Project IV 2024-25

Differential Forms, Integration and de Rham Cohomology

Martin Kerin

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Description From calculus and analysis, we know that the differentiability of a function is a local property. Thus, it is a property which generalises naturally to functions defined on topological spaces which look locally like Euclidean spaces, namely, smooth manifolds. The differential df of a smooth, real-valued function f on a manifold M is then a smooth function which takes in tangent vectors to M and spits out real numbers. A differential n-form ω on M is a natural generalisation of the differential of a function, which takes in n tangent vectors and spits out real numbers. It turns out that differential forms enable us to define a notion of integration on manifolds and, in particular, from this point of view, it is not difficult to establish a generalisation of Stokes' Theorem to smooth manifolds. Furthermore, the space of all differential forms on a manifold has a rich algebraic structure, which can be exploited to develop a cohomology theory (de Rham cohomology) that encodes the topology of the underlying manifold and agrees with well-known cohomology theories defined purely via topological means. As well as being interesting in their own right, differential forms are also a fundamental tool in the study of differential geometry, general relativity, mechanics, mathematical physics and beyond. In this project, we will develop the theory of differential forms, integration on manifolds and de Rham cohomology. A particular highlight along the way will be Stokes' Theorem on smooth manifolds, which easily leads to the the classical Stokes', Green's and Divergence Theorems on Euclidean space. There will be a lot of scope for individual choice regarding the focus of the project, due to the vast amount of material and sources available on differential forms. Prerequisites Essential: Topology III Recommended: Differential Geometry III or Geometry III Analysis III Corequisites Essential: Riemannian Geometry IV (for smooth manifolds and vector bundles) Recommended: Algebraic Topology IV (especially if you wish to prove the famous de Rham Theorem linking de Rham cohomology and singular cohomology) Resources John Lee, Introduction to Smooth Manifolds , Springer 2012 Ib Madsen and Jorgen Tornehave, From Calculus to Cohomology: de Rham Cohomology and Characteristic Classes, Cambridge University Press 1997 Michael Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1, 3rd Edition, Publish or Perish 1999

email: M Kerin