David Corfield intuitions about cohomology

Carrying

Carrying when performing addition concerns the cocycle necessary to extend $\mathbb{Z}/10 \mathbb{Z}$ to $\mathbb{Z}/100 \mathbb{Z}$.

A Cohomological Viewpoint on Elementary School Arithmetic, Daniel C. Isaksen, The American Mathematical Monthly, Vol. 109, No. 9. (Nov., 2002), pp. 796-805.

Impossible figures

See Roger Penrose, On the Cohomology of Impossible Figures, Leonardo Vol. 25, No. 3/4, Visual Mathematics: Special Double Issue (1992), pp. 245-247, JSTOR. So, $H^1(S^1, R^+) =R^+$. A coboundary would have multiplications around the triangle equal to 1.

Electricity

Voltage as cochain, see Baez TWF293 and Circuit theory. The idea goes back to Weyl. Bott’s training as an electrical engineer would have stood him in good stead. Bott in ‘On Induced Representations’ describes how he told Weyl about ideas, which Weyl had done in 1920s.

Social Choice

The Condorcet paradox that individually consistent comparative rankings can lead to global inconsistencies is a favorite topic in voting theory. Its best explanation cohomology is less popular. (Ghrist, ELEMENTARY APPLIED TOPOLOGY)

Pythagorean triples

Hilbert’s Theorem 90 has as a consequence the parameterisation of Pythagorean triples. The theorem is a consequence of a more general result, expressing the lack of one-dimensional cohomology in a galois group. $H^1(\mathbb{Z}_2, \mathbb{Q}(i)^{\ast}) = 0$, where $\mathbb{Z}_2$ is $Gal(\mathbb{Q}(i)/\mathbb{Q})$. Any element in $\mathbb{Q}(i)$ of norm 1 would provide a 1-cocycle, since the multiplicative cocycle condition requires $\phi (s t) = s \phi(t) \cdot \phi(s)$, for $\phi: \mathbb{Z}_2 \to \mathbb{Q}$. This must then be a 1-coboundary, so of the form $(a + i b)/(a - i b)$, where $a$ and $b$ are integers.

$H^1(Gal(\mathbb{Q}(i)/\mathbb{Q}), \mathbb{Q}(i)^{\ast})$ classifies $\mathbb{Q}$ vector spaces $V$, such that $\mathbb{Q}(i) \otimes_{\mathbb{Q}} V$ is isomorphic as an $\mathbb{Q}(i)$-vector space to $\mathbb{Q}(i)$, up to $\mathbb{Q}$-linear isomorphisms.

We could interpret the group cohomology results in terms of Eilenberg-MacLane spaces. So $H^1(RP^{\infty}, \mathbb{Q}(i)^{\ast}) = 0$. Whenever $|b.\bar{b}| = 1, b = \bar{a}/a$.

Mass in classical mechanics

Mass “has a cohomological significance, it parametrizes the extensions of the Galileo group.” (Santiago Garcıa, hep-th/9306040). See Café discussion.

Non-Locality and Contextuality

Shane Mansfield, video

Model theory

Angus MacIntyre, A History of Interactions between Logic and Number Theory

“bounds in polynomial ideals, a topic still developing because of the needs of a logic of cohomology [38, 31];”

“For me personally, the main surprise arising from the discovery of ACFA was how much there was to be done in terms of a model-theoretic reaction to the development of etale cohomology and its relatives.”

“in a different direction, one begins to see cohomological ideas coming up all over applied model theory, for example in o-minimality.They are certainly present in the work of Denef and Loeser, currently a high point of the subject. However, they are not so obvious in p-adic settings. There seems no obstruction in principle now in making an analysis of crystalline cohomology from a model theoretic perspective”

See,

Dieudonn´e, J. Jacques Herbrand et la th´eorie des nombres. (French) [Jacques Herbrand and the theory of numbers] Proceedings of the Herbrand symposium (Marseilles, 1981), 3–7, Stud. Logic Found. Math., 107, North-Holland, Amsterdam, 1982.

Macintyre, Angus . Weil cohomology and model theory. Connections between model theory and algebraic and analytic geometry, 179–199, Quad. Mat., 6, Aracne, Rome, 2000.

and

MathOverflow question: “In o-minimality, the ”spectral“ topology is often used: see e.g. many papers by Edmundo. A similar approach can be used in other topological structures, as long as the structure is definably connected; for structures that are (totally) definably disconnected (like the p-adics) you would need to come out with something different.”

Cocycles classify things

Informally, if $X$ is a topological space, then we think of an element of $H^i(X)$ as being represented by a codimension-$i$ subspace of $X$ that can move around freely in $X$. For example, suppose that $f$ is a continuous map from $X$ to an $i$-dimensional manifold. If X is a manifold and $f$ is sufficiently “well-behaved,” then the inverse image of a “typical” point in the manifold will be an $i$-codimensional submanifold of $X$, and as we move the point about, this submanifold will vary continuously, and will do so in a way that is similar to the way that a circle became two circles and a sphere became a torus earlier.

Interfaces in object oriented programming

An interface can be thought as a coboundary connecting the boundaries of two different systems, for instance one can think of CORBA scripts as coboundaries that compiling in Java on one side and in Cobol on he other side, give a way to exchange data between two different systems.