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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: | |||
# at what point it lies in the stack | |||
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 at what point in the stack | # what concepts lie below it | ||
Core lecture series: [https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic Lectures on Geometrical Anatomy of Theoretical Physics] | Core lecture series: [https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic Lectures on Geometrical Anatomy of Theoretical Physics] | ||
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==Set theory== | ==Set theory== | ||
[[File:LawsOfSets.png|thumb|Laws of set theory]] | |||
[https://www.youtube.com/watch?v=AAJB9l-HAZs&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=2 Lecture 02] | [https://www.youtube.com/watch?v=AAJB9l-HAZs&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=2 Lecture 02] | ||
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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. | 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 | 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== | ==Topological spaces== | ||
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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. | 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== | ==Applications== | ||
===Quantum mechanics on curved spaces=== | ===Quantum mechanics on curved spaces=== | ||
[https://www.youtube.com/watch?v=C93KzJ7-Es4&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=26 Lecture 26] | [https://www.youtube.com/watch?v=C93KzJ7-Es4&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=26 Lecture 26] | ||
===Spin structures=== | ===Spin structures=== | ||
[https://www.youtube.com/watch?v=Way8FfcMpf0&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=27 Lecture 27] | [https://www.youtube.com/watch?v=Way8FfcMpf0&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=27 Lecture 27] | ||
===Kinematical and dynamical symmetries=== | ===Kinematical and dynamical symmetries=== | ||
[https://www.youtube.com/watch?v=F3oGhXNhIDo&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=28 Lecture 28] | [https://www.youtube.com/watch?v=F3oGhXNhIDo&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=28 Lecture 28] | ||
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[[Category:Graph, Wall, Tome]] | |||
[[Category:Mathematics]] | |||