The concept of Duality - MathOverflow

The concept of Duality

2011-08-26T00:22:59Z

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come.

Wikipedia has a good page on several forms of "duality" in mathematics, which outlines several notions of duality (geometric, in convex analysis, topology, set theory, etc.) I am very interested in getting help with the following goal:

Collect an annotated list of various notions of duality that occur in mathematics, with the ultimate aim of describing the notions in a way that makes it easier to recognize and intuitively build connections between the various notions of duality. Also welcome are comments / answers that highlight how a particular notion of duality can be extremely useful (in proving theorems, in applications, for computational reasons, etc.)

Some additional context

I got thinking about this question after reading the following amazing paper: The concept of duality in convex analysis, and the characterization of the Legendre transform, by Shiri Artstein-Avidan and Vitali Milman, where the authors talk about duality in more abstract terms (though, largely in the setting of convex analysis). Motivated by their abstract treatment got me thinking whether such abstract treatments of duality have been investigated for other types of duality, which eventually led to this question.

Thus, in line with with the Avidan-Milman results, one may also ask similar questions about other types of duality (i.e., one tries to characterize why and how a chosen notion of duality is the only "natural" choice under a set of axiomatic requirements).

Answer by Gerald Edgar for The concept of Duality

2011-08-26T00:28:41Z

Projective geometry. Is that the first use of the term "dual" in mathematics?

Answer by Michael Thaddeus for The concept of Duality

2011-08-26T00:55:09Z

The (1) Fourier transform, (2) mirror symmetry, (3) electric-magnetic duality, and the (4) Pontrjagin and (5) Langlands dualities of Lie groups are all seen to be interrelated by the proposal of Strominger-Yau-Zaslow for mirror symmetry and the work of Kapustin-Witten (foreshadowed by Montonen-Olive) framing the geometric Langlands program in physical terms.

Answer by Joel David Hamkins for The concept of Duality

2011-08-26T01:32:38Z

There are various dualities arising in elementary logic:

the duality between $\forall$ and $\exists$, as expressed by the validity $$\neg\forall x\ \neg\varphi(x)\iff \exists x\ \varphi(x).$$
the duality between $\wedge$ (and) and $\vee$ (or), as expressed via the de Morgan laws $$\neg(p\wedge q)\iff (\neg p)\vee(\neg q).$$
the duality in modal logic between possibility and necessity, as expressed via $$\neg\Diamond\varphi\iff\square\neg\varphi,$$ (that is: $\varphi$ is not possible if and only if $\neg\varphi$ is necessary), a principle which has manifestations for any of the diverse interpretations of these modal operators satisfying this equivalence.

Each of these dualities arises in the conjugation of one logical quantifier or operation with $\neg$.

Answer by David Roberts for The concept of Duality

2011-08-26T01:57:49Z

Galois connections (nLab,wikipedia). This is really just an adjuction between one category and the opposite of another, where these categories are preorders. A Galois correspondence is when this adjunction is an equivalence of categories.

Stone duality (nLab,wikipedia). This is best explained by the linked page, but one I will point out is that one has as a small part of this duality, $FinSet \simeq FinBool^{op}$ (the category of finite sets is equivalent to the opposite of the category of finite boolean algebras), which has as a corollary, the category of Stone spaces is equivalent to that of profinite sets.

There is the nLab page duality, but one can see by searching the nLab, there are a number of other pages that people might find useful.

Answer by Alex for The concept of Duality

2011-08-26T02:13:55Z

How about the duality between proofs and models?

Answer by John Bentin for The concept of Duality

2011-08-26T08:32:13Z

Finite-dimensional linear spaces. A particular feature in this case is that the (algebraic) dual of a finite-dimensional vector space, namely the space of linear maps from the vector space into the base field, is isomorphic to the original space (since it is of the same dimensionality) but not canonically so. In contrast, the bi-dual (the dual of the dual) is canonically isomorphic to the original space, and so may be identified with it.

