XIth Oporto meeting on geometry, topology & physics
From July 12th to July 15th, 2002, Oporto, Portugal

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:

  1. Symmetries and Conservation Laws for Differential Equations of Mathematical Physics (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 AMS "Translation of Mathematical Monographs" series, vol. 182, Providence, Rhode Island,1999.
  2. I. S. Krasil'shchik, 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.
  3. I. S. Krasil'shchik, Some new cohomological invariants of nonlinear differential equations, Differential Geometry and Its Appl. 2 (1992) no. 4.
  4. I. S. Krasil'shchik and A. M. Verbovetsky, Homological methods in equations of mathematical physics, arXive: math.DG/9808130
  5. I. S. Krasil'shchik, and P. H. M. Kersten, 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:

  1. A. M. Vinogradov, Cohomological Analysis of Partial Differential Equations and Secondary Calculus; AMS "Translation of Mathematical Monographs" series, vol. 204, Providence, Rhode Island, 2001.
  2. A. M. Vinogradov, Introduction to Secondary Calculus, Contemporary Mathematics 219 (1998), pp 241-272, Amer.Math.Soc., Providence, Rhode Island.
  3. I. S. Krasil'shchik , A. M. Verbovetski, Homological Methods in Equations of Mathematical Physics, Advanced Texts in Mathematics, Open Education & Sciences, 1998.
  4. A. M. Vinogradov, From symmetries of partial differential equations towards secondary ("quantized") calculus, J. Geom. and Phys., 14 (1994), 146-194.
  5. A. B. Bocharov, S. V. Duzhin, et al, ( I. S. Krasil'shchik, 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.
  6. J. Nestruev, Smooth manifolds and observables (in Russian), MCCME, Moscow; English translation to appear in Springer GTM series.
  7. D. V. Alekseevski, V. V. Lychagin, A. M. Vinogradov, Basic ideas and concepts of differential geometry, Encyclopedia of Math. Sciences, 28 (1991), Springer-Verlag, Berlin.
  8. I. S. Krasil'shchik, V. V. Lychagin, 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.

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:

  1. J. Krasil'shchik and A. Verbovetsky, Homological methods in equations of mathematical physics, Advanced Texts in Mathematics, Open Education & Sciences, Opava, 1998, Diffiety Inst. Preprint Series: DIPS 7/98, arXiv: math.DG/9808130
  2. 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
  3. A. Verbovetsky, Remarks on two approaches to the horizontal cohomology: compatibility complex and the Koszul-Tate resolution, in Symmetries of Differential Equations and Related Topics edited by I. S. Krasil'shchik, Acta Applicandae Mathematicae, Volume 72, Issue 1-2, June 2002, p. 123-131, arXiv: math.DG/0105207