# Tag Info

46

Actually even schoolchildren calculate group co-cycle. (Without knowing that it is called like this). Cohomology occurs in everyday life as soon as one learns to count. 5+7 = 1 2 4 + 5 = 0 9 2 + 8 = 1 0 What is the function on which sends a pair (a,b) to the $0$ or $1$ depending result is greater than 9 or not ? ( e.g. f(5,7)= 1, f(4,5) = 0, ...

27

Robert Ghrist is all about applied topology: Sensor Network, Signal Processing, and Fluid Dynamics. (homepage: http://www.math.upenn.edu/~ghrist/index.html ). For instance, we want to use the least number of sensors to cover a certain area, such that when we remove one sensor, a part of that area is undetectable. We can form a complex of these sensors ...

16

I don't think you can draw something meaningful - I would be surprised if someone made a good drawing of the Frobenius morphism ;). That being said, here is an example (possibly misleading or unrelated to your research) I saw in the slides of Benedict Gross's lectures on the arithmetic of hyperelliptic curves. Take a prime $p$, say $p=57$, and an equation ...

15

Quantum field theory is outside the realm of pure mathematics, makes contact with the real world and features chain complexes and cohomology. The current paradigm for gauge theories such as the standard model is based on Yang-Mills theories coupled to matter. The quantisation of nonabelian (and, depending on your choice of gauge fixing function, also ...

11

I heard the following analogy when talking to some specialists in absolute de Rham theory. I think Deninger's name was mentioned at about the same time. One possible way to imagine a variety over $\mathbb{F}_p$ is as a manifold equipped with a distinguished vector field, which we call "Frobenius". The usual discrete Frobenius that admits integer powers is ...

11

My understanding, from conversations with Raoul Bott, is that his early work on electrical circuits and the Bott-Duffin theorem can be intepreted as exhibiting close connections between de Rham cohomology and the laws of electrical circuits, and that this is part of what led him into pure mathematics early in his career.

9

The mass of a classical mechanical system is an element in the (one-dimensional) second cohomology group of the Lie algebra of the Galilei group. See J. M. Souriau, Stucture des Systèmes Dynamiques, Chap. III, section (12.136). Or in english translation, search inside here for "total mass".

8

Something resembling de Rham complex with differential-algebraic flavor appears in (variant of) control theory, see, for example, G. Conte, C.H. Moog, A.M. Perdon, Algebraic Methods for Nonlinear Control Systems, 2nd ed., Springer, 2006. But, as far as I can tell, they do not use the word "cohomology" explicitly. Spencer cohomology (which is, essentially, a ...

7

The Aharonov–Bohm effect. Classically, you can't distinguish two electromagnetic potentials which are in the same cohomology class. From quantum viewpoint, they can be distinguished, because an electron changes its phase under parallel transport defined by the connection associated to a potential.

7

Anders Björner and László Lovász used bounds on the Betti numbers for the complement of a real subspace arrangement called the $k$-equal arrangement to give a complexity theory lower bound that agreed, up to a scalar multiple, with the previously known upper bound in: A. Björner and L. Lovász, Linear decision trees, subspace arrangements, and Mobius ...

6

Recently, it is realized that quantum many-body states can be divided into short-range entangled states and long-range entangled states. The quantum phases with long-range entanglements correspond to topologically ordered phases, which, in two spatial dimensions, can be described by tensor category theory (see cond-mat/0404617). Topological order in higher ...

6

The finite element method- a numerical method for solving PDE's- has a homological interpretation: MR2269741 (2007j:58002) Arnold, Douglas N.(1-MN-MA); Falk, Richard S.(1-RTG); Winther, Ragnar(N-OSLO-CMA) Finite element exterior calculus, homological techniques, and applications. (English summary) Acta Numer. 15 (2006), 1–155

3

This is not nearly as nice an example as the others, but I always imagined the line in characteristic five geometry as a countable set of points that glow like blue Christmas tree lights vaguely in the shape of a narrow paraboloid, with 0 at the vertex, with 1, 2,3, and 4 at the next "height", with the quadratic "irrationals" next, etc. although the makes it ...

3

http://sigact.org/Prizes/Godel/2004.html http://www.cs.brown.edu/~mph/HerlihyS99/p858-herlihy.pdf Maurice Herlihy and Nir Shavit won the 2004 Gödel Prize for topological analysis of asynchronous computation. Homology was involved.

1

There are some applications of topology/cohomology to combinatorics and combinatoric geometry. One of the earliest examples is surely Lovasz's proof of a bound for the chromatic number of the Kneser graph; he uses the Borsuk-Ulam theorem, which is usually proved by homological methods. A modern exposition can be found here. Another example is Tveberg's ...

1

There's a CST.SE thread of possible interest here: http://cstheory.stackexchange.com/questions/7958/papers-on-relation-between-computational-complexity-and-algebraic-geometry-topol It mentions stuff like Geometric Complexity Theory, a far-out program for proving P!=NP with algebraic geometry. Similarly here: ...

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