Answer by Noah Stein for The concept of Duality

2011-08-26T09:25:53Z

I enjoyed a series of talks by Bernd Sturmfels on some such interrelationships, which it looks like are written up in a paper by Rostalski and Sturmfels called "Dualities in Convex Algebraic Geometry."

Abstract: Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semideﬁnite programming and from sums of squares. This article compares three notions of duality that are relevant in these contexts: duality of convex bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the optimal value of a polynomial program is an algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary structure of the convex hull of a compact variety, we contrast this to Lasserre’s representation as a spectrahedral shadow, and we explore the geometric underpinnings of semideﬁnite programming duality.

Answer by F. C. for The concept of Duality

2011-08-26T10:38:44Z

Koszul duality is a useful duality. For example, one can cite

Koszul duality of quadratic algebras (due to Priddy) which is related to inversion of formal power series.
Koszul duality of quadratic operads (due to Ginzburg and Kapranov) which is related to reversion of formal power series or plethystic reversion.
Koszul duality of cyclic quadratic operads (due to Getzler and Kapranov) which is related to Legendre duality and Legendre transform.

One can see (1 and 2) that Koszul duality is often related to the notion of inversion $g \mapsto g^{-1}$ in a group.

Answer by Igor Khavkine for The concept of Duality

2011-08-26T11:05:16Z

I think that the obvious one between spaces (topological, differentiable, algebraic, etc.) and the rings of structure preserving functions on them should be mentioned.

Answer by unknown (google) for The concept of Duality

2011-08-26T11:07:24Z

Serre duality

Grothendieck duality

Verdier duality

Answer by Gil Kalai for The concept of Duality

2011-08-26T11:19:09Z

In the study of convexity and convex polyhedra there are three (related) important notions of duality

1) Polar duality

This is a map assigning to every convex set $K$ containing the origin its polar dual: $K^*$ which is the set of all points whose inner product with every point in $K$ is at most 1.

On polytopes it induces an order reversing map on the face lattices. This operation has subtle relations to mirror-symmetry and Koszul duality.

Web sources (1; 2; 3; 4)

2) Gale transform

This is an operation to move from n points in $R^d$ to n points in $R^k$ where k=n-d-1. It is especially useful if the original n points are in convex position to study the convex polytope they define. (Web-sources: 1, 2; 3; 4)

3) Linear programming duality

This is an operation to move from a linear programming problem to a dual problem which have the same solution.

(We sources: 1; 2; 3;)

Answer by Emil Jeřábek for The concept of Duality

2011-08-26T13:26:03Z

The Jónsson–Tarski duality between Boolean algebras with operators (in particular, modal algebras) and general frames. (A variant of this, called Esakia duality, has topological frames instead of general frames. There is also an analogous duality of Heyting algebras and intuitionistic frames, which I never remember whose name it bears.) This duality is the basis of the Kripke semantics for modal, intuitionistic, and other nonclassical logics.

Answer by Emil Jeřábek for The concept of Duality

2011-08-26T13:36:25Z

The Curry–Howard isomorphism between typed $\lambda$-calculus and intuitionistic proofs.

Answer by Haim for The concept of Duality

2011-08-26T14:15:10Z

The duality between measure and category in the set theory of the reals.

Answer by Denis Serre for The concept of Duality

2011-08-26T19:27:43Z

Duality is the corner stone of the theory of Distributions

Answer by Vijay D for The concept of Duality

2011-08-27T00:36:37Z

This answer has a heavy bias towards logical structures. The simplest notion I know is order-theoretic duality.

The dual of an order is the inverse relation of the order (less-than vs. greater-than, subset vs. superset)
Greatest lower bounds and least upper bounds (minimum vs. maximum, intersection vs. union, conjunction vs. disjunction)
Bottom and top
Least and greatest fixed points
Additive and multiplicative maps

In structures containing negation, we have De Morgan duality, such as the examples from logic given by Joel David Hamkins.

