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.

Seminar
Speaker: Nilabha Saha, IIT Bombay
Host: Ravi Raghunathan
Title: Elliptic curves
Time, day and date: 11:00:00 AM – 12:00:00 PM, Tuesday, May 05
Venue: Ramanujan Hall
Abstract: We survey the basic arithmetic theory of elliptic curves. After recalling the abstract definition as smooth projective curves of genus one equipped with a marked rational point and its equivalence to the concrete description via Weierstrass equations, we quickly describe the group law through the chord-and-tangent construction. We then turn to isogenies, which are morphisms of elliptic curves preserving the basepoint, and prove that every isogeny is automatically a group homomorphism. The dual isogeny is constructed using the pullback of divisors, and we establish the fundamental identity relating the composition of an isogeny with its dual to the multiplication-by-degree map. Finally, we construct the Weil pairing on the m-torsion subgroup, taking values in the m-th roots of unity, and establish its key properties: bilinearity, the alternating property, non-degeneracy, Galois equivariance, and compatibility with isogenies.
Familiarity with algebraic geometry at the level of varieties, function fields, and divisors for curves will be assumed.

Presentation
Speaker: Nilabha Saha, IIT Bombay
Host: Rekha Santhanam
Title: Extending to a Model Structure Is Not a First-Order Property
Time, day and date: 3:00:00 PM – 4:00:00 PM, Tuesday, May 05
Venue: Ramanujan Hall
Abstract: A model structure on a category equips it with three distinguished classes of morphisms (weak equivalences, cofibrations, and fibrations) satisfying axioms that enable homotopy-theoretic constructions. A natural question arises: given a category C and a chosen subcategory W of "weak equivalences," when can we complete the pair (C, W) to a full model structure?
In this talk, we present the main results of Droz and Zakharevich (2021), who answer this question completely when C is a poset. They characterise precisely which pairs (C, W) admit a model structure in terms of a functorial "choice of centers," and show that for countable posets this existence condition is first-order expressible. The main theorem, however, establishes that no such first-order characterisation exists in general: there are pairs that satisfy all the same first-order properties as extendable ones, yet do not themselves extend to a model structure.
We will develop the necessary background on model categories and state the relevant results from first-order logic (in particular, the Lowenheim--Skolem theorem) during the talk. A working knowledge of basic category theory (functors, limits, colimits) will be assumed. Prior familiarity with model structures may help in appreciating the content, but is not a hard prerequisite.

Analysis seminar
Speaker: Tanuj Gupta, Guru Gobind Singh Indraprastha University
Host: Prachi Mahajan
Title: DF index and global regularity of the complex Green operator
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, May 06
Venue: Online (https://meet.google.com/sbt-yseg-pgn)
Abstract: Let $\Omega\subset\mathbb{C}^n$, with $n \geq 3$, be a smooth bounded pseudoconvex domain satisfying the symmetric eigenvalue comparability condition $D(q_0)$ for some $1\leq q_0 \leq n-2$. We show that if the DF-index of $\Omega$ is one, then the complex Green operator $G_q$, associated with $\Omega$, is globally regular for $q$ in the range $\min\{q_0,\, n - 1 - q_0\} \leq q \leq \max\{q_0,\, n - 1 - q_0\}$.

Topology Seminar
Speaker: Udit Mavinkure, IIT Bombay
Host: Rekha Santhanam
Title: Quasicategories vs Segal spaces, Part 2
Time, day and date: 11:30:00 AM – 12:30:00 PM, Thursday, May 07
Venue: Room 215
Abstract: This is Part 2 in a series of 4 expository talks that will culminate in Joyal and Tierney's proof of two Quillen equivalences between the Joyal model structure for quasicategories and the Rezk model structure for complete Segal spaces, both of which present the homotopy theory of (oo,1)-categories.

In this second talk, we will look at quasicategories, which are simplicial sets satisfying the weak Kan condition (i.e. inner horns admit fillers). These generalize the notion of categories and furthermore are the fibrant objects in a model structure on the category of all simplicial sets where the weak equivalences generalize the notion of equivalences of categories. We will also look at relevant aspects of this model structure, called the Joyal model structure for quasicategories.