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.

Commutative Algebra seminar
Speaker: Sayed Sadiqul Islam, IIT Bombay
Host: Tony Puthenpurakal
Title: Zariski Closedness of Supports of Local Cohomology Modules
Time, day and date: 11:30:00 AM - 12:30:00 PM, Monday, October 12
Venue: Room No .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.

Mathematics Colloquium
Speaker: Parthe Pandit, IIT Bombay
Host: Parthanil Roy
Title: Precise Asymptotics of Statistical Linear Inverse Problems in High Dimensions
Time, day and date: 4:15:00 PM - 5:15:00 PM, Wednesday, October 14
Venue: Ramanujan Hall
Abstract: This is a tutorial-style talk on a class of algorithms studied under the name of Approximate Message Passing. These algorithms have a scalar State Evolution which enables characterizing the asymptotic behavior of their fixed points in a precise manner in the proportional high dimensional regime. This enables rigorous analysis of a large class of optimization formulations such as regularized M-estimation for generalized linear models, and structural risk minimization in machine learning.

Special Session
Speaker: Madhusudan Manjunath, IIT Bombay
Title: Mathematics for Well-Being (MW) IV: Math in the AI times cries for MW
Time, day and date: 10:30:00 AM - 11:30:00 AM, Thursday, October 15
Venue: Ramanujan Hall
Abstract: After a careful review of the current situation with AI involvement in Mathematics, we formulate key questions that arise in this context. We outline the guidance MW provides here. Special emphasis will be given to India's unique position to guide the community through this challenging time.

Geometry and Topology seminar
Speaker: Sumanta Das, IIT Bombay
Host: Rekha Santhanam
Title: Exotic Open Contractible 3-Manifolds and Asymptotic Limits
Time, day and date: 2:30:00 PM - 3:30:00 PM, Thursday, October 15
Venue: Ramanujan Hall
Abstract: In 1934, Whitehead proposed a proof of the Poincaré conjecture. It rested on the claim that every open contractible 3-manifold is homeomorphic to Euclidean 3-space. The following year he found the claim to be false. He then published a counterexample, now known as the Whitehead manifold. McMillan later produced uncountably many such manifolds, no two of them homeomorphic.

Telling Euclidean space apart from these exotic manifolds is delicate: they are contractible, so no ordinary homotopy invariant distinguishes them, and the whole of the difference sits at infinity, where they are wild. The invariant that does see it is the Brown-Grossman proper fundamental group, extracted from an inverse sequence of ordinary fundamental groups by a tool developed by Brown. We call this tool the asymptotic limit functor.

In this talk, we will explain the construction, and show that the asymptotic limit behaves considerably better than the inverse limit and its first derived functor. We will then use it to quantify the wildness algebraically: every open contractible 3-manifold not homeomorphic to Euclidean 3-space has exactly one end, and its Brown-Grossman proper fundamental group is uncountable and perfect. This contrasts sharply with the ordinary fundamental group of a manifold, which is always countable.

Joint work with Rekha Santhanam. Based on arXiv:2609.14063.

Algebraic Groups Seminar
Speaker: Shripad M. Garge, IIT Bombay
Title: -
Time, day and date: 5:00:00 PM – 6:00:00 PM, Thursday, October 15
Venue: Room 215
Abstract: Brief review of chapter 8, a review of chapter 9, and possibly the first section of chapter 10. 

Students' Seminar
Speaker: Harshita Agarwal, IIT Bombay
Host: Suman Kumar Sahoo
Title: Physics-Informed Neural Networks: A mathematical Introduction|
Time, day and date: 4:00:00 PM – 5:00:00 PM, Friday, October 16
Venue: Ramanujan Hall (https://meet.google.com/aon-mshf-ovk)
Abstract: Neural networks are usually trained using only examples. But often we also know a simple mathematical rule the answer should follow — even when we don't have much data to work with. This talk introduces Physics-Informed Neural Networks (PINNs): a simple way to teach a network to respect such a rule, using nothing more than ideas we already know — the chain rule and gradient descent. We'll build the idea step by step, from how a basic network learns, to how it can check a rule exactly, and finally see it all come together on one simple example. No physics or PDE background is needed; the talk stays friendly and non-technical throughout, explained through pictures and one running example rather than long derivations.

Commutative Algebra seminar
Speaker: Kaushik Khamari, IIT Bombay
Host: Tony Puthenpurakal
Title: On licci monomial ideals
Time, day and date: 11:30:00 AM - 12:30:00 PM, Monday, October 19
Venue: Room No .215
Abstract: In this talk, we discuss basic notions of complete-intersection liaison and licci ideals, which are ideals in the liaison class of a complete intersection. We then introduce universal linking sequences corresponding to an arbitrary sequence of links. As an application, we give a characterization of finite colength licci monomial ideals in a polynomial ring over a field, due to Huneke and Ulrich.

