Commutative Algebra seminar
Speaker: Sayed Sadiqul Islam, IIT Bombay
Host: Tony Puthenpurakal
Title: Zariski Closedness of Supports of Local Cohomology Modules III
Time, day and date: 11:30:00 AM - 12:30:00 PM, Monday, October 05
Venue: Room 215
Abstract: These talks are based on two recent papers, one by Katzman and Zhang and the other by Gintz and Zhang, concerning the Zariski closedness of supports of local cohomology modules in prime characteristic.

We shall show that the support of H_I^i(R/gR) is Zariski closed when R is a regular ring of prime characteristic and g is an element R . We shall also discuss an extension of this result to the case in which R has finitely many isolated singular points. Furthermore, we shall prove that if f_1 and f_2 form a regular sequence, then the support of H_I^j(R/(f_1,f_2)) is Zariski closed.

Plancherel measure seminar
Speaker: Devadatta Hegde, School of Mathematics, TIFR
Host: Ravi Raghunathan
Title: Paper of Waldspurger on the Plancherelmeasure for $p$-adic groups following Harish-Chandra
Time, day and date: 5:00:00 PM – 6:15:00 PM, Monday, October 05
Venue: Room 215
Abstract: -

PDE and Numerical Analysis seminar
Speaker: Amiya Pani, BITS Pilani, KK Birla Goa campus
Host: Harsha Hutridurga
Title: On Backward Time-Fractional Diffusion Problems: A Unified Approach
Time, day and date: 2:30:00 PM – 3:30:00 PM, Tuesday, October 06
Venue: Ramanujan Hall
Abstract: Inverse fractional diffusion problems, particularly backward problems, have received considerable attention in recent years. The resulting literature is diverse and sometimes fragmented, owing to the variety of models and numerical techniques employed. In this talk, we present a unified analysis of the quasi-boundary value method for solving backward problems associated with a broad class of time-fractional diffusion models. The regularized problem can be discretized using diverse spatial and temporal strategies. As a representative example, we analyze a piecewise linear finite element method in space, outline a rigorous error analysis for the semidiscrete scheme, and propose practical guidelines for selecting discretization and regularization parameters relative to the noise level. The unified nature of the analysis ensures its applicability to various fractional models, spatial discretizations, and more.

This is joint work with Samir Karaa and Mariam Al-Maskari (Sultan Qaboos University, Muscat, Oman).

Probability and Statistics Seminar
Speaker: Dr. Parichoy Pal Choudhury, KCDH, IIT Bombay
Host: Siuli Mukhopadhyay
Title: Statistical approaches to disease risk prediction with applications in risk-stratified disease prevention
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, October 07
Venue: Ramanujan Hall
Abstract: Risk-stratified disease prevention involves tailoring of health decisions about screening and prevention based on the individualized risk predictions. This requires a comprehensive understanding of the risk factors, including genetic variants, biomarkers, lifestyle/behavioral and environmental factors leading to the development of a model for predicting absolute risk of a disease of interest. Absolute risk model development requires information on relative risks of the risk factors, population-based age-specific disease incidence rates and competing event rates and population distributions of the risk factors. Such a model needs to be validated ideally in independent prospective cohorts before clinical applications. In this talk, I will describe the Individualized Coherent Absolute Risk Estimator (iCARE) software tool for implementing absolute risk estimation of a disease integrating multiple data sources leveraging the best information available for each of the input parameters and standardized approaches for risk model validation. I will describe multiple recent applications of this tool in the risk prediction of breast, bladder and ovarian cancers. Model validation in two-phase study settings often involve scenarios where expensive biomarkers (e.g., polygenic risk score) are measured in smaller subsample of a prospective cohort and subjects may be selected in the sub-sample using complex sampling designs. I will describe a simple partial risk score-based method for improving precision of model validation statistics (e.g., AUC) using the partial risk factors from the full cohort and complete risk factors from the subsample. I will show two applications of this method: (1) in lung cancer risk prediction with questionnaire-based risk factors and inflammation biomarkers and (2) in breast cancer risk prediction with questionnaire-based risk factors and PRS. If time permits, I will briefly discuss additional ongoing projects.

