Talk
Speaker: Aditya Dwivedi, IIT Bombay
Host: Rekha Santhanam
Title: A généralisation of the HKR theorem for simplicial vector spaces.
Time, day and date: 11:00:00 AM – 12:00:00 PM, Friday, May 01
Venue: Room 215
Abstract: Hochschild homology, introduced by Hochschild in 1945 remains an important invariant of associative algebras. The theory of Associative algebras has connections to sevelars areas in commutative algebra, algebraic geometry and algebraic topology. In their 1962 paper, Hochschild, Konstant and Rosenberg proved a landmark result in this area which is now known as the HKR-theorem. Several generalisations have been done of the HKR theorem to highlight connections with the de Dham complex, Andre Quillen homology and cyclic homology.

In this talk, we present the work of Pirashvilli(2000) proving a version of HKR theorem for simplicial k vector spaces. The proof defined stable homotopy groups of such objects and the classical HKR theorem follows from the general case.

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.

Algebraic groups seminar
Speaker: Shripad M. Garge, IIT Bombay
Title: Schur Weyl duality V
Time, day and date: 4:00:00 PM - 5:30:00 PM, Wednesday, May 13
Venue: Room No .105
Abstract: We complete our discussion on the Schur-Weyl duality.

Number theory seminar
Speaker: Sampurna Pal, IIT Kanpur
Host: Sumit Kumar
Title: Moments of Degree Three L-functions
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, May 14 
Venue: Ramanujan Hall
Abstract: In this talk, we will discuss a non-trivial upper bound of the standard GL(3) L-functions, uniform in the t-aspect and the depth aspect. This is based on an ongoing work with Mayukh Dasaratharaman, Sumit Kumar, and Wing Hong Leung.

IPDF Talk
Speaker: Supriyo Jana, IITMadras
Host: G.K. Srinivasan
Title: Dynamics and integrability of polynomial vector fields on certain compact manifolds
Time, day and date: 11:00:00 AM - 12:00:00 PM, Tuesday, May 19
Venue: Ramanujan Hall
Abstract: We present a study of polynomial vector fields on algebraic manifolds, including the n-dimensional sphere and product of spheres. We provide a complete algebraic characterization of polynomial vector fields on $S^n$ of arbitrary degree, and investigate their first integrals and invariant algebraic sets. Sharp bounds on the number of invariant meridian and parallel hyperplanes are established. We also determine when a Hamiltonian vector field on Euclidean space induces a vector field on these manifolds. Darboux theory of integrability is employed to show that cubic Kolmogorov vector fields on the product of 1-sphere and 2-sphere, while not Hamiltonian, admit rational first integrals.

We conclude by motivating the study of polynomial vector fields on real projective spaces. We outline several open questions concerning the characterization of such vector fields, the classification of global phase portraits on $RP^2$, and the extension of Darboux integrability theory to the projective setting.

IPDF Talk
Speaker: Annot Kumar Yadav
Host: Tony J P
Title: Bounds on Hilbert coefficients and reduction number
Time, day and date: 3:30:00 PM – 4:30:00 PM, Wednesday, May 20
Venue: Online (meet.google.com/qwt-rmfj-hgm) 
Abstract: Let $(A, m)$ be a Cohen–Macaulay local ring and $I$ an $m$-primary ideal. In this talk, we study how Hilbert coefficients, reduction numbers, and the Ratliff–Rush filtration govern the depth and homological properties of associated graded rings and related blow-up algebras. Our work is motivated by a conjecture of Maria Evelina Rossi, which predicts a linear upper bound for the reduction number in terms of multiplicity and the first Hilbert coefficient.
Using higher Hilbert coefficients, particularly the third Hilbert coefficient, we obtain improved bounds on reduction numbers under suitable depth conditions. In dimension three, our results provide the best-known bound for a class of integrally closed ideals. We also establish inequalities for the third Hilbert coefficient and show that equality cases correspond to favourable structural properties of the Ratliff–Rush filtration. These results extend to $I$-admissible filtrations, where Rossi’s conjecture is verified in certain extremal cases.
Finally, we study the behaviour of the Ratliff–Rush filtration modulo superficial elements and obtain computable criteria for its compatibility with superficial elements. Overall, the talk highlights how numerical invariants detect structural properties of blow-up algebras.
This talk is based on joint work with Kumari Saloni.

