Commutative Algebra seminar
Speaker: Kaushik Khamari, IIT Bombay
Host: Tony Puthenpurakal
Title: On Cohen-Macaulay Property of Affine Semigroup Rings
Time, day and date: 3:00:00 PM - 4:00:00 PM, Thursday, April 02
Venue: Ramanujan Hall
Abstract: In this talk, we discuss the Cohen-Macaulay property of affine semigroup rings. In the process, we describe the relationship between local cohomology modules of these rings and the homology of certain simplicial complexes. As an application, we present a proof of Hochster's Theorem, which asserts that normal semigroup rings are Cohen-Macaulay. We follow the paper of N.V. Trung and L.T. Hoa, "Affine Semigroups and Cohen-Macaulay Rings generated by monomials.

Algebraic groups seminar
Speaker: Patrick Polo, IIT Bombay
Host: Shripad Garge
Title: Properties of Harish Chandra induction and restriction
Time, day and date: 4:00:00 PM - 5:30:00 PM, Thursday, April 02
Venue: Room No .215
Abstract: We will explain how properties like transitivity of induction and Mackey formula rely on structural properties of parabolic subgroups and their unipotent radicals.

Statistics Seminar
Speaker: Savita Pareek, Post doctoral fellow MIT Sloan
Host: Siuli Mukhopadhyay
Title: Semiparametric Dynamic Copula Models using Rolling-window Portfolio Optimization
Time, day and date: 4:30:00 PM – 5:30:00 PM, Thursday, April 02
Venue: Ramanujan Hall
Abstract: The mean-variance portfolio model, based on the risk-return trade-off for optimal asset allocation, remains fundamental in portfolio optimization. However, its reliance on restrictive assumptions about asset return distributions limits its applicability to real-world data. Parametric copula structures provide a novel way to overcome these limitations by accounting for asymmetry, heavy tails, and time-varying dependencies. Existing methods have been shown to rely on fixed or static dependence structures, thus overlooking the dynamic nature of the financial market. 
In this study, a semiparametric model is proposed that combines nonparametrically estimated copulas with parametrically estimated marginals to allow all parameters to dynamically evolve over time. A novel framework was developed that integrates time-varying dependence modelling with flexible empirical beta-copula structures. Marginal distributions were modeled using the skewed generalized t-family. This effectively captures asymmetry and heavy tails and makes the model suitable for predictive inferences in real-world scenarios. Furthermore, the model was applied to rolling windows of financial returns from the USA, India, and Hong Kong economies to understand the influence of dynamic market conditions. The approach addresses the limitations of models that rely on parametric assumptions. By accounting for asymmetry, heavy tails, and cross-correlated asset prices, the proposed method offers a robust solution to optimize diverse portfolios in an interconnected financial market. Through adaptive modeling, it allows for better management of risk and return across varying economic conditions, leading to more efficient asset allocation and improved portfolio performance.

Student Seminar
Speaker: Rajendra Meena, IIT Bombay
Host: Suman Kumar Sahoo
Title: The Brahmagupta-Pell Equation: Solutions and Structure
Time, day and date: 5:35:00 PM – 6:35:00 PM, Thursday, April 02
Venue: Ramanujan Hall
Abstract: The Brahmagupta–Pell equation x 2 − dy2 = 1, where d is a non-square integer, is a classical Diophantine equation that has fascinated mathematicians for centuries due to its simple form yet surprisingly rich structure. A natural question is whether such an equation admits non-trivial integer solutions, and if so, how one can describe all of them. In this talk, we address these questions by proving the existence of non-trivial solutions and introducing the notion of a fundamental solution, from which all other solutions can be generated. We also describe all solutions explicitly and briefly discuss method for solving the generalized equation x 2 − dy2 = n, where n ∈ Z. The ideas will be illustrated with examples. If time permits, we will also mention the classical Indian” Chakravala method” for computing the fundamental solution.