• Compactification and Derived Representations
  • Fixed Points and Invariants
  • Linearization and Matrix Representations with Lie Groups
  • Abelianization and Completions
  • Sieves, Filtrations, Spectral Sequences
  • Projections, Ramifications, and Cohomology
  • Extensions, Closures, and Automorphisms
  • Quotients and Reductions
  • Diffusion, Gaussian Isotropy Group, Skeleton Category
  • Inverse Limits, Profinite Groups, Absolute Galois Groups
  • Energy Minimization, Moduli Spaces, and Flat Connections
  • Instantons and Poles in Extended Supersymmetry
  • Complexification, Embeddings, and Motivic Stabilizations
  • Covariant, Geometric, Deformation Quantization (Gluing and Intersections)
• Linearization of Abelian Categories and Commutative Geometry
  • Zariski Topology, Borel subgroups
  • Fields, Rings, Spectrum of Prime Ideals
  • Adeles and Ideles
  • Abelian Categories, Schemes
  • Homotopy Abelianization of Homology
  • Grothendieck Completion and K-Theory
• Linearization of Automorphisms Under Complexification
  • Connections as automorphisms
  • Galois Group, Galois Representations
  • Absolute Galois Group
  • Moduli stack of elliptic curves and modular forms
  • Automorphic forms
  • Moduli spaces, Sheaves, Stacks, Cohomology of Shtukas
• Linearization of Non-Abelian Manifolds Under Extended Supersymmetry
  • Differential Algebras, Lie gauges, Differential Galois Theory
  • Connections as differential forms on tangent bundles
  • Quotient spaces and Moduli Spaces
  • Non-Abelian Yang-Mills theory and Lagrangian Mechanics
  • Irreducible connections, Instantons and Monopoles
  • Torsion, Holonomy, spectral and mass gaps
  • Donaldson Theory and Exotic R4
  • Floer Homology
  • ADHM Monad Construction (Penrose twistor theory)
  • Seiberg-Witten Theory and Invariants
• Symplectic Geometry as weak Abelianization
  • Solder form, cotangent bundle
  • Hamiltonian Mechanics vs Lagrangian Mechanics
  • Symplectification, symplectic reduction
  • Interpreting symplectic reduction with Galois theory
  • Symplectic connection, deformation quantization
  • Covariant phase space
• Linearization with Graphs Under Complexification
  • CW complex
  • Chains, Cochains, Cycles, Cocycles
  • Simplicial complex
  • Lefschetz fixed-point theorem
  • Cellular complex
  • Laplacians and spectral theory
  • Triangulated categories
  • Quivers and Representations
  • Dynkin Diagrams, Gabriel’s theorem, Root systems
  • Dessin d’enfant, Belyi’s theorem, moduli spaces
• Linearization with Cohomology Derived Invariants
  • de Rham Cohomology
  • Betti cohomology
  • Étale Cohomology
  • Weil cohomology
  • Crystalline cohomology
  • Hodge theory
  • Chern–Weil theory
• Grothendieck’s Dream - A Universal Cohomology and Derived Invariant
  • Chain-complex
  • Exact and Closed Sequences, Spectral Sequences, Filtrations
  • Stable infinity-category
  • Derived Categories, Coherent Sheaves
  • Projective categories, Pure and Mixed Motives
  • Derived Category of Motives
  • Derived Invariants
• Furthest extent
  • Grothendieck-Teichmüller space
  • Motivic Galois group, cosmic Galois group, renormalization
  • Motivic stablization of Symplectic manifolds
  • Shimura Variety, L-functions, and Zariski closure
  • Nekrasov Instanton Partition Function, Young Diagrams
  • Triangulated categories of mixed motives
  • Motives over simplicial schemes
  • Probability by Homology, Gromov p-widths
  • Weinstein symplectic category
  • Tannakian Formalism
  • Kähler Manifolds, hyperkähler manifold, Calabi–Yau manifold
  • Amplituhedron, poles, factorization, gauge quivers
  • Homological mirror symmetry
  • Topological Recursion

