Open Questions: Mathematics and Physics
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See also: Symmetry 
Superstring theory 
Mtheory 
Quantum field theory 
Geometry and topology 
Noncommutative geometry 
Quantum geometry 
Mathematical analysis and differential equations 
Chaos theory and dynamical systems
Since gauge fields, including in particular the electromagnetic field,
are fiber bundles, all gauge fields are thus based on geometry.
To us it is remarkable that a geometrical concept formulated without
reference to physics should turn out to be exactly the basis of one,
and indeed maybe all, of the fundamental interactions of the physical
world.
T. T. Wu and C. N. Yang

Introduction
The distinguished physicist Eugene Wigner wrote a wellknown essay
entitled "The Unreasonable Effectiveness of Mathematics in the Natural
Sciences". (From which this quote is taken.)
Many other scientists have, like Wigner, gone beyond agreeing with
Galileo that mathematics is the language of science to noting that
there is something a bit eerie and uncanny about this situation,
especially in the physical (and increasingly, the biological) sciences.
How does it happen, for example, that the abstract geometry developed
by the mathematician G. F. B. Riemann in the 1850s turned out to be
exactly the tool Einstein needed in 1915 to formulate his
general theory of relativity? This was not, after all, a case of
building a tool to suit the problem (which is what Newton did when
he invented calculus). It was, instead, a matter of Einstein's finding
an existing, quite sophisticated mathematical theory which, when
pressed into service for describing gravitation, turns out to make
some astonishing physical predictions  from the precession of Mercury's
perihelion to black holes  new examples of which are
still being confirmed today.
At present, we have no satisfactory answers to this question. That is,
it remains in the province of philosophy  it is still an "open
question". Looking at some of the current ways in which mathematics
and physics interact will emphasize the remarkable interconnection  and
deepen the mystery of the relationship.
It should not be thought, however, that all is sweetness and light between
mathematicians and physicists. The chances that the two disciplines will
merge anytime in the foreseeable future is quite remote, because the
characteristic modes of reasoning employed by mathematicians and physicists
are quite different.
Although this is a large oversimplification, it is still not unfair to
say that the ultimate goal of physicists is to discover basic laws of
nature. There are no hardandfast rules about how such discovery may
be done. Pure guessing, for instance, is allowed. All that matters,
ultimately, is that the resulting "law" stand up to all attempts to
find experimental or observational evidence which might falsify some
logical prediction of the supposed law. In other words, if there are
experimental or observational tests which can be made, and if the
theory passes all of them, it is accepted  until it either fails a
test or some other equally successful theory comes along with is
manifestly simpler.
Therefore, it does not fundamentally matter to physicists (in general)
if steps taken in deriving the theory or law from starting premises and
known facts are lacking in mathematical rigor. If the steps are "intuitively"
correct and the answer is right, that's enough. Quantum field theory, for
instance, is a prime example. The relativistic theory of electromagnetism,
known as quantum electrodynamics (QED) produces answers which agree with
experiment to more than 10 decimal places. Yet the techniques used to
derive the theory (such as "path integrals" and "renormalization") have
yet be put on a rigourous mathematical foundation.
Physicists are content to leave the development of such rigorous foundations
to the mathematicians. Indeed, there are a number of important examples
where work of this sort provides important open questions facing
mathematics.
From the mathematicians' point of view (again oversimplifying), rigor in
formulating a theory and making deductions is all important. This attitude
goes right back to Euclid, over 2500 years ago, whose science is so
durable that it alone among its contemporaries is still taught almost
unchanged in schools today. As far as mathematicians are concerned, it is only
such formalization and rigor that guarantees the correctness of results.
Whether or not the results can be applied or put to experimental tests is all
but irrelevant. (And yet, Euclidean geometry is anything but useless or
irrelevant, 2500 years later.)
But in fact, mathematicians and physicists usually get along well enough
when they have occasion to interact. Physicists are sensitive to
considerations of mathematical elegance and aesthetics. From Maxwell's
equations, through special and general relativity, to contemporary
superstring theory there are numerous examples of this. If given the
chance, physicists are often guided in their thinking by mathematical
aesthetics.
Likewise, mathematicians cannot fail to be impressed by physical laws
which seem to meet all experimental challenges  such as quantum field
theory and the standard model of particle physics  in spite of less than
rocksolid mathematical foundations. Cases like this provide the
inspiration to attempt a more rigorous formulation of the theory, and a
seemingly endless supply of difficult open questions.
It's worth noting that there certainly have been many cases when
physicists and mathematicians have resisted or resented encroachments from
the other side on "their" territory. For example, a leading American
physicist, John Slater, referred contemptuously to the application
of group theory in quantum mechanics as the "Gruppenpest". He wrote,
as late as 1975,
As soon as [my] paper became known, it was obvious that a great many
other physicists were as disgusted as I had been with the grouptheoretical
approach to the problem. As I heard later, there were remarks made such as
"Slater has slain the 'Gruppenpest'". I believe that no other piece of
work I have done was so universally popular.
Poor Slater. From today's perspective, it's clear that the ideas of
symmetry and group theory are absolutely pervasive in physics.
Mathmematicians have also been guilty of chauvinism. The number theorist
G. H. Hardy, for instance, liked to boast that none of his work had any
practical applications at all. But number theory (primes and factorization)
and the related theory of elliptic curves are now of fundamental
importance in cryptography. And many number theorists now suspect 
with perhaps justified pride  that deep results yet to come in their
field may underlie physics at the most fundamental level.
Here are some of the more esoteric mathematical topics, developed
initially for their own intrinsic interest which have
been found essential in formulating important parts of modern physics:
 Riemannian geometry of manifolds
 fiber bundles
 abstract harmonic analysis
 Lie groups, algebras, and representation theory
 automorphic and modular functions
 topological vector spaces
 von Neuman algebras
 spinors and twistors
 symplectic geometry
 instantons
 quantum groups
 quantum geometry
 algebraic geometry
 Riemann surface theory
 index theorems
 knot theory
 noncommutative geometry
 spin networks
The mathematician Simon Donaldson had been doing some extremely
important work on the geometry of 4manifolds. This work involved a
many concepts from physics, such as "instantons" and YangMills
equations. Among other things, it led to the very surprising conclusion
that 4dimensional Euclidean space (R^{4})
had an infinite number of distinct "differentiable structures". That
is, R^{4} could be regarded as a differentiable
manifold in infinitely many different ways. What's so surprising is that
R^{n}, for every other n, has a unique
differentiable structure. This aroused the interest of physicists, because
they need to know which is the right structure for doing physics.
Donaldson's work also led to the construction of new "invariants" of
4manifolds, which are numeric or algebraic objects that help classify
manifolds. (The Euler number is an example.) The physicist Nathan Seiberg,
who had been working with supersymmetry, became interested in this research.
In 1993 he collaborated with Witten on an investigation of supersymmetric
gauge theory related to Donaldson's theory. What they found were new
equations that greatly simplified what Donaldson had discovered.
Site indexes

