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.