GEOMETRY, TOPOLOGY & PHYSICS

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.

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.

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.

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.

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.

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.

http://www.diffiety.ac.ru

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