Lecture 13 - Lie groups and their Lie algebras (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: Lie groups; dimension of a Lie group; examples of Lie groups: n-dimensional translation group, unitary group U(1), general linear GL(n,R), orthogonal group O(p,q); pseudo-inner products on a vector space; Lie group homomorphism and isomorphism; proof that the left translation map is a diffeomorphism; push-forward of the left translation map; left-invariant vector fields; proof that the space of left-invariant vector fields is isomorphic to the tangent space at the identity; proof that the left-invariant vector fields form a Lie algebra, the Lie algebra of the Lie group. Lie algebra homomorphisms and isomorphic Lie algebras.

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Lecture 12 - Grassmann algebra and de Rham cohomology (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: differential n-forms; orientable manifolds; degree of a differential form; pull-back of a differential form; wedge (or exterior) product of differential forms; local expression of a differential form; proof that the pull-back distributes over the wedge product; Grassmann algebra; Grassmann numbers; proof that the wedge product is graded commutative; exterior derivative; Lie bracket (or commutator) of vector fields; example: exterior derivative of a differential one-form; proof that the exterior derivative is graded additive; commutation of the exterior derivative with the pull-back; Maxwell's electrodynamics and Maxwell's equations expressed using differential forms; symplectic forms and classical mechanics; closed and exact forms; proof that d^2=0; symmetrisation and anti-symmetrisation of indices with examples; every exact form is closed; kernel and image of a linear map; Z^n and B^n; Poincaré lemma; cohomology groups.

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Lecture 11 - Tensor space theory II: over a ring (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: vector fields as smooth sections of the tangent bundle; vector fields as linear maps on the space of smooth maps; push-forward of a smooth map as a map between tangent bundles; push-forward of a vector field; structure of the set of vector fields; rings: commutative, unital and division (or skew) rings; examples; modules of a unital ring; examples of modules admitting and not admitting a basis; Zorn's lemma; partial orders and partially ordered sets (posets); total order and totally ordered sets; upper bounds; proof that every module over a division ring (and hence every vector space) admits a Hamel basis; direct sum of modules; finitely generated, free and projective modules; homomorphism of modules (or linear maps); Serre-Swan-et al.'s theorem; pull-back of forms; tensor fields as multilinear maps; tensor product of tensor fields.

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Lecture 10 - Construction of the Tangent Bundle (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: cotangent space and tensor space at a point of a manifold; differential of a smooth map; gradient of a real function on a manifold; dual coordinate-induced basis and gradients of coordinate functions; push-forward and pull-back of smooth maps at a point; push-forward of tangent vectors and pull-back of covectors; immersions and immersed submanifolds; embedding and embedded submanifolds; Whitney's theorem; definition of tangent bundle; proof that the tangent bundle is a smooth manifold.

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Lecture 09 - Differential structures: the pivotal concept of tangent vector spaces (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: the space of smooth maps on a manifold; smooth curves on a manifold; directional derivative operator; tangent vectors at a point and tangent space at a point; proof that the sum of tangent vectors is a tangent vector; alternative definitions of tangent space (via equivalence classes of smooth curves, derivations at a point on germs of functions, and physical tangent vectors); algebras over an algebraic field; associative, unital and commutative algebras; Lie algebras, Lie bracket and Jacobi identity; commutator; derivations on an algebra; detailed examples; proof that derivations on a algebra constitute a Lie algebra; proof of equality of manifold dimension and tangent space dimension: dim M = dim TpM; coordinate-induced basis of tangent spaces; change of coordinates under a change of coordinate-induced bases.

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