I do not know if 'duality' is the right term, but I think of adjunctions as duals too. To add to the answer of David Roberts:

Conjunction and implication (both with one argument fixed) are adjoints
Existential and universal quantification are adjoints to a certain form of substitution
Strongest postconditions and weakest liberal preconditions in programming language semantics
Sets of models and sets of formulae
A lattice and its image under a closure operator

In settings with a notion of time, there are temporal dualities from the interaction of the past and the future. There are several examples in temporal and modal logics.

Some representation theorems for lattices are ancestors of dualities. For example, Stone's representation theorem for Boolean algebras is now usually referred to as a duality. There are various dualities relating families of lattices with families of discrete structures.

Complete, atomic, Boolean algebras and powersets [Lindenbaum and Tarski]
Finite distributive lattices and finite posets [Birkhoff]
Completely distributive, algebraic lattices and posets [Raney, others I cannot recall]
Boolean algebras with operators and sets with relations [Jónsson and Tarski]
Distributive algebras with operators and ordered sets with relations [Gehrke and Jónsson (though there may be earlier work)]

The list goes on. Such results are sometimes called discrete dualities. There is much recent work on discrete duality in terms of what are called canonical extensions. These duality results often include a topological component.

Boolean algebras and Stone spaces [Stone]
Distributive lattices and Priestley spaces [Priestley]
Heyting algebras and Esakia spaces [Esakia]
Topological representations of arbitrary lattices [Urquhart]
Extensions of Stone and Priestley duality to lattices with operators
Dualities arising in Modal logic [Goldblatt]

One 'analogy between analogies' is that of a dualising object. The term schizophrenic object has also been used in this context.

Porst and Tholen's article Concrete Dualities discusses some of these and other dualities and the connection to adjunctions. Other references are Peter Johnstone's book Stone Spaces and Clarke and Davey's book Natural Dualities for the Working Algebraist.

Answer by Pait for The concept of Duality

2011-08-27T12:50:39Z

In control theory there exists the duality controllability and observability. It is very well understood in the context of linear control theory, not so much for nonlinear systems. It is related to the linear space duality between vectors and functionals, but more work in understanding it from a more general perspective would be welcome.

Answer by Murphy for The concept of Duality

2011-08-27T15:16:33Z

The duality between projective modules and injective modules, also the duality between divisible abelian groups and free abelian groups.

Answer by Spice the Bird for The concept of Duality

2011-08-29T01:05:08Z

One "duality principle" that occurs in category theory is that of Isbel Duality. My feeling is (feel free to correct me if I am wrong) is that this encapsulates stone duality, Gelfand duality, and the duality of affine schemes and commutative rings in the same disscusion. Let $\mathcal{C}$ be a (small) category. Then the presheaves on $\mathcal{C}$ and the co-presheaves on $\mathcal{C}$ are somehow dual to one another. Conceptually, one thinks of the presheves as spaces and co-presheaves as quantities. This is something that I am trying to understand myself. Some nice articals at n-lab are:

http://ncatlab.org/nlab/show/space+and+quantity

http://ncatlab.org/nlab/show/Isbell%20duality

Answer by David Corfield for The concept of Duality

2011-09-01T08:23:03Z

What would be useful here is a list of mechanisms lying behind these appearances of duality. So we have (at least)

Duality pairing
Dualizing object
Maximal fixed subcategories of an adjunction
Arrow reversal

Then we could look at any relations between these mechanisms, such as between 2 and 3, maps into a dualizing object form the functors for an adjunction.

Atiyah in his talk Duality in Mathematics and Physics says

"Fundamentally, duality gives two different points of view of looking at the same object. There are many things that have two different points of view and in principle they are all dualities."

So perhaps we need

5 . Something is seen in two different ways

The Dynkin diagram for $SL_n$ is a string of $n-1$ dots, we can view it from either end as point, line, plane, etc. Put another way, the symmetry of the diagram corresponds to an outer automorphism which account for the duality of projective geometry.

I wonder if 'deeper' dualities come from more intricate processes of seeing something from two points of view. Frenkel gives a very accessible talk What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common? where he explains that the duality of Geometric Langlands arises from compactifying a 6d quantum field theory in two different ways onto 2d surfaces.

