Seminar
Speaker: Ujjwal Kumar Mishra, IIT Bombay
Host: Rekha Santhanam
Title: The Puppe Sequence of Cofibrations and Its Applications
Time, day and date: 10:00:00 AM – 11:00:00 AM, Monday, May 04
Venue: Room 215
Abstract: In this talk, we present the derivation of the Puppe sequence for cofibrations and demonstrate its utility in constructing long exact sequences in cohomology. We will begin with a foundational discussion on cofibrations and their properties. Following this, we provide an overview of the essential geometric constructions required for the proof, specifically mapping cones, Moore spaces, and Eilenberg-MacLane spaces.

Presentation Seminar
Speaker: Annu, IIT Bombay
Host: Saurav Bhaumik
Title: Tangent bundle and cotangent bundle on smooth manifold.
Time, day and date: 3:30:00 PM – 4:30:00 PM, Monday, May 04
Venue: Room 215
Abstract: This presentation provides a systematic introduction to the concepts of tangent and cotangent structures on smooth manifold, along with vector and covector field.The tangent space TpM at a point p ∈ M is a vector space whose dimension equals the dimension of the manifold M. Building on this, the tangent bundle T M is defined, and it is shown that T M is smooth manifold of dimension 2dim(M ).
            The study then proceeds to the dual notion of the tangent space, namely the cotangent space T*pM, defined as the dual space of the tangent space. We prove that T*p M is also a vector space of dimension dim(M ), and further construct the cotangent bundle T*M demonstrating that it is a smooth manifold of dimension 2dim(M).
        Finally, the presentation introduces vector field and convector field. It is shown that both vector field and covector field form modules over C^∞(M ).

Presentation Seminar
Speaker: Sagnik Roy, IIT Bombay
Host: Saurav Bhaumik
Title: Vector Bundles and Serre Swann Theorem
Time, day and date: 4:40:00 PM – 5:40:00 PM, Monday, May 04
Venue: Room 215
Abstract: The goal of the talk is to explore the geometric and algebraic foundations of vector bundles, culminating in the proof of the Serre-Swan Theorem, which relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules are like vector bundles". We begin by formalizing $k$-vector bundles over Hausdorff spaces and their smooth counterparts over $C^{\infty}$ manifolds. After establishing the mechanics of local trivializations and transition functions, we systematically construct new bundles using pullbacks, Whitney sums, and smooth functors.

A central focus is the module of smooth sections, $\Gamma(\pi)$, which is shown to be a $C^{\infty}(M)$-module. We demonstrate that the assignment of a vector bundle to its module of sections acts as a covariant functor. By analyzing subbundles, orthogonal complements via scalar products, and embeddings into trivial bundles, we establish that this functor is fully faithful.
 Ultimately, we establish that for a connected manifold $M$, the category of vector bundles is equivalent to the category of finitely generated projective $C^{\infty}(M)$-modules.

This theorem was originally proved by Jean-Pierre Serre in 1955. Serre's version was more algebraic in nature, vector bundles on an algebraic variety over an algebraically closed field (of any characteristic). The version I intend to prove later is a analytic invariant of Serre's theorem which was proved by Richard Swan in 1962.