2010 Spring MATH 268-01
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Bulletin Course Description
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|---|
Lie groups and related topics, Hodge theory, index theory, minimal
surfaces, Yang-Mills fields, exterior differential systems, harmonic
maps, symplectic geometry. Instructor: Staff
(Instructor named in bulletin description above may not be current. For current instructor, see listing below.)
|
| Title |
DIFFERENTIAL GEOMETRY |
| Department |
MATH |
| Course Number | 2010 Spring 268 |
| Section Number |
01 |
| Primary Instructor |
Bray,Hubert |
| Prerequisites |
Prerequisite: Mathematics 267 or consent of instructor. |
| Synopsis of course content |
|---|
| This
course will cover selected topics in three areas: scalar curvature and
its relationship to global properties of manifolds, geometric analysis
applications to general relativity including black holes and
singularities, and geometric studies of the axioms of general
relativity and how this relates to dark energy and dark matter, which
together are thought to compose 96% of the mass of the universe. Other
topics may be covered as requested by students. |
| Occasional homework assignments |
| participation, homeworks, and potentially a short presentation. |
Students may be asked to present certain topics.
Please email bray@math.duke.edu if you have any questions. |
Topics covered this year will include purely geometric topics about Ricci
and scalar curvature as well as geometric areas of general relativity. For
example, we will discuss the Positive Mass Theorem in general relativity
which, in physics terms, is the statement that nonnegative energy densities
everywhere "add up" to a total mass of the system which is also nonnegative.
Remarkably, the proof of this statement is pure differential geometry.
Also, we will explore the relationship between minimal surfaces (surfaces
which locally minimize their areas, like soap bubbles in the case of a
volume constraint) and black holes as well as the Penrose Conjecture which
states that the mass of a system containing a collection of black holes is
at least the square root of the total surface area of the horizons of the
black holes divided by 16 pi. So far most of the progress on this problem,
including my proof of the Riemannian Penrose Conjecture, has been pure
differential geometry. We will also discuss dark energy (73% of the mass of
the universe), responsible for the -accelerating- expansion of the universe,
which is again easily explained by a cosmological constant term in the
geometric equations governing general relativity. More recently I have been
interested in dark matter (23% of the mass of the universe) and have been
studying geometric models for this currently mysterious form of matter which
dominates the mass of galaxies and may be at least partly responsible for
spiral structure in galaxies. We will discuss all of these topics and more,
including purely geometric topics like the Yamabe problem and Ricci flow.