Statistics and Probability seminar
Speaker: Sagnik Nandy, The Ohio State University, Columbus
Host: Koushik Saha
Title: Degree-Heterogeneous Networks: Optimality, Inference, and Privacy
Time, day and date:  10:00:00 AM – 11:00:00 AM, Tuesday, March 10
Venue: https://meet.google.com/wjs-deag-ysh Online
Abstract: Networks arising in real relational datasets exhibit strong degree heterogeneity: the propensity of nodes to participate in interactions varies widely. When the full set of edges is unavailable or too sensitive to release, degree summaries often become the primary data object, motivating the classical $\beta$-model and its higher-order extension to hypergraphs. In this talk, I develop a statistical theory for degree-heterogeneous $r$-uniform hypergraphs using the hypergraph $\beta$-model. First, I characterize sharp estimation rates for the maximum likelihood estimator (MLE) of the model parameters in both $\ell_2$ and $\ell_\infty$ losses, highlighting an effective sample size scaling of order $n^{r-1}$ per node parameter. I then prove that these rates are minimax optimal. Next, I study inference where I characterize minimax detection thresholds for testing the presence of degree heterogeneity.
Finally, I turn to privacy. Using edge differential privacy, I quantify the statistical price of privacy for estimation in $\beta$ models and show a sharp separation between local and central privacy regimes. Simulations illustrate the predicted privacy–utility tradeoffs, and an application to the Enron email hypergraph demonstrates the impact of different regimes of privacy on link prediction in a real organizational communication network.
(Work based on joint papers with B. Bhattacharya and B. Mandal.)

Analysis Seminar
Speaker: Dr. Pritam Ganguly, ISI Kolkata
Host: Santanu Dey
Title:  From Uncertainty to Rellich-type estimates: A journey through $ mathbb{R}^n$, the Heisenberg group, and symmetric spaces.
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, March 11
Venue: Ramanujan Hall
Abstract: In this talk, we explore two classical yet evolving themes in analysis: uncertainty principles and Rellich-type estimates for eigenfunctions of the Laplacian. We begin with a theorem of Ingham, which characterizes the optimal decay of the Fourier transform of a compactly supported function on the real line. This result captures a sharp form of the (qualitative) uncertainty principle. We then examine an analogue in the setting of the Heisenberg group by investigating the optimal operator-valued decay of the group Fourier transform of compactly supported functions.

In the second part, we turn to a classical result of Rellich concerning solutions of the Helmholtz equation, a fundamental equation in mathematical physics. In his 1943 paper, Rellich proved that in the exterior region $\Omega = \{x \in \mathbb{R}^n \mid |x| > R_0\},$ there are no nontrivial solutions $ f $ of the Helmholtz equation $\Delta f +\lambda f=0, \qquad \lambda > 0,$ such that $ f \in L^2(\Omega) $. His proof is based on establishing the $L^2$-asymptotic growth of solutions over annuli. We discuss this result and its extension to $L^p$ in the context of rank one Riemannian symmetric spaces of non-compact type, highlighting how exponential volume growth and the dependence of the $L^p$-spectrum of the Laplace--Beltrami operator on $p$ lead to genuinely non-Euclidean phenomena.

Statistics and Probability seminar
Speaker: Sourish Das, Chennai Mathematical Institute
Host: Parthanil Roy
Title: Jacobi Prior: An Alternative Bayesian Method for Supervised Learning
Time, day and date: 02:30:00 PM – 03:30:00 PM, Wednesday, March 11
Venue: Ramanujan Hall
Abstract: The Jacobi prior offers an alternative Bayesian framework for predictive modelling, designed to achieve superior computational efficiency without compromising predictive performance. This scalable method is suitable for image classification and other computationally intensive tasks. Compared to widely used methods such as Lasso, Ridge, Elastic Net, uniLasso, the MCMC-based Horseshoe prior, and non-Bayesian machine learning methods including Support Vector Machines (SVM), Random Forests, and Extreme Gradient Boosting (XGBoost), the Jacobi prior achieves competitive or better accuracy with significantly reduced computational cost. The method is well-suited to distributed computing environments, as it naturally accommodates partitioned data across multiple servers. We propose a parallelisable Monte Carlo algorithm to quantify the uncertainty in the estimated coefficients. We establish that the Jacobi estimator is asymptotically close to, and asymptotically equivalent to, the posterior mode under the Jacobi prior.  To demonstrate its practical utility, we conduct a comprehensive simulation study comprising seven experiments focused on statistical consistency, prediction accuracy,  scalability, sensitivity analysis and robustness study. We further present three real-data applications: credit risk modelling using U.S. Small Business Administration (SBA) loan default data, multi-class classification of stars, quasars, and galaxies using Sloan Digital Sky Survey data, and spinal degeneration classification using sagittal MRI scans from the RSNA 2024 Lumbar Spine Degenerative Classification Challenge. In the spine classification task, we extract last-layer features from a fine-tuned ResNet-50 model and evaluate multiple classifiers, including Jacobi-Multinomial logit regression, SVM, and Random Forest. The Jacobi prior achieves state-of-the-art results in recall and predictive stability, especially when paired with domain-specific features. In addition, it substantially outperforms competing methods in runtime by large margins, leading to significant reductions in computational cost. This highlights its potential for scalable, high-dimensional learning in medical image analysis.

