User:Jakob.scholbach/sandbox

Det

• 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.

• [1]
• Dodgson condensation - The theory of determinants in the historical order of development Volume 3 by Thomas Muir, p. 17
• Chio condensation, Cramers rule https://hal.archives-ouvertes.fr/hal-01500199/file/HA.pdf
• The Linear Algebra a Beginning Graduate Student Ought to Know by Jonathan Samuel Golan, p 208
• [2]

Duality notes

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

quadratic reciprocity, cubic reciprocity

caption

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 $B=End_{A}E$. Then $A=End_{B}E$, 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 $Top\rightleftarrows Frm$ 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 ($L\subset R^{n}$, dual lattice $L^{*}=\{f\in Hom(R^{n},S^{1})(\cong R^{n}),f(L)=0\}$.	Tannaka-Krein; variant for group schemes Cartier duality: Z/p is dual to \mu_p
Dual pair	vector spaces V x W \r K, $V\subset W^{*};W\subset V^{*}$		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	$C\subset R^{n};C^{*}=\{u,<u,v>\geq 0forallv\in C$	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

• Alvis–Curtis duality
• Araki duality
• Beta-dual space
• Dual basis in a field extension
• Dual (category theory)
• Dual graph
• Dual polygon
• Dual polyhedron
• Dual problem
• Dual q-Hahn polynomials
• Dual q-Krawtchouk polynomials
• Dual topology: topology on a pair of topologically dual locally convex vector spaces
• Dual wavelet
• Duality (optimization), also called Lagrange duality
• Duality (order theory)
• Duality of stereotype spaces
• Duality theory for distributive lattices
• Dualizing complex
• Eckmann–Hilton duality
• Esakia duality
• Fenchel's duality theorem
• Haag duality
• Jónsson–Tarski duality
• Langlands dual
• Local Tate duality
• Opposite category
• Poitou–Tate duality
• Morita duality? [3]
• duality in measure theory ? [4]

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