representation theory

geometric representation theory

Ingredients

• linear algebra, algebra, higher algebra

Definitions

representation, 2-representation, ∞-representation

• group, ∞-group

• group algebra, algebraic group, Lie algebra

• vector space, n-vector space

• affine space, symplectic vector space

• action, ∞-action

• module, equivariant object

• bimodule, Morita equivalence

• induced representation, Frobenius reciprocity

• Hilbert space, Banach space, Fourier transform, functional analysis

• orbit, coadjoint orbit, Killing form

• unitary representation

• geometric quantization, coherent state

• socle, quiver

• module algebra, comodule algebra, Hopf action, measuring

Geometric representation theory

• D-module, perverse sheaf,

• Grothendieck group, lambda-ring, symmetric function, formal group

• principal bundle, torsor, vector bundle, Atiyah Lie algebroid

• geometric function theory, groupoidification

• Eilenberg-Moore category, algebra over an operad, actegory, crossed module

• reconstruction theorems

Theorems

• Borel-Weil-Bott theorem

• Beĭlinson-Bernstein localization

• Kazhdan-Lusztig theory

• BBDG decomposition theorem

• mathematics

  • math resources

  • history of mathematics

• Structural Foundations

  • logic

    • internal language
    • classical mathematics
    • evil
    • constructive mathematics
    • predicative mathematics
    • category:foundational axiom

  • set theory

    • structural set theory

  • category theory

    • Categories and Sheaves

    • topos theory

      • Sheaves in Geometry and Logic

  • higher category theory

    • higher topos theory

      • (∞,1)-topos theory

        • models for ∞-stack (∞,1)-toposes
        • cohomology

    • homotopy theory

      • stable homotopy theory
      • rational homotopy theory

• Topology and Geometry

  • geometry (general list), topology (general list)

  • general topology

  • differential topology

  • differential geometry

    • synthetic differential geometry
    • symplectic geometry

  • algebraic geometry

  • noncommutative algebraic geometry

  • noncommutative geometry (general flavour)

  • higher geometry

• Algebra

  • universal algebra

  • higher algebra

  • homological algebra

  • group theory, ring theory

    • representation theory

  • algebraic approaches to differential calculus

  • counterexamples in algebra

• analysis

  • nonstandard analysis

  • functional analysis

    • operator algebras

  • Fourier transform

• Lie theory

  • higher Lie theory

• probability theory

• discrete mathematics