All code and datasets used in this paper are available at: 
https://github.com/sourish-cmi/Jacobi-Prior/ 
 

Mathematics Colloquium
Speaker: Sabyasachi Mukherjee, Tata Institute of Fundamental Research
Host: Parthanil Roy
Title: Fatou-Sullivan dictionary: Where rational dynamics meets Kleinian groups
Time, day and date: 04:00:00 PM – 05:00:00 PM, Wednesday, March 11
Venue: Ramanujan Hall
Abstract: Two central themes in holomorphic dynamics are the iteration of rational maps and the action of Kleinian groups on the Riemann sphere. Although these theories developed along largely independent paths, they exhibit striking conceptual parallels—a phenomenon often referred to as Sullivan’s Dictionary. As early as the 1920s, Fatou envisioned that these two classes of dynamical systems could be unified within the framework of iterated algebraic correspondences.

After surveying fundamental results in rational dynamics and Kleinian groups, we will present concrete realizations of Fatou’s vision by constructing algebraic correspondences that unite the behaviors of complex polynomials and Fuchsian groups in a single dynamical plane. We will also discuss examples of rational Julia set realizations of Kleinian limit sets, such as classical Apollonian-like gaskets. Timepermitting, we will outline the main analytic and algebraic ideas underlying this program, and highlight several applications to other areas of mathematics.

Topology seminar
Speaker: Bittu Singh, IIT Bombay
Host: Rekha Santhanam
Title: Rational Homotopy theory
Time, day and date: 11:45:00 AM – 1:00:00 PM, Thursday, March 12
Venue: Room 215
Abstract: Dennis Sullivan introduced a framework in which the rational homotopy type of a topological space is encoded by commutative differential graded algebras (CDGAs). Earlier, Daniel Quillen established an equivalence between the rational homotopy category and an algebraic category, although this equivalence arises through a sequence of intermediate categorical constructions. In their paper “On PL De Rham Theory and Rational Homotopy Type”, A. K. Bousfield and Victor K. A. M. Guggenheim gave a precise formulation of Sullivan’s equivalence using the language of model category theory.
In this talk, we outline this approach and discuss the relationship between topological spaces and CDGAs in rational homotopy theory.

Commutative algebra seminar
Speaker: Prof. R. V. Gurjar
Host: Tony J P
Title: Singularities of plane curves III
Time, day and date: 3:00:00 PM – 4:00:00 PM, Thursday, March 12
Venue: Ramanujan Hall
Abstract: We will consider reduced power series f(X,Y) over an algebraically closed field k of characteristic 0. Following topics about R=k[[X,Y]]/(f) will be discussed. Algebraicity results of Artin, Hironaka-Rossi, Samuel, Basic intersection theory of place curves, Conductor ideal, Value semi-group of R, Gorenstein's result, Dedekind's Conductor Formula, Jung's formula connecting the Milnor number and conductor, Applications of Dedekind's formula. We will give Abhyankar's proofs of Dedekind and Gorenstein results. If time permits, Abhyankar-Moh result about generators of the value semi-group will be discussed.

Algebraic groups seminar
Speaker: Hariom Sharma, IIT Bombay
Host: Shripad Garge
Title: Subgroups of maximal rank
Time, day and date: 04:00:00 PM – 05:00:00 PM, Thursday, March 12
Venue: Room no. 215
Abstract: We continue with the third chapter of Digne Michel and study quasi-closed subsets of root systems.