Math Forum Internet Mathematics Library: Physics
 Alphabetized list of links with extensive annotations.

Open Directory Project: Mathematical Physics
 Categorized and annotated mathematical physics links. A version of this
list is at
Google, with entries sorted in "page rank" order.

The Net Advance of Physics: Geometry and Topology
 An index of tutorial and research articles
located at the
physics preprint archive. Areas covered include
differential geometry, differential forms, geometric probability,
noncommutative and quantum geometry, topology,
topological field theory, SeibergWitten theory.

The Net Advance of Physics: Group Theory and Algebra
 An index of tutorial and research articles
located at the
physics preprint archive. Topics include Clifford algebras,
quaternions, Grassmann algebra, Hopf algebras, Lie algebras, quantum
groups, noncommutative geometry, group theory.

Galaxy: Mathematical Physics
 Categorized site directory. Entries usually include
descriptive annotations.
Sites with general resources

This Week's Finds in Mathematical Physics
 John Baez writes
an excellent and useful column on recent news, papers, and books about
mathematical physics that he's come across. The column appears
roughly twice a month. It can be searched by issue number, keyword,
or through a large table of contents.

String Theory Mathematics
 Excellent (but brief) summaries of the mathematical topics
that are essential in string theory, from the undergraduate level
to advanced research. Part of the
Official String Theory Web Site.

Annotated Bibliography for SeibergWitten Theory
 Good bibliography of books and papers, by Jonathan Poritz.

CalabiYau Home Page
 Resource for information about CalabiYau manifolds.