Compactification and Derived Representations

Fixed Points and Invariants

Linearization and Matrix Representations with Lie Groups

Abelianization and Completions

Sieves, Filtrations, Spectral Sequences

Projections, Ramifications, and Cohomology

Extensions, Closures, and Automorphisms

Quotients and Reductions

Diffusion, Gaussian Isotropy Group, Skeleton Category

Inverse Limits, Profinite Groups, Absolute Galois Groups

Energy Minimization, Moduli Spaces, and Flat Connections

Instantons and Poles in Extended Supersymmetry

Complexification, Embeddings, and Motivic Stabilizations

Covariant, Geometric, Deformation Quantization (Gluing and Intersections)

Linearization of Abelian Categories and Commutative Geometry

Zariski Topology, Borel subgroups

Fields, Rings, Spectrum of Prime Ideals

Adeles and Ideles

https://math.stackexchange.com/questions/25090/whats-the-significance-of-tates-thesis/25125#25125 https://math.stackexchange.com/questions/290847/what-do-ideles-and-adeles-look-like

Abelian Categories, Schemes

Homotopy Abelianization of Homology

Grothendieck Completion and K-Theory

Linearization of Automorphisms Under Complexification

Connections as automorphisms

Galois Group, Galois Representations

Absolute Galois Group

Moduli stack of elliptic curves and modular forms

Automorphic forms

Moduli spaces, Sheaves, Stacks, Cohomology of Shtukas

Linearization of Non-Abelian Manifolds Under Extended Supersymmetry

Differential Algebras, Lie gauges, Differential Galois Theory

Connections as differential forms on tangent bundles

Quotient spaces and Moduli Spaces

Non-Abelian Yang-Mills theory and Lagrangian Mechanics

Irreducible connections, Instantons and Monopoles

Torsion, Holonomy, spectral and mass gaps

Donaldson Theory and Exotic R4

Floer Homology

ADHM Monad Construction (Penrose twistor theory)

Seiberg-Witten Theory and Invariants

Symplectic Geometry as weak Abelianization

Solder form, cotangent bundle

Hamiltonian Mechanics vs Lagrangian Mechanics

Symplectification, symplectic reduction

Interpreting symplectic reduction with Galois theory

Symplectic connection, deformation quantization

Covariant phase space

Linearization with Graphs Under Complexification

permutation/symmetry group of roots

CW complex

Chains, Cochains, Cycles, Cocycles

Simplicial complex

Lefschetz fixed-point theorem

Cellular complex

Laplacians and spectral theory

Triangulated categories

Quivers and Representations

Dynkin Diagrams, Gabriel’s theorem, Root systems

Dessin d’enfant, Belyi’s theorem, moduli spaces

Linearization with Cohomology Derived Invariants

de Rham Cohomology

Betti cohomology

Étale Cohomology

Weil cohomology

Crystalline cohomology

Hodge theory

Chern–Weil theory

Grothendieck’s Dream - A Universal Cohomology and Derived Invariant

Chain-complex

Exact and Closed Sequences, Spectral Sequences, Filtrations

Stable infinity-category

Derived Categories, Coherent Sheaves

Projective categories, Pure and Mixed Motives

Derived Category of Motives

Derived Invariants

Furthest extent

Grothendieck-Teichmüller space

Motivic Galois group, cosmic Galois group, renormalization

Motivic stablization of Symplectic manifolds

Shimura Variety, L-functions, and Zariski closure

Nekrasov Instanton Partition Function, Young Diagrams

Triangulated categories of mixed motives

Motives over simplicial schemes

Probability by Homology, Gromov p-widths

Weinstein symplectic category

Tannakian Formalism

Kähler Manifolds, hyperkähler manifold, Calabi–Yau manifold

Amplituhedron, poles, factorization, gauge quivers

Homological mirror symmetry

Topological Recursion