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Noncommutative geometry, as developed by Connes starting in the early ’80s ..., extends the tools of ordinary geometry to treat spaces that are quoti… - Matilde Marcolli
" "Noncommutative geometry, as developed by Connes starting in the early ’80s ..., extends the tools of ordinary geometry to treat spaces that are quotients, for which the usual “ring of functions”, defined as functions invariant with respect to the equivalence relation, is too small to capture the information on the “inner structure” of points in the quotient space. Typically, for such spaces functions on the quotients are just constants, while a nontrivial ring of functions, which remembers the structure of the equivalence relation, can be defined using a noncommutative algebra of coordinates, analogous to the non- commuting variables of quantum mechanics.
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About Matilde Marcolli
Matilde Marcolli is an Italian mathematical physicist. She was an Invited Speaker at the International Congress of Mathematicians held in 2010 in Hyderabad.
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The general discourse of scientists about science is marred by beliefs of the Ancient Greeks in the kalos kai agathos: that which is beautiful must also be good, and conversely. This leads inevitably to portraits of scientists as cartoonish heroes: the more profound and significant the science, ... In fact what is truly heroic about science is the fact that it does uncover beautiful truths about the universe despite the ugliness and brutality of the human beings involved.
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It turns out that noncommutative geometry is a very good framework for theories of (modified) gravity coupled to matter. The main idea behind gravity and particle physics models based on noncommutative geometry is that "all forces become gravity" on an noncommutative space. In other words, it is only from the point of view of a slice of the geometry consisting of an ordinary spacetime manifold that we see a difference between gravity and the other forces, while from the point of view of the overall (noncommutative) geometry they are all seen together as gravity. As we will see, the main construction is not unlike the idea of "extra dimensions" many people are familiar with from string theory, except for the fact that the extra dimensions in these models are not only small, but also noncommutative, while the extended dimensions of spacetime maintain their commutative nature.
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