M$ a
Cohen-Macaulay $A$-module. We study certain lower bounds for the Hilbert
coefficients of $M$ and investigate the conditions under which the
associated graded module $G_{\mathfrak{m}}(M)$ is Cohen--Macaulay. Let
$M_i$ denote the $i$-th syzygy of $M$, and suppose that $(A,
\mathfrak{m})$ is a complete intersection ring. We examine the asymptotic
behavior of $e_1(M_i)$, the first Hilbert coefficient of $M_i$, and
$\operatorname{reg}(G_{\mathfrak{m}}(M_i))$, the Castelnuovo--Mumford
regularity of the associated graded module, for sufficiently large $i$.
Furthermore, for an $\mathfrak{m}$-primary ideal $I$ and an $I$-stable
filtration $F = \{I_n\}_{n \ge 0}$, we analyze the situation when $\dim
A(F)/A(I) = \dim A$, where $A(F)$ denotes the Rees algebra with respect to
the filtration $F$.

Mathematics Colloquium
Speaker: Saikat Mazumdar, IIT Bombay
Title: Conformal Powers of the Laplacian, Q-Curvature, and the Compactness Problem
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, October 15
Venue: Ramanujan Hall
Abstract: The conformal powers of the Laplacian, known as the GJMS operators, are a family of conformally invariant differential operators with leading term a power of the Laplacian, and the Q-curvature is the associated scalar invariant. They arise naturally and provide a unifying framework connecting important developments in geometry, analysis, and mathematical physics.
In this talk, I will consider the question of compactness of constantQ-curvature metrics on a closed Riemannian manifold. This is the higher-order analogue of the compactness problem for conformal metrics with constant scalar curvature. I will outline the background, motivations, and key challenges, and present some of my results in this direction. This is a recent work with Bruno Premoselli (ULB Brussels).

Geometry and Topology seminar
Speaker: Pritthijit Biswas, IIT Bombay
Host: Rekha Santhanam
Title: Brill-Noether loci inside the moduli space of stable vector bundles over curves
Time, day and date: 11:30:00 AM - 12:30:00 PM, Thursday, October 16
Venue: Ramanujan Hall
Abstract: Let X be a smooth projective curve of genus g over the field C. Let M_{X}(2, L)denote the moduli space of stable rank 2 vector bundles on X with fixed determinant L of degree 2g-1. Consider the Brill-Noether subvariety W ^{1}_{X} (2, L) of M_{X} (2, L) which parametrises stable vector bundles having at least two linearly independent global sections. In this talk, for generic X and L, I would outline a proof of the fact that W ^{1}_{X} (2, L) is stably-rational when g = 3 and unirational when g = 4. This is a part of joint work with Jaya NN Iyer.

Student Seminar
Speaker: Saibur Alom, IIT Bombay
Host: Santanu Dey
Title: Fourier Transform of Hilbert Transform on R^n
Time, day and date: 2:00:00 PM – 2:45:00 PM, Thursday, October 16
Venue: Room 113

Student Seminar
Speaker: Subir Dakshi, IIT Bombay
Host: Santanu Dey
Title: The Hardy-Littlewood maximal function
Time, day and date: 2:45:00 PM – 3:30:00 PM, Thursday, October 16
Venue: Room 113

Commutative algebra seminar
Speaker: Prof. R. V. Gurjar
Date: 16 Oct 2025
Day Thursday
Venue: Ramanujam Hall
Time 4-5 pm
Title: Resolution of singularities of algebraic or analytic varieties.

Abstract.
First I will show how to resolve curve singularities (this must be
well-known to many). Then discuss Jung-Hirzebruch argument to get the
equation of a surface in three variables in a very simple form, and then
resolve its singularities. This will complete the proof of resolution of
singularitiies of surfaces. Next I will discuss some easy but useful
results about resolution type results (resolution of indeterminacies,
principalization, general notion of resolution of singularities, and their
relations to each other).

Statistics/Probability Seminar
Speaker: Mayukh Choudhury, IIT Bombay
Host: Debraj Das
Title: Asymptotic Theory of $K$-fold Cross-validation in Lasso and the Validity of Bootstrap
Time, day and date: 5:00:00 PM – 6:00:00 PM, Thursday, October 16
Venue: Ramanujan Hall (meet.google.com/bgr-tffn-cem)
Abstract: Lasso is one of the widely used regularization methods in regression. Statisticians usually implement Lasso in practice by choosing the penalty parameter in a data-dependent way, the most popular being the $K-$fold cross-validation (or $K-$fold CV). However, inferential properties, such as the variable selection consistency and $n^{1/2}-$consistency, of the $K-$fold CV based Lasso estimator and validity of the Bootstrap approximation are still unknown. In this talk, we will discuss about $n^{1/2}-$consistency of the $K$-fold CV based penalty and utilizing that we explore the aforementioned inferential properties of the underlying Lasso estimator. Additionally, we establish the validity of Bootstrap in approximating the distribution of the $K-$fold CV based Lasso estimator. We validate our Bootstrap method in finite samples based on simulations.

Analysis seminar
Speaker: Prof. S. Thangavelu
Host: Sanjoy Pusti
Title: On Hardy's inequality for fractional powers of the sublaplacian
Time, day and date: 4:00:00 PM – 5:15:00 PM, Friday, October 17
Venue: Ramanujan Hall
Abstract: In these lectures we plan to give an introduction to Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group. We will recall the case of the standard Laplacian on $ \R^n $ and give a proof of Hardy's inequality for $ (-Delta)^s $ using extension problem and trace Hardy inequality. We will then develop necessary background for studying the modified extension problem for the sublaplacian, establish an analogue of trace Hardy inequality and deduce Hardy's inequality.