Mathematics Colloquium
Speaker: Saurabh Kumar Singh
Host: Sumit Kumar
Title: Counting special points on quadratic surfaces.
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, May 20
Venue: Ramanujan Hall
Abstract: We show that the modern version of the circle method powered by the equidistribution of quadratic roots allows us to count special points on quadratic surfaces. For example, we obtain asymptotic for integer points on quadratic surfaces with prime coordinates and in short intervals.

PDE Seminar
Speaker: Pardeep Sharma
Host: Harsha Hutridurga
Title: Impulsive inverse problem for the Navier-Stokes equations.
Time, day and date: 10:00:00 AM – 11:00:00 AM, Friday, May 22
Venue: Online (https://zoom.us/j/95223583387?pwd=i49UhXlwzHiLr6aLK6tqnlesWKIc8C.1)
Abstract: Attached (https://drive.google.com/file/d/1R6eEC2IhOQldZlImzdyH2z3OfWwTYANF/view?usp=sharing)

PDE Seminar
Speaker: Dr. Rakib Mondal
Host: Sivaji Ganesh Sista
Title: Rarefaction Wave Interaction and Existence of Global Solutions for a Hyperbolic System of PDEs
Time, day and date: 11:30:00 AM – 12:30:00 PM, Friday, May 22
Venue: Online (https://meet.google.com/ika-sdnw-yuf)
Abstract: In this talk, we discuss the first-order hyperbolic systems of partial differential equations and their mathematical challenges due to lack of regularity. In particular, we introduce a simplified 1-D system of balance laws governing blood flow with time-dependent body force. We discuss the interaction of two centered rarefaction waves and show that the interaction gives rise to a Goursat boundary value problem (GBVP). By transforming the system into Riemann-invariant coordinates, we discuss the existence and uniqueness of a global $C^1$ solution to the GBVP.

We further extend this system of balance laws considering both time-space dependent body force and the viscosity resistance, and prove the existence and stability of global weak entropy solutions. We establish a priori uniform $L^\infty$ bounds for a sequence of solutions to the regularized two-parameter viscosity-flux approximated systems. Using the vanishing-diffusion limit and the compensated compactness framework, we show the strong convergence and establish the global weak entropy solutions to the original Cauchy problem with large initial data.

IPDF Talk
Speaker: Rahul Bhardwaj, IIT Ropar
Host: Suman Kumar Sahoo
Title: Reconstruction Results for Inverse Problems in Integral Geometry and Hyperbolic PDEs
Time, day and date: 4:00:00 PM – 5:00:00 PM, Friday, May 22
Venue: Online (https://zoom.us/launch/jc/93166883629) (Meeting ID: 931 6688 3629 Passcode: 506739)
Abstract: In this talk, we address two inverse problems arising in integral geometry and hyperbolic partial differential equations (PDEs). The first problem concerns the reconstruction of a vector field in the plane from a set of generalised V-line transforms, namely longitudinal and transverse V-line transforms. Specifically, we focus on recovering a vector field which is supported inside a disk of radius R centered at the origin. This recovery process is accomplished through a combination of the aforementioned generalized V-line transforms, using two different techniques for different sets of data. The second problem concerns an inverse problem for a semi-linear wave equation posed on a bounded subset of Euclidean space. The objective is to reconstruct the coefficients from the associated Dirichlet-to-Neumann map. The analysis is based on the method of higher-order linearization.

IPDF talk
Speaker: Mr. Akurathi Jaya Naga Sri, University of Hyderabad
Host: Debraj Das
Title: Semiparametric linear transformation models for survival data with covariate measurement error
Time, day and date: 4:00:00 PM – 5:00:00 PM, Tuesday, May 26
Venue: Online (https://meet.google.com/vgm-gsjw-fqc)
Abstract: Measurement error in covariates frequently arises in biomedical, epidemiological, and clinical studies, and ignoring such errors may lead to biased estimation and misleading statistical inference.
In this talk, I will discuss semiparametric approaches for analyzing censored and competing risks survival data when covariates are measured with error. The proposed methods combine flexible modeling frameworks with robust estimation procedures and establish desirable asymptotic properties of the estimators. Simulation studies and real data applications illustrate the practical performance of the proposed approaches.