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Andrius Kulikauskas
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Study duality as the basis of logic, and mathematics as ways of altering duality.
(internal structure mirrors external structure  duality of category theory)
I am studying the various case of duality in math. I imagine that at the heart is the duality between zero and infinity by way of one as in God's Dance.
Duality breaking
The duality between zero and infinity, between nothing and everything, is broken in many subtle ways. Here are some examples:
 By definition, a topological space includes both an entire set X and the empty set. However, the intermediary sets are closed under arbitrary unions, but only finite intersections. What would happen if they were closed under infinite intersections?
 Perhaps similarly, having in mind the Zariski topology, ideals of a ring are defined with respect to multiplication (union) but not addition (intersection).
Equivalence and uniqueness
In Math, there is an everpresent tension between the notions of equivalence class and uniqueness. If something is mathematically significant, it should in some sense be unique. But math is a model and so, as such, can never be entirely unique but represents a variety of cases. Thus it is ever natural to define equivalence classes, especially in math itself. For example, a rational number is an equivalence class that establishes a proportion.
Dualities. Duality arises from a symmetry between two ways of looking at something where there is no reason to choose one over the other. For example:
 Square roots of i. There are two square roots of 1. One we call +i, the other i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over 1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.)
 A rectangular matrix can be written out from left to right or right to left. So we have the transpose matrix.
 Normality says conjugate invariancy: gN = Ng.
 Opposite category? Morphisms can be organized from left to right or from right to left. The opposite category turns all of the arrows around.
 Colimits and limits
 Monomorphisms ("onetoone") and epimorphisms (forcing "onto").
 Coproducts and products
 Initial and terminal objects
 Wikipedia: In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). If we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices.
 Wikipedia: Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.)
 Wikipedia: Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called Eckmann–Hilton duality.
 Adjoint bendrai ir Adjoint functors. Wikipedia: It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. A construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them.
 Switching of "existing" and "nonexisting", for example, edges in a graph. This underlies Ramsey's theorem. Tao: "the Ramseytype theorem, each one of which being a different formalisation of the newly gained insight in mathematics that complete disorder is impossible."
 Coordinate systems can be organized "bottom up" or "top down". This yields the duality in projective geometry.
 Root systems relate reflections (hyperplanes) and root vectors. Given a root R, reflecting across its hyperplane, every root S is taken to another root S, and the difference between the two roots is an integer multiple of R. But this relates to the commutator sending the differences into the module based on R.
 Analysis provides lower and upper bounds on a function or phenomenon which helps define the geometry of this space.
 We can look at the operators that act or the objects they act upon. This brings to mind the two representations of the foursome.
 This is related to the duality between left and right multiplication. Examples include Polish notation.
 Faces of an object and corners of an object. (Why are they dual?)
 Coxeter groups are built from reflections. Reflections are dualities.
 Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other.
 Galois theory: field extensions (solutions of polynomials) and groups
 Lie groups: solutions to differential equations..
Read nLab: Duality. Here are examples to consider:
 Duality (projective geometry). Interchange the role of "points" and "lines" to get a dual truth: The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". (Compare with the construction of an equilateral triangle and the lattice of conditions.)
 AtiyahSinger index theorem...
 RiemannRoch theorem
 Covectors and vectors
 Cotangent space and tangent space
 de Rham cohomology links algebraic topology and differential topology
 Modularity theorem.
 Langlands program
 general Stokes theorem: duality between the boundary operator on chains and the exterior derivative
 Hilbert's Nullstellensatz
 Class field theory provides a onetoone correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field.
 Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.
 One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject.[79] This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complexvalued function.
 Meromorphic function is the quotient of two holomorphic functions, thus compares them.
 Isbell duality relates higher geometry with higher algebra.
 Topos links geometry and logic.
 For integers, decomposition into primes is a "bottom up" result which states that a typical number can be compactly represented as the product of its prime components. The "top down" result is that this depends on an infinite number of exceptions ("primes") for which this compact representation does not make them more compact.
 The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is there a problem to which F is the most efficient solution? The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves.
 https://en.m.wikipedia.org/wiki/Coherent_duality https://en.m.wikipedia.org/wiki/Serre_duality https://en.m.wikipedia.org/wiki/Verdier_duality https://en.m.wikipedia.org/wiki/Poincaré_duality
 https://en.m.wikipedia.org/wiki/Dual_polyhedron
 a very general comment of William Lawvere[2] is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G.
 division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint.
 Tensor products are adjoint to a set of homomorphisms.
 Duality  parity  išsiaiškinimo rūšis. Įvairios simetrijos  išsiaiškinimo būdų sandaros.
 In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module or monster vertex algebra, constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the noghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras.
List of dualities (Wikipedia)
 Alexander duality
 Alvis–Curtis duality
 Araki duality
 Betadual space
 Coherent duality
 De Groot dual
 Dual abelian variety
 Dual basis in a field extension
 Dual bundle
 Dual curve
 Dual (category theory)
 Dual graph
 Dual group
 Dual object
 Dual pair
 Dual polygon
 Dual polyhedron
 Dual problem
 Dual representation
 Dual qHahn polynomials
 Dual qKrawtchouk polynomials
 Dual space
 Dual topology
 Dual wavelet
 Duality (optimization)
 Duality (order theory)
 Duality of stereotype spaces
 Duality (projective geometry)
 Duality theory for distributive lattices
 Dualizing complex
 Dualizing sheaf
 Esakia duality
 Fenchel's duality theorem
 Haag duality
 Hodge dual
 Jónsson–Tarski duality
 Lagrange duality
 Langlands dual
 Lefschetz duality
 Local Tate duality
 Poincaré duality
 Twisted Poincaré duality
 Poitou–Tate duality
 Pontryagin duality
 Sduality (homotopy theory)
 Schur–Weyl duality
 Serre duality
 Spanier–Whitehead duality
 Stone's duality
 Tannaka–Krein duality
 Verdier duality
 AGT correspondence
 A "transformation group" is a group acting as transformations of some set S. Every transformation group is the group of all permutations preserving some structure on S, and this structure is essentially unique. The bigger the transformation group, the less structure: symmetry and structure are dual, just like "entropy" and "information", or "relativity" and "invariance".
Duality of silence (topdown) and speaking (bottomup).

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