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Det

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  • Dieudonné det etc. https://core.ac.uk/download/pdf/82261662.pdf
  • F.F. Knudsen and D. Mumford,The projectivity of the moduli space of stable curves I.Preliminaries on “det” and “Div”, Math. Scand.39(1976), no. 1, 19–55.

Duality notes

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Riesz–Markov–Kakutani representation theorem: linear functionals on a space vs. measures.

quadratic reciprocity, cubic reciprocity


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duality explanation, context conceptual explanation application generalization
Schur–Weyl duality representations of symmetric group and GLn see A categorical approach to classical and quantumSchur–Weyl duality http://www.maths.usyd.edu.au/u/pubs/publist/preprints/2010/davydov-17.pdf ; Double centralizer theorem, Thm 4.54 in http://math.mit.edu/~etingof/replect.pdf,: Let A, B be two subalgebras of the algebra End E of endomorphisms of a finite dimensional vector space E, such that A is semisimple, and . Then , more... ??
Spanier–Whitehead duality (also called S-duality (homotopy theory)) embed a topological space into some S^n, stable homotopy type; Dold: "Fixed point index and fixed point theorem for Euclidean neighborhood retracts"; explained in Ponto; Shulman "Traces in symmetric monoidal categories" example: Alexander duality
Stone's duality restricts from an adjunction to sober spaces and spatial locales Example: Duality theory for distributive lattices, gives rise to spectral spaces
Tannaka–Krein duality generalization of Pontryagin for nonabelian groups, . reconstruct a group from its unitary representations equivalence of categories?
Pontryagin duality dualizing object, Hom(-, S^1); can also be proven by Poincaré-Lefschetz duality and topological means, see http://www.encyclopediaofmath.org/index.php/Pontryagin_duality Fourier transform (case R^n) in harmonic analysis; reciprocal (or dual) lattice in crystallography (, dual lattice . Tannaka-Krein; variant for group schemes Cartier duality: Z/p is dual to \mu_p
Dual pair vector spaces V x W \r K, example (by Hahn-Banach) locally convex topological vector space; duality between Lp spaces; Riesz representation theorem: identifies a Hilbert space with its dual.; Convex conjugate: f sometimes = f**, described by Fenchel–Moreau theorem, basis of duality (optimization), in general only weak duality
Dual curve tangents to a curve in P^2; curve in dual projective plane Legendre transform, see

Arnold, Vladimir Igorevich (1988), Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, ISBN 3-540-96649-8

dual space vector spaces or topological vector space dual object (also called dualizable object) other examples: dual bundle, distribution (mathematics)
dualizing sheaf algebraic geometry Adjunction formula
Poincaré duality Twisted Poincaré duality (local coefficients), Lefschetz duality (manifolds with boundary), Verdier duality for singularities
Coherent duality chain of adjoints, see Balmer, Dell'Ambrogio, Sanders. Grothendieck-Neeman duality and the Wirthmüller isomorphism Serre duality, hence Riemann-Roch theorem
Verdier duality chain of adjoints as in Coherent duality
dual object examples: Dual representation
Hopf algebras vs. group schemes equivalence of categories
Dual abelian variety equivalence of categories
dual cone adjunction between the category of subsets of R^n; C** is the closure of the smallest convex cone containing C; equivalence on the subcategory of closed convex cones. toric geometry, e.g. dual of <e_2, 2e_1-e_2> spanned by <f_1, f_1 + f_2, f_1+2 f_2>, hence C[U,V,W]/(V^2-UW), see Fulton

Coactions may be thought of as "actions of the dual group even when there isn't any dual group [5]