Mathematical Physics Electronic Journal
 Free, refereed online journal.
Surveys, overviews, tutorials

Mathematical physics
 Article from
Wikipedia.
See also
Gauge theory,
Gauge field theory

YangMills Theory
 Brief description of the problem at the
Clay Mathematics Institute site, by Arthur Jaffe and Edward
Witten.
(A more complete description is available as a PDF file.)

Twists and Supersymmetry
 Slide presentation given by Arthur Jaffe at the
2001: A Spacetime Odyssey conference.

Preparation for Gauge Theory
 "Class lecture notes at a beginning graduate level on the
mathematical background needed to understand classical gauge theory."
By George Svetlichny.
 Geometry of Solitons
ChuuLian Terng; Karen Uhlenbeck
Notices of the AMS, January 2000, pp. 1725
 Solitons are solitary wave solutions of certain nonlinear partial
differential equations, like the KortewegdeVries and sineGordon
equations. Although solutions of nonlinear equations can't be added to
produce additional solutions, sometimes solutions will be asymptotic
to linear combinations of solutions.
[Article in PDF format]
 Magic, Mystery, and Matrix
Edward Witten
Notices of the AMS, October 1998, pp. 11241129
 A fairly nontechnical survey article on Mtheory
by the person responsible for the "second superstring revolution".
The main historical developments and concepts are outlined, including
quantum field theory, quantum gravity, string theory, and Mtheory.
[Article in PDF format]
 The Symmetries of Solitons
Richard S. Palais
Bulletin of the AMS, October 1997, pp. 339403
 The history of the investigation of solitons is presented in
this long technical survey article. In particular, the evolution of
equations of solitons have many conserved quantities, which suggests
the existence of unobvious symmetries in their solutions.
[Abstract, references, downloadable text]
 Solitary Waves
Russell Herman
American Scientist, JulyAugust 1992, pp. 350361
 Solitary waves, or solitons, were first observed in 1834 and
long thought rare. However, they have turned out to be common in
nature. The absence of dispersion in solitons is a result of
nonlinearities in the differential equations that govern them.
 Fiber Bundles and Quantum Theory
Herbert J. Bernstein; Anthony V. Phillips
Scientific American, July 1981, pp. 123137
 Fiber bundles are a mathematical construct that can be
applied in the differential geometry of manifolds to generalize
the notion of curvature. In physics they may be applied to model
the spin rotation of neutrons in a magnetic field.
 Solitons
Claudio Rebbi
Scientific American, February 1979, pp. 92116
 Solitons are wavelike solutions of certain differential
equations, which propagate but do not dissipate or disperse.
They may represent predicted phenomena such as magnetic monopoles
in elementary particle theory. When a soliton solution is
limited in temporal as well as spatial extent it is called an
instanton and can represent quantum tunneling.
 M. Waldschmidt; P. Moussa; J.M. LUck; C. Itzykson, eds.  From
Number Theory to Physics
 Normally we wouldn't list this sort of book here, as it's
at a very advanced level (and expensive). It is a collection
of expository papers intended for professionals in mathematics
and physics. But its subject matter includes many topics which
indicate the striking applicability of some very "deep" classical
mathematical ideas to modern physics  such as zeta functions,
abelian varieties, elliptic curves, modular forms, algebraic
number theory, and padic numbers.
 Michael Monastyrsky  Riemann, Topology, and Physics
Birkhäuser, 1987
 This is really two (short) books in one: a biography and
overview of the work of G. F. B. Riemann, one of the outstanding
mathematicians of the 19th century, and a monograph on
"topological themes in contemporary physics". The second part,
as the mathematician/physicist Freeman Dyson says in the Foreword,
illustrates "one of the central themes of science, the mysterious
power of mathematical concepts to prepare the ground for physical
discoveries which could not have been foreseen or even imagined
by the mathematicians who gave the concepts birth."
 Yu. I. Manin  Mathematics and Physics
Birkhäuser, 1981
 The author offers an extended essay (but not a long book)
on the relation between mathematics and physics. Some of the
concepts involved are advanced ones, but the reader isn't asked
to follow detailed proofs. In the author's own words, "This
book describes how mathematics associates to some important
physical abstractions (models) its own mental constructions,
which are far removed from the direct impressions of experience
and physical experiment."
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