XIth OPORTO MEETING on
GEOMETRY, TOPOLOGY & PHYSICS
From July 12th to July 15th, 2002
COURSES (Detailed)
I. ANDERSON
Group Invariant Solutions and Symmetric Criticality
Lie's method of symmetry reduction for
finding the group invariant solutions to partial
differential equations is widely recognized as one of the most general and effective methods for
obtaining exact solutions of non-linear partial differential equations. In recent years Lie's
method has been described in a number of excellent texts and survey articles (see, for
example, Bluman and Kumei [3], Olver [5], Ovsiannikov [6], Stephani [9]) and has been
systematically applied to differential equations arising in a broad spectrum of disciplines
(see,for example, Ibragimov [4] or Rogers and Shadwick [8]). It is therefore quite surprising
that Lie's method as it is conventionally described, does not provide an appropriate
theoretical framework for the derivation of such celebrated invariant solutions as the
Schwarzschild solution of the vacuum Einstein equations, the instanton and monopole solutions in
Yang-Mills theory or the Veronese map for the harmonic map equations.
In my first lecture I
shall describe the elementary steps needed to correct this deficiency in the classical Lie
method, and to give a precise formulation of the reduced differential equations for the group
invariant solutions which arise from this generalization.
In my second and third lectures I
shall discuss some recent efforts to systematically study of the interplay between the formal
geometric properties of a system of differential equations, such as the conservation
laws, symmetries, Hamiltonian structures, variational principles, local solvability, formal
integrability and so on, and those same properties of the reduced equations for the group
invariant solutions.
Two problems merit special attention. First, one can interpret the principle
of symmetric criticality (Palais [7], [8]) as the problem of determining those group actions
for which the reduced equations of a system of Euler-Lagrange equations are derivable from a
canonically defined Lagrangian. The obstructions to the validity of the symmetric criticality
principle will be described within the framework of group actions on jet spaces.
Secondly,there
do not appear to be any general theorems in the literature which insure the local existence of
group invariant solutions to differential equations; as one step in this direction, it is
possible to give simple conditions under which a system of differential equations of
Cauchy-Kovalevskaya type remain of Cauchy-Kovalevskaya type under reduction.
If time permits, I
shall briefly review other group theoretic techniques for the reduction of differential
equations.
References
- I.M.Anderson and M..Fels, ``Symmetry Reduction of Variational Bicomplexes
and the principle of symmetry criticality", Amer.J.Math.112 (1997),609 - 670.
- I.M.Anderson, M..Fels and Charles Torre, ``Group Invariant Solution without
Transversality", Amer.J.Math. 212 (2000),653-686.
- G.W.Blumanand and S.Kumei, ``Symmetries and Differential Equations", Applied Mathematical
Sciences, 81, Springer-Verlag, New York-Berlin, 1989.
- N.H.Ibragimov, ``CRC Handbook of Lie Group Analysis of Differential Equations", Volume
1, Symmetries, Exact Solutions and Conservation Laws., CRC Press,Boca Raton,Florida,1995.
- P.J.Olver, ``Applications of Lie Groups to Differential Equations", (Second
Ed.), Springer, New York,1986.
- L.V.Ovsiannikov, ``Group Analysis of Differential Equations",
Academic Press, New York,1982.
- R.S.Palais, ``The principle of symmetric criticality",
Comm. Math. Phys. 69 (1979), 19-30.
- R.S.Palais, ``Applications of the symmetric criticality
principle in mathematical physics and differential geometry", Proc.U.S.-China Symp. on
Differential Geometry and Differential Equations II,1985.
- C.Rogers and W.Shadwick, ``Nonlinear
boundary value problems in science and engineering", Mathematicsin Science and
Enginering, vol.183, Academic Press, Boston,1989.
- H.Stephani, ``Differential Equations and their
Solutions using symmetries", Cambridge University Press, Cambridge,1989.
Ph.A. GRIFFITHS
Abel's Differential Equations
This year marks the bicentenary of the birth of Niels Abel. In hindsight,
one may now see
that the general form of Abel's differential equations for the rational
motion of configurations
of points on an algebraic curve was in some ways the decisive event in the
development of
the theory of algebraic curves in the nineteenth century. The extension of
Abel's differential
equations to configurations of points on higher dimensional algebraic
varieties is one of the
central problems in modern algebraic geometry. For my short course, I
propose to give
three talks centered around Abel's differential equations.
The first talk will be
concerned with the legacy of Abel in algebraic geometry. This will be from a
historical perspective, looking at how Abel was led to his theorem in the
first place, and then discussing what some of its repercussions have been in
modern mathematics. This will lead naturally into the introduction of Abel's
differential equations in higher dimensions and once again the entrance of
arithmetic considerations into geometry.
