Math 2.0

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Topological groups, Lie groups

307 ideas in this subject. The map below lists the main ideas of the subject by area, in teaching order. 219 of its 224 ideas have a page so far.

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Topological groups: basic notions 36 of 36

  1. Topological group definition
  2. Continuous multiplication definition
  3. Semitopological group definition
  4. Paratopological group definition
  5. Topological monoid definition
  6. Topological semigroup definition
  7. Topological ring definition
  8. Topological vector space definition
  9. Continuous group homomorphism definition
  10. Isomorphism of topological groups definition
  11. Open subgroup definition
  12. Open subgroups are closed statement
  13. Quotient of a topological group definition
  14. Identity component definition
  15. Totally disconnected group definition
  16. Uniform structure of a topological group concept
  17. Completion of a topological group definition
  18. Birkhoff–Kakutani theorem statement
  19. Hausdorff topological group concept
  20. Topological homogeneity concept
  21. Pettis lemma statement
  22. Steinhaus theorem statement
  23. Polish group definition
  24. Topological transformation group definition
  25. Homogeneous space definition
  26. Profinite group definition
  27. Profinite completion definition
  28. Pro-p group definition
  29. p-adic Lie group definition
  30. Lazard's theorem on p-adic analytic groups statement
  31. Solenoid example
  32. Cantor group example
  33. Hilbert's fifth problem problem
  34. Gleason–Yamabe theorem statement
  35. No small subgroups definition
  36. Hilbert–Smith conjecture conjecture

Locally compact groups and Haar measure 41 of 43

  1. Locally compact group definition
  2. Haar measure definition
  3. existence of Haar measure statement
  4. uniqueness of Haar measure statement
  5. Modular function definition
  6. Unimodular group definition
  7. Lattice in a locally compact group definition
  8. Fundamental domain definition
  9. Group algebra of a locally compact group definition
  10. Convolution definition
  11. Group C*-algebra definition
  12. Group von Neumann algebra definition
  13. Unitary representation definition
  14. Gelfand–Raikov theorem statement
  15. Positive-definite function on a group definition
  16. Gelfand–Naimark–Segal construction definition
  17. Unitary dual definition
  18. Plancherel theorem for groups statement
  19. Induced representation of a locally compact group definition
  20. Mackey machine concept
  21. Mackey imprimitivity theorem statement
  22. Amenable group definition
  23. Følner condition definition
  24. Property (T) definition
  25. Haagerup property definition
  26. Gelfand pair definition
  27. Spherical function definition
  28. Hecke algebra of a locally compact group definition
  29. Kazhdan–Margulis theorem statement
  30. Borel–Harish-Chandra theorem statement
  31. Margulis arithmeticity theorem statement
  32. Margulis normal subgroup theorem statement
  33. Mostow rigidity theorem statement
  34. Borel density theorem statement
  35. Ratner's theorems statement
  36. Raghunathan's conjecture statement
  37. Mahler's compactness theorem statement
  38. Zassenhaus neighbourhood concept
  39. Margulis lemma statement
  40. Tits alternative statement
  41. Structure theory of totally disconnected locally compact groups concept
  42. van Dantzig's theorem statement
  43. Cayley–Abels graph definition

Locally compact abelian groups and Pontryagin duality 17 of 17

  1. Locally compact abelian group definition
  2. Character of an abelian group definition
  3. Pontryagin dual definition
  4. Pontryagin duality theorem statement
  5. Circle group example
  6. Circle group example
  7. Torus as a group example
  8. Bohr compactification definition
  9. Fourier transform on locally compact abelian groups definition
  10. Plancherel theorem for abelian groups statement
  11. Bochner's theorem statement
  12. Poisson summation formula statement
  13. Harmonic analysis on groups concept
  14. Structure theorem for compactly generated LCA groups statement
  15. Adele ring example
  16. Tate's thesis concept
  17. Wiener's Tauberian theorem statement

Compact groups 13 of 13

  1. Compact group definition
  2. Peter–Weyl theorem statement
  3. Weyl's unitary trick concept
  4. Complete reducibility of representations of compact groups statement
  5. Schur orthogonality for compact groups statement
  6. Tannaka–Krein duality statement
  7. Maximal torus definition
  8. Conjugacy of maximal tori statement
  9. Weyl group of a compact Lie group definition
  10. Weyl integration formula statement
  11. Weyl character formula for compact groups statement
  12. Borel–Weil–Bott theorem statement
  13. Kronecker's approximation theorem statement

