Geometry

Each section corresponds to a video in the lecture series below. The pedagogy follows a conceptual stack of layered mathematical structure from first principles. The initial utility of this page is that it will ideally allow for a user to ctrl-f any technical term and find out:

1. at what point it lies in the stack
2. what concepts lie below it

Core lecture series: Lectures on Geometrical Anatomy of Theoretical Physics

Blogpost from which section descriptions are taken: Blogpost

High quality notes to accompany the lecture series: Simon Rea's Notes

concept stack

Logic

Lecture 01

Introduction to logic; propositions and predicates; truth tables; tautologies and contradictions; negation, and, or, implication, nand connectives; existential and universal quantifiers, logical equivalence of propositions; negation of quantifiers; order of quantifiers; axiomatic systems; formal proofs; consistency and completeness.

Set theory

Laws of set theory

Lecture 02

Lecture 03

epsilon-relation (member relation); Zermelo-Fraenkel axioms of set theory; Russel's paradox; existence and uniqueness of the empty set (standard textbook proof and formal proof); axioms on the existence of pair sets and union sets; examples; finite unions; functional relation and image; principle of restricted and universal comprehension; axiom of replacement; intersection and relative complement; power sets; infinity; the sets of natural and real numbers; axiom of choice; axiom of foundation.

definition of maps (or functions) between sets; structure-preserving maps; identity map; domain, target and image; injective, surjective and bijective maps; isomorphic sets; classification of sets: finite and countably and uncountably infinite; cardinality of a set; composition of maps; commutative diagrams; proof of associativity of composition; inverse map; definition of pre-image and properties of pre-images (with proof); equivalence relations: reflexivity, symmetry, transitivity; examples; equivalence classes and quotient set; well-defined maps; construction of ℕ, ℤ, ℚ, ℝ (natural, integer, rational and real numbers); successor and predecessor maps; nth power set; addition and multiplication of numbers; canonical embeddings.

Topological spaces

Lecture 04

Lecture 05

topologies and topological spaces; examples; chaotic and discrete topologies; coarser (or weaker) and finer (or stronger) topologies; open subsets; open balls; standard topology on R^d with proof; induced (or subset) topology with proof; product topology; sequences, converge and limit points; open neighbourhoods; definitely constant sequences; continuity of maps between topological spaces; examples; homeomorphisms and homeomorphic spaces.

Separations properties: T1, T2 (Hausdorff), T2 an a half; covers and open covers, subcovers and finite subcovers; compact spaces; Heine-Borel theorem (compact if and only if closed and bounded); open and locally finite refinements; paracompactness; metrisable spaces and Stone's theorem; long line (or Alexandroff line); partition of unity subordinate to an open cover; examples; connectedness and proof that M is connected if and only if M and the empty set are the only subsets which are both open and closed; path-connectedness and proof that path-connectedness implies connectedness; homotopic curves on a topological space; concatenation of curves; fundamental group; group isomorphism; topological invariants and classification of topological spaces; examples: 2-sphere, cylinder, 2-torus.

Topological manifolds

Lecture 06

topological manifolds; manifold dimension; submanifolds; product manifold; bundles of topological manifolds; Möbius strip; total space, base space, projection map and fibres; product bundles; fibre bundles; examples; (cross-) section of a bundle; subbundles and restricted bundles; bundle morphisms and isomorphisms; local bundle isomorphisms; trivial and locally trivial bundles; pull-back of a bundle; sections on a bundle pull back to the pull-back bundle; charts, component and coordinate functions; atlases and C^0-compatibility; chart transition maps; maximal atlases.

Differentiable structures

Lecture 07

refining a maximal atlas; C^k and smooth compatibility of charts; Cauchy-Riemann equations; differentiable atlas; compatibility of differentiable atlases; examples; proof of well-definedness of the definition of differentiability of maps; smooth maps and diffeomorphisms; diffeomorphic manifolds; classification of smooth structure on manifolds; Betti numbers.

Tensor space theory: over a field

Lecture 08

algebraic fields; vector spaces over an arbitrary field; vector (or linear) subspaces; linear maps; linear isomorphisms and isomorphic vector spaces; Hom-spaces; endomorphisms and automorphisms; dual vector space and linear functionals (covectors/one-forms); bilinear and multilinear maps; tensors and tensor product; examples; equivalence of endomorphisms and (1,1)-tensors; Hamel bases; linear independence and spanning set; dimension; double dual; dual bases and isomorphism of a vector space and its dual in finite dimensions; components of vectors and tensors; change of basis formulas; Einstein's summation convention and examples; column and row vectors and matrices; change of components under a change of basis; bilinear forms; permutations, symmetric group, transpositions, and signature of a transposition; totally anti-symmetric tensor; n-forms; volume-form and volume; determinant of an endomorphism.

Tangent vector spaces

Lecture 09

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.

Tangent bundle

Lecture 10

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.

Tensor space theory: over a ring

Lecture 11

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.

Grassmann algebra and deRham cohomology

Lecture 12

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.

Lie groups and their Lie algebras

Lecture 13

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.