The second talk
will look at the subject from the point of view of classical complex
analysis. This talk will be "elementary," showing that if one approaches
certain very natural geometric questions naively, one encounters a
post-modern, algebraic/number-theoretic object.
The third talk will be on Abel's
differential equations per se, giving a discussion of what these
differential equations are and how one integrates them modulo a central
conjecture in arithmetic algebraic geometry.
N. KAMRAN
The Dirac Equation in Kerr Geometry
According to a celebrated uniqueness theorem for the Einstein
field equations, the exterior gravitational and electromagnetic fields of
a charged rotating black hole in equilibrium are described by one of the
exact solutions belonging to the three-parameter family of non-extreme
Kerr-Newman geometries. The rigorous analysis of the long-term behavior
of external fields in this geometry is an interesting challenge for
mathematicians working in General Relativity, since it is one of the keys
to the understanding of the formation of event horizons and singularities in non spherically symmetric gravitational collapse. The long-term dynamics
of the solutions of the Dirac equation for a massive spin 1/2 fermion
field in Kerr-Newman geometry is now fairly well understood. This is
notably to the remarkable global properties of the background
metric, which ensure that the Dirac operator admits a first-order
generalized symmetry of a very special type, arising from the complete
integrability of the geodesic flow. In particular, we will show that the
probability of locating a fermion in any compact spatial region tends to
zero as t tends to infinity, and we will prove that the pointwise rate of
decay is in t^{-5/6} for sufficiently generic Cauchy data, thus slower
than the rate of t^{-3/2} which holds in Minkowski space. These results
have been obtained in collaboration with Felix Finster, Joel Smoller and
Shing-Tung Yau.
References:
- Finster, Felix; Kamran, Niky; Smoller, Joel; Yau, Shing-Tung; Nonexistence
of time-periodic solutions of the Dirac equation in an axisymmetric black
hole geometry, Comm. Pure Appl. Math. 53 (2000), no. 7, 902--929.
gr-qc/9905047.
- Finster, Felix; Kamran, Niky; Smoller, Joel; Yau, Shing-Tung; Decay rates
and probability estimates for Dirac particles in the Kerr-Newman black
hole geometry, Comm. Math. Phys. (2002), to appear, 54 pages.
gr-qc/0107094.
- Finster, Felix; Kamran, Niky; Smoller, Joel; Yau, Shing-Tung; The long
term dynamics of Dirac particles in the Kerr-Newman black hole geometry,
preprint.
gr-qc/0005088.
J. KRASIL'SHCHIK
Cohomological Theory of Recursion Operators
Lecture 1.
Cohomological Theory of Recursion Operators
A notion of algebra with flat connection is introduced. For such algebras,
a cohomological theory based on the Froelicher-Nijenhuis bracket
is constructed. The theory is applicable both to classical commutative
algebras
and to graded commutative algebras. The latter is also defined in a purely
algebraic way.
Applied to infinitely prolonged equations, this theory, in particular,
provides methods
for computation of recursion operators and is closely related to the
Vinogradov
C-spectral sequence and horizontal cohomology with coefficients
introduced
by A. Verbovetsky.
Lecture 2.
Coverings and Computation of Recursion Operators
For nonlocal extensions of nonlinear PDE introduced by means of the
theory of coverings, computational formulas for recursion operators are
deduced (in the form of overdetermined systems of linear differential
equations). Solving these equations gives a test for "weak" integrability
of the initial PDE. Several examples are considered and relations to
Hamiltonian structures are briefly discussed.
References:
- Symmetries of Differential Equations in Mathematical Physics and Natural
Sciences
(a monograph by A. V. Bocharov, S. V. Duzhin, et al. edited by A. M.
Vinogradov
and I. S. Krasil'shchik). Factorial Publ. House, 1997 (in Russian). English
translation
in the AMS Monograph Series, 1999.
-
Krasil'shchik, I.S., Algebras with flat connections and symmetries of
differential
equations, in Lie Groups and Lie Algebras: Their Representations,
Generalizations
and Applications, Kluwer Acad. Publ., Dordrecht / Boston / London, 1998,
pp. 407-424.
- Krasil'shchik, I.S., Some new cohomological invariants of nonlinear
differential
equations, Differential Geometry and Its Appl. 2 (1992) no. 4.
- Krasil'shchik, I.S. and Verbovetsky A.M., Homological methods in
equations
of mathematical physics, arXive:
math.DG/9808130
- Krasil'shchik, I.S., and Kersten P.H.M., Symmetries and Recursion
Operators for Classical and Supersymmetric Differential Equations,
Kluwer Acad. Publ., Dordrecht, 2000.
