Open Mathematics Courses

Microlocal sheaves

A research course developing conic sheaves and kernel operations into specialization, microlocal morphisms and microsupport geometry. Each lesson states its hypotheses, supplies proofs or links the precise prerequisite argument, and includes worked examples.

  1. SH02-COURSE-CONTRACT — How to use this course
  2. SH02-PREREQ-OPEN — Open prerequisites and exact import contracts
  3. SH02-PREREQ-PROOFS — Supporting verifications for open prerequisites
  4. Open extensions and ambient supports
  5. SH02-SIX-BRIDGE — Proper supports and the bounded classical comparison
  6. Proper-support composition through a change of base
  7. SH02-CA — Extending sections on convex sets
  8. SH02-SUP-EXH-001 — Recovering sheaf cohomology from closed pieces
  9. SH02-NCD — Continuing cohomology through a moving boundary
  10. SH02-EXCEPTIONAL-OPERATIONS — Exceptional inverse image from a finite resolution
  11. SH02-UR — Finite resolutions without a lower bound
  12. SH02-MD — Local orientations, dimension, and integration
  13. Transport along a scaling action
  14. Operations on orbits with their induced topology
  15. Composing sheaf operators through an intermediate space
  16. SH02-SPH — Inverse operators from complementary hemisphere boundaries
  17. Fourier kernels as radial averaging
  18. Fourier duality through a test complex on the base
  19. SH02-UFD — Duality when the dual leaves the original transform category
  20. SH02-FTC. Comparing traces after Fourier transformation
  21. Moving Fourier kernels across maps and products
  22. SH02-LFT. A linear map inside the Fourier comparison
  23. SH02-FGC-UNIT. The graded line in a Fourier support comparison
  24. SH02-FTE-UNIT. Transposing the complete Fourier trace comparison
  25. SH02-NDF-UNIT. The geometric normalization of Fourier adjunctions
  26. SH02-MEP-UNIT. Following the comparison maps into the normal bundle
  27. SH02-GAM — Directional neighborhoods and a sheaf projector
  28. SH02-FSB — Formal stabilization over arbitrary neighborhood sets
  29. SH02-CB-UNIT — Finite local data and sheaf biduality
  30. SH02-DA-UNIT — Supports, characteristic classes, and finite duality
  31. Boundary conditions, incidence operators, and trace identities
  32. Convex tests and radial comparisons
  33. SH02-NG-UNIT — Normal geometry as a family with a central fibre
  34. Reading a sheaf at the normal scale
  35. Covector tests of normal limits
  36. SH02-SA — What a normal limit preserves
  37. SH02-MST — Detecting and removing directional obstructions
  38. SH02-MSD — Comparing sheaves through local support tests
  39. Local morphisms in cotangent directions
  40. SH02-MHPR — Product recovery for microlocal composition
  41. SH02-MHPC — Transporting both inputs before forming internal Hom
  42. SH02-SUB-UNIT — Reading a subset through its sheaf
  43. SH02-MO-UNIT — Transporting directional obstructions
  44. Analytic closures of the original strata and conormal exclusion
  45. SH02-FAG-UNIT — Analytic geometry for finite maps
  46. Polar sections, integral equations and generic Lipschitz algebras
  47. Polar multiplicity in linear sections and Whitney strata
  48. Characteristic values from compact analytic cells
  49. Relative conormals at the zero fibre of a holomorphic function
  50. Finite nearby-cycle models for an arbitrary holomorphic function
  51. Holomorphic critical support and graph covectors
  52. Finite holomorphic maps with arbitrary weak coefficients
  53. Finite resolutions and proper supports on locally compact spaces
  54. Derived module coefficients and their spectrum model
  55. Compact supports, covariant Verdier duality and modern composition
  56. Bounded section recognition from injective mapping complexes
  57. Modern proper base change and arbitrary stable coefficients
  58. The top orientation class on a connected manifold
  59. Compatible triangulation for the original analytic Whitney strata
  60. SH02-NMC-UNIT — Normal Morse data and change of coefficients
  61. SH02-PNM-UNIT — Perverse degrees and normal Morse complexes
  62. SH02-IHM-UNIT — An isolated holomorphic test and its Morse filtration
  63. The original isolated holomorphic pair, its whole cluster and integer count
  64. SH02-FH-UNIT — Directional information under a finite holomorphic map
  65. Directional constraints under algebraic and geometric constructions
  66. SH02-MC — Categories and operations in one cotangent direction
  67. SH02-AE-UNIT — Covectors at a boundary and at infinity
  68. SH02-CHE-UNIT — Cotangent directions that survive a limiting operation
  69. SH02-UCE — Uniform tests for an unbounded Hom complex
  70. Required proof completions for the bounded involutivity route
  71. Ordinary geometry at the bounded involutivity scope
  72. Proper Euclidean embeddings and the compact-control reduction
  73. SH02-INV — Hamiltonian motion forced by a sheaf
  74. SH02-LFI — Local models and change of ambient manifold
  75. SH02-MA — Continuing coefficients and transporting local morphisms
  76. SH02-RR — From directional tests to further microlocal theories

Original text CC0 1.0 unless the lesson states other terms.