Plancherel Measure seminar
Speaker: Dipendra Prasad, IIT Bombay
Title: Plancherel Measure
Time, day and date: 5:00:00 PM – 6:00:00 PM, Monday, October 19
Venue: Room 215
Abstract: following Waldspurge

Algebraic Groups Seminar
Speaker: Shripad M. Garge, IIT Bombay
Title: -
Time, day and date: 5:00:00 PM – 6:00:00 PM, Thursday, October 22
Venue: Room 215
Abstract: tenth chapter of Digne Michel

Students' Seminar
Speaker: Surjeet Singh, IIT Bombay
Host: Suman Kumar Sahoo
Title: Curvature of Curves and Surfaces
Time, day and date: 3:30:00 PM – 4:30:00 PM, Friday, October 23
Venue: Ramanujan Hall (https://meet.google.com/meet/dvs-xapw-vbk)
Abstract: Curvature is a fundamental concept in differential geometry that describes how a curve bends and how a surface behaves locally. The aim of this seminar is to introduce the notion of curvature, beginning with curves and then extending the discussion to surfaces in Euclidean space.
 
We first discuss the curvature of a regular plane curve and interpret it geometrically as a measure of its bending. Using this notion as a starting point, we introduce parametrized surfaces and discuss their tangent spaces and unit normal fields. We then define the normal curvature of a surface in a given tangent direction and introduce Gaussian curvature as an intrinsic measure of the surface's local geometry. Through suitable examples, we illustrate these concepts and examine how curvature distinguishes different types of surfaces.
 
The seminar aims to develop an intuitive and mathematical understanding of curvature through definitions, geometric interpretations, and examples, highlighting the transition from the curvature of curves to the curvature of surfaces.

Commutative Algebra seminar
Speaker: Kaushik Khamari, IIT Bombay
Host: Tony Puthenpurakal
Title: On licci monomial ideals II
Time, day and date: 11:30:00 AM - 12:30:00 PM, Monday, October 26
Venue: Ramanujan Hall
Abstract: In this talk, we discuss basic notions of complete-intersection liaison and licci ideals, which are ideals in the liaison class of a complete intersection. We then introduce universal linking sequences corresponding to an arbitrary sequence of links. As an application, we give a characterization of finite colength licci monomial ideals in a polynomial ring over a field, due to Huneke and Ulrich.

Number Theory Seminar
Speaker: Arshay Seth, School of Mathematics, TIFR
Host: Ravi Raghunathan
Title: Euler products inside the critical strip
Time, day and date: 11:30:00 AM – 12:30:00 PM, Tuesday, October 27
Venue: Room 113
Abstract: Even though Euler products of L-functions are generally valid only to the right of the critical strip, there is a strong sense in which they should persist even inside the critical strip.  Indeed, the behaviour of Euler products inside the critical strip is very closely related to both the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture.  In this talk, we will give an introduction to this topic and then discuss recent work on establishing asymptotics for partial Euler products of L-functions in the critical strip.  We will end by giving applications of these results to questions related to Chebyshev's bias, and also discuss a new conjectural local-global principle for higher genus curves. This talk is partly based on joint works with Alina Bucur, Kiran Kedlaya and Shin-ya Koyama.

Mathematics Colloquium
Speaker: Jean-Marc Deshouillers, University of Bordeaux
Host: Keshav Aggarwal
Title: Coprimality of consecutive elements in regular sequences
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, October 28
Venue: Ramanujan Hall
Abstract: ``Regular sequences'' are integral sequences $ \mathcal{A}_f$ of the shape $\{ \lfloor f(n) \rfloor, n \in \mathbb{N}\}$, where $f$ is a smooth function from $[1, + \infty)$ to itself with polynomial growth (for example the Segal-Piatetski-Shapiro sequences where $f(n)=n^c$). We shall study the following questions: are there infinitely many arbitrarily large blocks of consecutive elements of $ \mathcal{A}_f$ which are pairwise coprime? If so, do they occur often? To illustrate the results in the case of SPS sequences, we show that the sequence $SPS(c)$ contains arbitrary large blocks of pairwise coprime elements when $1<c<2$. Moreover, those blocks of given length occur with positive density. For $c>2$ although we expect the same result to hold, we can prove it only for $c$ rational less than $111/55$. Proofs involve combinatorial constructions and harmonic analysis.\\ \textit{From joint works with M. Drmota and C. M\"{u}llner, with S. Naik and a work in progress with H. Iwaniec.}