Classification of Lie algebras and Dykin diagrams

Lecture 14

complex Lie algebras; abelian Lie algebras; the trivial Lie algebra; ideal of a Lie algebra; trivial ideals; simple and semi-simple Lie algebras; derived subalgebra; solvability; direct and semi-direct sum of Lie algebras; Levi's theorem on the decomposition of finite-dimensional complex Lie algebras; adjoint map and ad; proof that ad is a Lie algebra homomorphism; Killing form; proof of the invariance (or associativity, or anti-symmetry) of the Killing form; a Lie algebra is semi-simple if and only if the Killing is non-degenerate; structure constants; components of adjoint maps and the Killing form in terms of the structure constants; Cartan subalgebra, rank of a Lie algebra and Cartan-Weyl basis; roots and fundamental roots; proof that the restriction of the Killing form on a Cartan subalgebra is a pseudo inner product; real inner product; length and angle between roots; Weyl transformations and Weyl group; Cartan matrix; bond number; Dynkin diagrams and classification of finite-dimensional semi-simple complex Lie algebras.

Lie group SL(2,C) and its algebra

Lecture 15

the complex special linear group SL(2,C): as a set, as a group, as a topological space, as a topological manifold, as a complex differentiable manifold, as a Lie group; the Lie algebra sl(2,C) of the Lie group SL(2,C); detailed calculation of the structure constants of sl(2,C); determination of the Lie bracket between left-invariant vector fields on SL(2,C).

Dykin diagrams from Lie algebras and vice versa

Lecture 16

proof that sl(2,C) is simple; Cartan subalgebra of sl(2,C); roots and fundamental roots of sl(2,C); the Dynkin diagram of sl(2,C); the A2 Dynkin diagram; detailed reconstruction of A2 from its Dynkin diagram.

Representation theory of Lie groups and their algebras

Lecture 17

representations of a Lie algebras; representation spaces and dimension of a representation; examples of representations; homomorphism and isomorphism of representations; trivial and adjoint representations; faithful representations; direct sum and tensor product representations; invariant subspaces, reducible and irreducible representations; highest weights; Killing form associated to a representation; Casimir operator; proof that the Casimir operator commutes with the representation; Schur's lemma; worked examples; automorphism group; representation of Lie groups; Adjoint representation.

Reconstruction of a Lie group from its algebra

Lecture 18

integral curves to a vector field; maximal integral curves; complete vector fields; every vector field on a compact manifold is complete; exponential map; the image of exp is the connected component of the Lie group containing the identity; examples: orthogonal group, special orthogonal group; (restricted) Lorentz group: proper/improper orthochronous/non-orthochronous transformations; Lorentz algebra; one-parameter subgroups; flow of a vector field; the exponential map commutes with smooth maps.

Principal fibre bundles

Lecture 19

left and right Lie group actions; example: actions from representations; proof: right actions from left actions; equivariance of smooth maps; orbits, orbit space and stabilisers; free and transitive actions; examples; smooth and principal bundles; detailed example: the frame bundle; principal bundle morphisms and isomorphisms (or diffeomorphisms); trivial bundles; proof that a bundle is trivial if and only if it admits a global section.

Associated fiber bundles

Lecture 20

associated fibre bundle to a principal bundle; detailed example: the frame bundle; scalar and tensor densities on a manifold; associated bundle maps and isomorphisms; trivial associated bundles; restrictions and extensions of a principal bundle; examples.

Connections and 1forms

Lecture 21

vertical and horizontal subspaces at a point; decomposition in vertical and horizontal parts; connection on a principal bundle; connection one-form; properties of connection one-forms with proof.

Local representations of a connection on the base manifold: Yang-Mills fields

Lecture 22

Yang-Mills field as pull-back of a connection one form along a local section; local trivialisations of a principal bundle; local representation of a connection one-form; Maurer-Cartan form; example: the Yang-Mills fields on the frame bundle, Christoffel symbol; example: calculation of the Maurer-Cartan form of the general linear group GL(n,R); patching Yang-Mills fields on different domains; the gauge map; example: the gauge map on the frame bundle.

Parallel transport

Lecture 23

horizontal lifts of a curve to the principal bundle; ODE characterising horizontal lifts; explicit solution in the case of a matrix Lie group; path-ordered exponential; parallel transport map; loops and holonomy groups; horizontal lifts to the associated bundle; parallel transport map on the associated bundle; covariant derivative of a section.

Curvature and torsion on principal bundles

Lecture 24

exterior covariant derivative; curvature two-form; characterisation of the curvature two-form with proof; Yang-Mills field strength; First Bianchi identity; solder (ing) form; torsion two-form; Second Bianchi identity

Covariant derivatives

Lecture 25

proof of the equivalence of local sections and G-equivariant functions; linear actions on associated vector fibre bundles; matrix Lie group; construction of the covariant derivative for local sections on the base manifold.

Applications

Quantum mechanics on curved spaces

Lecture 26

Spin structures

Lecture 27

Kinematical and dynamical symmetries

Lecture 28

This article is a stub. You can help us by editing this page and expanding it.