Alg-top in CS

In a post below, I mentioned algebraic topology in computer science. A nice application of alg-top is for study of concurrency in distributed systems. For instance, one approach is to consider execution traces of a computational system being represented by time-directed paths through a space, and then to use alg-top methods to ask and answer questions about the structure of this space.

This approach leads naturally to a concept of homotopies of paths, equivalence classes of paths which may be transformed into one another via other paths in the space. What is different from traditional homotopy theory is that the paths are directed, and so these are referred to as directed homotopies or dihomotopies. Paths which are not dihomotopically equivalent represent execution traces on which there are events not reachable from one to another. For more on this, start with the GETCO conference pages.

New uses for Grothendieck topologies

I’ve been saying for a while that the big problems in computer science (eg, P vs NP; a theory of distributed systems; effective GO-playing machines; etc) need radical new methods, and I have suggested algebraic topology as a likely source of ideas. Joel Friedman has just applied some alg-top to boolean complexity, motivated by the P vs. NP problem, in Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity.

Geometry in art

Greetings, all, and thank you, Walt, for inviting me to post here. I hope you find my take on the mathematical arts interesting enough to read and respond to my posts!

Since I’m a believer in the importance of context, let me say that I’m posting this from Brussels, Belgium, where I happen to be for a meeting. This afternoon I caught a major exhibition of the arts of the first Russian Avant Garde, held at Bozar, the Center for Fine Arts in Brussels. Why am I reporting this on a site devoted to mathematics, I hear you cry? Well, I think a typical pure mathematician would be struck by the geometrical nature of cubist, futurist or constructivist art, and particularly that of the Russians who are the focus of this exhibit. The cubists sought to reveal an object from all perspectives simultaneously, the futurists to capture the dynanism of machines and the colours of metals, and the constructivists to distill visual art to its essential and abstract forms and colours.

Indeed, our typical mathematician would not be mistaken in seeing geometry in this art. In the last decades of the 19th century and the early years of the 20th, there was widespread public interest in the ideas which had recently revolutionized geometry — non-Euclidean geometry, David Hilbert’s axiomatization of geometry (1899), and ideas of “the fourth dimension”. Two of the leading artists of this period, Kazimir Malevich and Piet Mondrian, both sought to represent these new ideas from geometry in their art, and said so explicitly.

If this topic interests you, there is some further reading below.

I’ll have more to say on Hilbert and the intellectual trouble that his axiomatization of geometry caused the philosopher Gottlob Frege in a later post.

References:

M. Dabrowski [1992]: Malevich and Mondrian: nonobjective form as the expression of the “absolute'”, pp. 145-168, in: G. H. Roman and V. H. Marquardt (Editors): The Avant-Garde Frontier: Russia Meets the West, 1910-1930. University Press of Florida, Gainesville, FL, USA.

L. D. Henderson [1983]: The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton University Press, Princeton, NJ, USA.

D. Hilbert [1899]: Grundlagen der Geometrie. pp. 3-92, in: Festschrift zur Feier der Enthullung des Gauss-Weber-Denkmals in Gottingen. Teubner, Leipzig, Germany. (Translated by E. J. Townsend as, “Foundations of Geometry”, Open Court, Chicago, IL, USA. 1910.)