A. VINOGRADOV
A Panorama of Secondary Calculus
Infinitely prolonged differential equations supplied with some natural
geometrical structures are the simplest examples of diffieties. In the
theory of PDE's they play the same role as algebraic varieties in the
theory of algebraic equations. Informally, Secondary Calculus may be
viewed as Primary (="usual") Calculus respecting underlying geometrical
structures on diffieties. From the perspective of Secondary Calculus the
standard "differential" mathematics appears to be its 0-dimensional case.
In particular, each standard concept of usual Calculus has one or more
secondary analogues. For instance, secondary vector fields are nothing
but higher symmetries of PDE's, secondary functions are horizontal de
Rham cohomologies, secondary differential forms coincide with the first
term of the C-spectral sequence, etc. From one point of view Secondary
Calculus can be seen as a general theory of (nonlinear) PDE's, and from
another as a natural mathematical background of Quantum Field Theory and
its generalizations. Objects of Secondary Calculus are natural
differential complexes "growing" on diffieties and their morphisms are
homotopy classes of differential chain maps connecting them. It seems
plausible that this is a mathematical paraphrase of the "quantum
behaviour" in physics.
In the course we will start from a formalization of the observability
mechanism in classical physics that leads to Primary Calculus
(=Differential Calculus over commutative algebras). Then it will be
shown how a mathematical version of the Bohr Correspondence Principle
leads in its turn to Secondary Calculus. In the second part of the
course some basic results, recent developments and perspectives will be
discussed. Special attention will be given to the Secondarization
Problem, a mathematical analog of the Quantization Problem.
References:
- A.M.Vinogradov, Cohomological Analysis of Partial Differential
Equations and Secondary Calculus; AMS "Translation of
Mathematical Monographs" series, vol. 204, Providence, Rhode
Island, 2001.
- A.M.Vinogradov, Introduction to Secondary Calculus, Contemporary
Mathematics 219 (1998), pp 241-272, Amer.Math.Soc., Providence, Rhode
Island.
- Krasil'shchik I. S., A.M.Verbovetski, Homological Methods in
Equations of Mathematical Physics, Advanced Texts in Mathematics, Open
Education & Sciences, !998.
- A.M.Vinogradov, From symmetries of partial differential equations
towards secondary ("quantized") calculus, J. Geom. and Phys., 14 (1994),
146-194.
- Bocharov A.B., Duzhin S.V., et al, (Krasil'shchik I. S.,
A.M.Vinogradov, ed.), Symmetries and conservation laws of Differential
Equations in Mathematical Physics, Factorial Publ. House, Moscow, 1997;
English translation in AMS "Translation of Mathematical Monographs"
series, vol. 182, Providence, Rhode Island,1999.
- J.Nestruev, Smooth manifolds and observables (Russian), Izd-vo
M.C.N.M.O., Moscow (Russian); English translation to appear in Springer
GTM series.
- Alekseevski D. V., Lychagin V. V., A.M.Vinogradov, Basic ideas and
concepts of differential geometry, Encyclopedia of Math. Sciences, 28
(1991), Springer-Verlag, Berlin.
- Krasil'shchik I. S., Lychagin V. V., A.M.Vinogradov, Geometry of Jet
Spaces and Nonlinear Differential Equations, Advanced Studies in
Contemporary Mathematics, 1 (1986), Gordon and Breach, New York, London.
xx+441 pp.
Material on the web can be found on the site of the Diffiety Institute:
http://www.diffiety.ac.ru
A. VERBOVETSKY
Antifield Formalism and the Secondary Calculus
These two talks will survey the horizontal cohomology theory of
differential equations and its relation to antifield, antibracket
machinery for Lagrangian field theory. Two approaches to computing
the horizontal cohomology, one based on the compatibility complex, and
another based on the Koszul-Tate resolution, will be reviewed. We
will look at the Hamiltonian formalism, antibracket (= functional
Schouten bracket), functional Poisson bracket,
Tyutin-Voronov-Shahverdiyev operators in the context of the geometry
of differential equation.
References:
- J. Krasil'shchik and A. Verbovetsky, Homological methods in
equations of mathematical physics, Advanced Texts in Mathematics, Open
Education & Sciences, Opava, 1998, arXiv:
math.DG/9808130
- A. Verbovetsky, Notes on the horizontal cohomology, in Secondary
Calculus and Cohomological Physics (M. Henneaux, I. S. Krasil'shchik,
and A. M. Vinogradov, eds.), vol. 219 of Contemporary Mathematics,
Amer. Math. Soc., 1998, arXiv:
math.DG/9803115
- A. Verbovetsky, Remarks on two approaches to the horizontal
cohomology: compatibility complex and the Koszul-Tate resolution, Acta
Appl. Math. 72 (2002), 123-131, arXiv:
math.DG/0105207