Answer by Gil Kalai for The concept of Duality

2011-09-11T06:30:18Z

A very simple and important notion of duality is the following.

Start with a collection $F$ of subsets of a ground set $X$.

Now, define the blocker $F^*$ of $F$ as follows:

$F^*=${$ X \backslash A: A \notin F $}.

In words, we take the complements of all sets not in $F$.

This notion is very important in combinatorial optimization and polyhedral combinatorics. It is also a simple manifestation of Alexander duality from algebraic topology.

Addendum (Adam Bjorndahl):

This construction can be viewed as a generalization of the quantifier duality $$\forall \equiv \lnot \exists \lnot.$$

As above, fix a set $X$. For $F \subseteq 2^{X}$, define the formula $(\text{F}x) \ \phi(x)$ to mean that $$\{x \in X : \phi(x)\} \in F.$$ So $(\text{F}x) \ \phi(x)$ might be read "for $F$-many $x$, property $\phi$ holds". Three special cases deserve some attention.

When $F = \{X\}$, we recover the usual "for all" quantifier. Succinctly, $\forall = \{X\}$.
Dualizing, we obtain $$\lnot (\text{F}x) \lnot \phi(x) \iff \{x \in X : \lnot \phi(x)\} \notin F;$$ thus if $A = \{x \in X : \phi(x)\}$, we have $$\lnot (\text{F}x) \lnot \phi(x) \iff A \in F^{*} \iff (\text{F}^{*}x) \phi(x),$$ where $F^{*}$ is the blocker of $F$.
Finally, if $U \subset 2^{X}$ is an ultrafilter on $X$, then $$\lnot (\text{U}x) \lnot \phi(x) \iff (\text{U}x)\phi(x),$$ which exhibits ultrafilters as self-dual quantifiers, a perspective I find appealing.

Answer by Tony Huynh for The concept of Duality

2011-09-11T07:26:11Z

I am personally fond of matroid duality. Let $M$ be a matroid with ground set $E$. The dual of $M$ is the matroid with ground set $E$ and whose bases are the complements of bases of $M$. It is easy to verify that the dual of $M$ is indeed a matroid and we immediately have that $M^{dd}=M$.

Matroid duality illustrates that deletion and contraction are actually dual operations. That is, deletion corresponds to contraction in the dual and vice versa.

It also nicely generalizes duality for planar graphs. That is, if $G$ is a planar graph, and $M(G)$ is the cycle matroid of $G$, then $M^d(G)=M(G^d)$ (here, $G^d$ is the planar dual of $G$).

Finally, here is a proof for free of Euler's Formula via matroid duality. Let $G$ be a connected planar graph with edge set $E$. Suppose that $G$ has $v$ vertices, $e$ edges and $f$ faces. Let $r$ be the rank function of the cycle matroid of $G$ and let $r_d$ be the rank function of the dual of the cycle matroid of $G$. Then

\[ e=r(E)+r_d(E)=(v-1)+(f-1). \]

Answer by Marc Palm for The concept of Duality

2012-02-27T15:47:15Z

Differential geometry: Eigenvalues of Laplace operators $\Leftrightarrow$ length of closed geodesics

representation theory: irreducible representations $\Leftrightarrow$ conjugacy classes in a group

Number theory: primes $\Leftrightarrow$ zeros of $L$ functions

Quantum mechanics: particles $\Leftrightarrow$ waves

Argument principle in complex analysis: contour integrals $\Leftrightarrow$ residues

Index theory: topological index $\Leftrightarrow$ analytic index

Algebraic geometry: algebraic cycles $\Leftrightarrow$ motives

Most of them can be found in: www.claymath.org/cw/arthur/pdf/52.pdf

Trace fomulas like Poisson summation formula, Arthur's trace formula, Selberg's trace formula, Gutzwiller trace formula, Lefschetz trace formula, Weil's explicit formula quantify these relations.

There is always a sort of Fourier uncertainty involved, so a one-to-one correspondance between "geometric objects" and "spectral objects" is not available, except perhaps for the symmetric group $S_n$ via Young diagrams.