Tangent Tuesday:talk by Tharun and Nalin, IISERTVM on 1st Sept

We are back with the next episode of Tangent Tuesday! This time, we have two talks exploring differential forms and smooth manifolds. First, we will look at how differential forms provide a rigorous foundation for vector calculus and give real meaning to the infinitesimals we integrate in physics. After that, we will shift to topology to discuss Sard's theorem and see how Whitney's Embedding Theorem proves that any abstract smooth manifold can be embedded into standard Euclidean space.

Speaker: Tharun (B22)
Title of the talk: The Language of Differential Forms
Date: 1 September 2026 | Tuesday
Time: 8:45 pm – 9:30 pm
Venue: PSB 3101

Abstract: Vector calculus, developed in the late 19th century by Heaviside and Gibbs, gave us the language of vector fields, divergence, curl, and the classical integral theorems. But it wasn't until Cartan, Poincare, Hodge and others introduced exterior differential calculus that all of these theorems were unified into a single elegant statement: the generalized Stokes' theorem. Differential forms were originally conceived to give rigorous meaning to infinitesimals: objects that are meant to be integrated. What does it mean when you write dA=dx dy? and why do you also write dA= r dr dθ? Think of all the times physicists have written dW=F dx. In this talk, we will make such notions rigorous and call them "differential forms". We will develop forms from the ground up, see how classical vector calculus identities are elegantly recast using the exterior derivative and Hodge duality, and briefly explore the geometric intuition behind integration. Prerequisites: linear algebra (up to dual spaces), multivariable calculus, and some familiarity with vector calculus. Exposure to basic electrodynamics via vector calculus is preferred, but not necessary.

Speaker: Nalin (B22)
Title of the talk: Whitney Embedding Theorem and Sard's Theorem
Date: 1 September 2026 | Tuesday
Time: 9:40 pm – 10:25 pm
Venue: PSB 3101

Abstract: This presentation explores two fundamental results about smooth manifolds: Sard’s theorem and Whitney’s embedding theorem. We begin by introducing critical points, critical values, and regular values of smooth maps. Sard’s theorem states that the set of critical values of a smooth map has measure zero, showing that regular values are abundant. We then discuss how this theorem provides a powerful method for establishing the existence of points and directions with desirable regularity properties. The second part concerns Whitney’s embedding theorem, which asserts that every smooth n-dimensional manifold can be smoothly embedded in a finite-dimensional Euclidean space—in particular, in ℝ2n. The main ideas behind the proof will be outlined, including the construction of an initial embedding using coordinate charts and smooth bump functions, followed by a reduction of the ambient dimension through suitable projections. This illustrates how Sard’s theorem supports the general-position arguments underlying Whitney’s result and reveals the close relationship between abstract smooth manifolds and concrete submanifolds of Euclidean space.