Lie groups: basic notions 21 of 22

  1. Lie group definition
  2. Lie algebra of a Lie group definition
  3. Left-invariant vector field definition
  4. Exponential map of a Lie group definition
  5. One-parameter subgroup definition
  6. Adjoint representation of a Lie group definition
  7. Maurer–Cartan form definition
  8. Baker–Campbell–Hausdorff formula statement
  9. Lie's third theorem statement
  10. Lie group–Lie algebra correspondence statement
  11. Closed subgroup theorem statement
  12. Lie subgroup definition
  13. Universal covering group definition
  14. Matrix Lie group definition
  15. Lie's theorems on transformation groups statement
  16. Cartan's theorem on continuous homomorphisms statement
  17. Haar measure on a Lie group concept
  18. Lie group action definition
  19. Lie groupoid definition
  20. Lie supergroup definition
  21. Diffeomorphism group example
  22. Loop group example

Classical Lie groups 24 of 24

  1. General linear group example
  2. Special linear group example
  3. Orthogonal group example
  4. Special orthogonal group example
  5. Unitary group example
  6. Special unitary group SU(n) example
  7. Symplectic group example
  8. Compact symplectic group example
  9. Indefinite orthogonal group example
  10. Rotation group SO(3) example
  11. SU(2) example
  12. SL(2,R) example
  13. SL(2,C) example
  14. Lorentz group example
  15. Poincaré group example
  16. Euclidean group example
  17. Heisenberg group example
  18. Spin group example
  19. Pin group example
  20. Metaplectic group example
  21. Exceptional Lie group G2 example
  22. Exceptional Lie group E8 example
  23. Table of Lie groups example
  24. List of simple Lie groups example

Structure of Lie groups 27 of 28

  1. Semisimple Lie group definition
  2. Simple Lie group definition
  3. Reductive Lie group definition
  4. Solvable Lie group definition
  5. Nilpotent Lie group definition
  6. Compact Lie group definition
  7. Levi–Malcev decomposition statement
  8. Cartan decomposition of a Lie group statement
  9. Iwasawa decomposition of a Lie group statement
  10. KAK decomposition statement
  11. Langlands decomposition statement
  12. Polar decomposition statement
  13. Bruhat decomposition of a Lie group statement
  14. Parabolic subgroup definition
  15. Maximal compact subgroup definition
  16. Cartan–Iwasawa–Malcev theorem statement
  17. Lie algebra of a compact group is reductive statement
  18. Complexification of a Lie group definition
  19. Real form of a Lie group definition
  20. Symmetric space definition
  21. Hermitian symmetric space definition
  22. Flag manifold definition
  23. Nilmanifold definition
  24. Solvmanifold definition
  25. Malcev completion concept
  26. Cohomology of Lie groups concept
  27. Hopf's theorem on cohomology of compact Lie groups statement
  28. Bott periodicity theorem statement

Representation theory of Lie groups 21 of 22

  1. Representation of a Lie group definition
  2. Representation theory of SU(2) example
  3. Representation theory of SL2(R) example
  4. Representation theory of the Lorentz group example
  5. Wigner's classification statement
  6. Admissible representation definition
  7. Harish-Chandra module definition
  8. Harish-Chandra character definition
  9. Harish-Chandra's regularity theorem statement
  10. Discrete series representation definition
  11. Principal series representation definition
  12. Langlands classification statement
  13. Stone–von Neumann theorem statement
  14. Kirillov orbit method concept
  15. Oscillator representation definition
  16. Theta correspondence concept
  17. Selberg trace formula statement
  18. Arthur–Selberg trace formula statement
  19. Automorphic form definition
  20. Ramanujan–Petersson conjecture conjecture
  21. Selberg's 1/4 conjecture conjecture
  22. unitary dual problem problem

Transformation groups and discrete subgroups of Lie groups 19 of 19

  1. Proper group action definition
  2. Properly discontinuous action definition
  3. Discrete subgroup definition
  4. Fuchsian group definition
  5. Kleinian group definition
  6. Modular group example
  7. Fundamental domain of the modular group example
  8. Arithmetic group definition
  9. Bieberbach theorems statement
  10. Clifford–Klein form definition
  11. Geodesic flow on a locally symmetric space example
  12. Zimmer program concept
  13. Zimmer's conjecture conjecture
  14. Oppenheim conjecture statement
  15. Littlewood conjecture conjecture
  16. Lattice in a Lie group definition
  17. Wang's finiteness theorem statement
  18. Weil's local rigidity theorem statement
  19. Calabi–Weil rigidity statement

More ideas in this subject

Concepts 9

Theorems and results 8

Conjectures 1

Definitions 106

This subject is MSC class 22. See it on zbMATH Open.