String Theory, at 20, Explains It All (or Not)

The New York Times ^ | December 7, 2004 | Dennis Overbye

[snarks_when_bored]

The "Chern class", I should have said.

[PatrickHenry]

Science list Ping! This is an elite subset of the Evolution list.
See the list's description in my freeper homepage. Then FReepmail me to be added or dropped.

[Ernest_at_the_Beach]

Some very serious mathematics research going on :........

_________________________________________________________________

These are topics in mathematics at the current cutting edge of superstring research.

K-theory
	Cohomology is a powerful mathematical technology for classifying differential forms. In the 1960s, work by Sir Michael Atiyah, Isadore Singer, Alexandre Grothendieck, and Friedrich Hirzebruch generalized coholomogy from differential forms to vector bundles, a subject that is now known as K-theory.     Witten has argued that K-theory is relevant to string theory for classifying D-brane charges. D-brane objects in string theory carry a type of charge called Ramond-Ramond charge. Ramond-Ramond fields are differential forms, and their charges should be classifed by ordinary cohomology. But gauge fields propagate on D-branes, and gauge fields give rise to vector bundles. This suggests that D-brane charge classification requires a generalization of cohomology to vector bundles -- hence K-theory. Overview of K-theory Applied to Stringssby Edward Witten D-branes and K-theory by Edward Witten
Noncommutative geometry (NCG for short)
	Geometry was originally developed to describe physical space that we can see and measure. After modern mathematics was freed from Euclid's Fifth Axiom by Gauss and Bolyai, Riemann added to modern geometry the abstract notion of a manifold M with points that are labeled by local coordinates that are real numbers, with some metric tensor that determines an extremal length between two points on the manifold.     Much of the progress in 20th century physics was in applying this modern notion of geometry to spacetime, or to quantum gauge field theory.     In the quest to develop a notion of quantum geometry, as far back as 1947, people were trying to quantize spacetime so that the coordinates would not be ordinary real numbers, but somehow elevated to quantum operators obeying some nontrivial quantum commutation relations. Hence the term "noncommutative geometry," or NCG for short.     The current interest in NCG among physicists of the 21st century has been stimulated by work by French mathematician Alain Connes. Two Lectures on D-Geometry and Noncommutative Geometry by Michael R. Douglas Noncommutative Geometry and Matrix Theory: Compactification on Tori by Alain Connes, Michael R. Douglas, Albert Schwarz String Theory and Noncommutative Geometry by Edward Witten and Nathan Seiberg. Non-commutative spaces in physics and mathematics by Daniela Bigatti Noncommutative Geometry for Pedestrians by J.Madore

[jmc813]

Do you know of any web sites out there that presents string theroy in laymans' terms?

[Shryke]

May I ask what the &*)! you are talking about?

[snarks_when_bored]

Physicists who work at the fundamental level fully understand that General Relativity breaks down at the Planck scale and also that Quantum Field Theory isn't feeling so good at that scale, either. That's why they're searching for an over-arching theory that transcends (and subsumes) both GR and QFT. Superstring Theory, it was hoped, was such a theory. That hasn't proved to be the case...yet.

[Southack]

"Physicists who work at the fundamental level fully understand that General Relativity breaks down at the Planck scale..."

It's more than that. GR contradicts QM in regards to universal Gravity and energy.

...And even Einstein himself had to delete his Universal Constant from GR after 4 decades of scientific denial...

[Junior]

If true, it could be used like a searchlight to illuminate some of the deepest mysteries physicists can imagine, like the origin of space and time in the Big Bang and the putative death of space and time at the infinitely dense centers of black holes.

Majikthise: I mean, what's the use of our sitting around half the night arguing about whether there may...

Vroomfondel: Or may not.

Majikthise: - be a God, if this machine only goes and gives you his phone number in the morning?

[Ernest_at_the_Beach]

An interesting reference here:

Noncommutative Geometry for Pedestrians

[Alamo-Girl]

Thanks for the ping!

[snarks_when_bored]

Some links:

Leonard Susskind (one of the discoverers of String Theory) talks about the String Theory Landscape and the Anthropic Principle

Responses to Susskind's talk on the Landscape and the Anthropic Principle (scroll down)

Susskind stomps on poor Lee Smolin, the Loop Quantum Gravity guy

The entire Edge site is embarrassingly fun to read:

Edge

[Ernest_at_the_Beach]

Recreational Mathematics Mathematical Art Mathematical Images
Topology Topological Structures
Math Contributors Fowler

Alexander's Horned Sphere

The above topological structure, composed of a countable union of compact sets, is called Alexander's horned sphere. It is homeomorphic with the ball , and its boundary is therefore a sphere. It is therefore an example of a wild embedding in . The outer complement of the solid is not simply connected, and its fundamental group is not finitely generated. Furthermore, the set of nonlocally flat ("bad") points of Alexander's horned sphere is a Cantor set.

The horned sphere as originally drawn by Alexander (1924) is illustrated above.

The complement in of the bad points for Alexander's horned sphere is simply connected, making it inequivalent to Antoine's horned sphere. Alexander's horned sphere has an uncountable infinity of wild points, which are the limits of the sequences of the horned sphere's branch points (roughly, the "ends" of the horns), since any neighborhood of a limit contains a horned complex.

A humorous drawing by Simon Frazer (Guy 1983, Schroeder 1991, Albers 1994) depicts mathematician John H. Conway with Alexander's horned sphere growing from his head.

Antoine's Horned Sphere

Albers, D. J. Illustration accompanying "The Game of 'Life."' Math Horizons, p. 9, Spring 1994.

Alexander, J. W. "An Example of a Simply Connected Surface Bounding a Region Which Is Not Simply Connected." Proc. N. A. S. 10, 8-10, 1924.

Guy, R. "Conway's Prime Producing Machine." Math. Mag. 56, 26-33, 1983.

Hocking, J. G. and Young, G. S. Topology. New York: Dover, 1988.

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 80-81, 1976.

Schroeder, M. Fractals, Chaos, Power Law: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 58, 1991.

[snarks_when_bored]

And here's a link to a widely read technical article by Leonard Susskind:

The Anthropic Landscape of String Theory (PDF format)

[beezdotcom]

Yeah, but I want to differentiate between things that we know HOW to test, but CAN'T until we achieve some new technological milestone (like a more powerful supercollider, or probes that are light-years away), vs. those things for which we can't even devise a good gedanken experiment to validate/invalidate.

[snarks_when_bored]

Throwing in the kitchen sink horned sphere, eh?

[Ernest_at_the_Beach]

That was the universal counter example to a lot of stuff in topology of simple surfaces ( proves 3 dimensions is very different than two dimensions..... )

[snarks_when_bored]

...And even Einstein himself had to delete his Universal Constant from GR after 4 decades of scientific denial...

The cosmological constant is back...it seems to account for the accelerating cosmic expansion being attributed to 'dark energy'. Check out the Sepetember, 2004, issue of Scientific American.

It's more than that. GR contradicts QM in regards to universal Gravity and energy.

I'm not sure what you mean by this.

[snarks_when_bored]

'September' (of course)

[Ernest_at_the_Beach]

Thanks.

[Francis McClobber]

"A day without string is chaos."

Rudolph Smuntz