Geometric Analysis Talk
Speaker: Dr. Ramesh Mete, IIT Bombay
Host: Saikat Mazumdar
Title: On the existence of coupled extremal Kähler metrics on ruled surfaces.
Time, day and date: 3:00:00 PM – 4:00:00 PM, Wednesday, October 07
Venue: Ramanujan Hall
Abstract: We investigate the existence conditions for coupled extremal Kähler metrics on minimal ruled surfaces over a genus 2 Riemann surface. Using the Calabi ansatz to reduce the coupled extremal equations to ordinary differential equations, we prove that a pair of coupled extremal metrics exists for normalized Kähler classes $\Omega_a$ and $\Omega_b$ if and only if their parameters $(a,b)$ belong to an explicitly defined open region $\mathcal{S}_{ext}$ in the positive real quadrant. Furthermore, we show that this existence region inherently contains the diagonal segment corresponding to classical extremal Kähler metrics.

Ph.D. Defence
Speaker: Mr. Rishi Das, IIT Bombay
Host: Harsha Hutridurga
Title: Robust Mathematical Analysis and Numerical Methods for Flow-Transport Problems
Time, day and date: 4:00:00 PM – 5:00:00 PM, Wednesday, October 07
Venue: Ramanujan Hall (https://zoom.us/j/92827897299?pwd=55qKZRjyi6JM8dPTupMAGekskLkkk3.1)
Abstract: Mixed finite element methods naturally give rise to saddle-point structures. We exploit this intrinsic structure to construct parameter-dependent weighted spaces and establish robust well-posedness for three models arising in fluid flow and transport through porous media: the Darcy system, the Darcy—Forchheimer system, and the coupled Darcy--Forchheimer and advection-diffusion-reaction system. For each model, we develop parameter-robust discretization, derive a priori error estimates with constants independent of the physical parameters, and construct robust block-diagonal preconditioners within the operator preconditioning framework. Numerical results illustrate the robustness of the proposed methods across a wide range of physically relevant parameter regimes.

Seminar
Speaker: Harsha Hutridurga, IIT Bombay
Title: Two Materials and One Surprisingly Hard Problem in Optimal Design
Time, day and date: 3:00:00 PM – 4:00:00 PM, Thursday, October 08
Venue: Ramanujan Hall
Abstract: Given fixed amounts of two materials, how should they be arranged for the best performance? Surprisingly, in many natural settings no optimal arrangement exists: better and better designs oscillate on ever finer scales and never converge to a clean shape. Resolving this led to homogenization, relaxation, and the study of which effective properties mixtures can achieve. I will survey this story, from laminates and the Hashin-Shtrikman bounds to modern topology optimization, and walk through one short argument showing how such bounds arise. I will close with open problems, including three-phase conductivity and elasticity, that are easy to state and still await new ideas.

Algebraic Groups Seminar
Speaker: Shripad M. Garge
Title: Digne-Michel
Time, day and date: 5:00:00 PM – 6:00:00 PM, Thursday, October 08
Venue: Room 215
Abstract: -
 

Students' Seminar
Speaker: Ankita Dargad, IIT Bombay
Host: Suman Kumar Sahoo
Title: Ferry Cover with Connectivity Constraints
Time, day and date: 4:30:00 PM – 5:30:00 PM, Friday, October 09
Venue: Ramanujan Hall (https://meet.google.com/cxf-egsq-igz)
Abstract: The classical Ferry Cover problem asks for the minimum boat capacity needed to transport all vertices of a graph across a river such that no edge remains on either bank at any time - a requirement that the banks induce stable (independent) sets. We study a natural generalization in which the banks must satisfy an arbitrary graph property. For hereditary properties such as acyclicity or planarity, we show that the structural characterization of small-boat and large-boat graphs established by Csorba, Hurkens, and Woeginger extends directly.
We then turn to the connected-bank variant, where the property of interest - connectedness - is not hereditary: both banks must induce connected subgraphs throughout the transfer. We provide a complete characterization of graphs that can be transferred with a boat of size one (boat-1 graphs): a connected graph is boat-1 if and only if its block-cut tree is a path. This characterization yields a linear-time recognition algorithm. As a consequence, we show that every biconnected graph is boat-1, since such graphs admit an st-numbering. We also develop an efficient algorithm for determining the boat number of trees. Our work opens new directions for river-crossing problems under non-hereditary bank constraints.