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From: OsherD on 10 Apr 2010 01:57
From Osher Doctorow

Igor Nikolaev of the Fields Institute for Mathematical Sciences,
Toronto Canada (one of the mathematical analogs of the Nobel Prize for
physics or its granting institutions) has proven some important cases
of the Langlands Conjecture in "Langlands reciprocity for the even
dimensional noncommutative tori," arXiv: 1004.0904 v1 [math.QA] 6 Apr
2010. QA here refers to "Quantum Algebra".

Roughly speaking, he proves for 0 and 1 dimensions and for Abelian
Galois Groups the Conjecture that the Galois Extensions (see below) of
the field of rational numbers come from the even dimensional
noncommutative tori with real multiplication. The Langlands Program
is by Robert Langlands, now Professor Emeritus at the Princeton
Institute for Advanced Study in New Jersey USA. At the heart of the
Langlands program are the irreducible infinite dimensional
representations of the Lie group GL(n), and it is known that the n-
dimensional noncommutative tori classify such representations.

An example of a Galois Extension of the field of rational numbers is
adjoining sqrt(2) to the rational number field.

Langlands pioneered in relating number theory to algebraic
representation theory, and the relevance of this to physics in this
thread can be seen or guessed from similar relationships between
number theory and geometric objects such as Fibonacci and Pell numbers
and pentagons, octagons, icosahedrons in the previous post. Tori also
relate to distinguishing integrable systems from Chaos, another topic
of this thread from previous posts.

Osher Doctorow
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