Zoran Skoda
hom10connDmod

One idea what to teach in academic year 2011-2012 is

The course, if decided, would be recommended to students with interests in one or more among the following areas: differential geometry, algebraic geometry, representation theory, harmonic analysis (esp. invariant differential operators, oscillating integrals, microlocal analysis), homotopy theory, mathematical physics (gauge theories, integrable systems, classical mechanics, gravity, string theory), differential equations (ODE, PDE, ψ\psiDO, index theory, microlocal analysis), noncommutative geometry.

Key words: Weyl algebra, regular differential operator, fibre bundle, sheaf, connection, curvature, Maurer-Cartan equation, Kahler differential, ring of dual numbers, crystal of quasicoherent modules, crystal of schemes, Grothendieck connection, D-module, D-scheme, jet space, connection, holonomy, monodromy, parallel transport, fundamental group, local system, fiber functor, classifying space, universal bundle, characteristic class, Chern-Weil theory, de Rham space, Chen iterated integral, formal connection, holonomic D-module, Bernstein-Beilinson localization theorem.

Description

Connections are usually taught in differential geometry, although they exist in various versions in many other setups, including in algebraic geometry, analytic geometry, (non)commutative algebra and so on. Connections supply a rule of a parallel transport of vectors in a vector bundle along smooth curves in the base manifold. They are often given by infinitesimal quantities like covariant derivative and the field of horizontal planes. Theory of connections on fibre bundles is one of the most basic and important notions in modern geometry. For example a Riemannian manifold comes with a canonical connection, called the Levi-Civita connection. This connection defines the equations of geodesics, curvature and so on. Various important tensors of Einstein general relativity theory are defined with help of various derivatives of the connection form. Another example of a connection is basic constituent of the Yang-Mills theories in physics. Operator of parallel transport along a given curve is called the holonomy. All possible holonomies at a point form a subgroup of the automorphism group of the fiber over that point, called the holonomy subgroup. If it has a smaller dimension than the whole automorphism group we talk about special holonomy. Manifolds with special holonomy are important in differential geometry.

In terms of curvature, one defines very important functorial choice of classes attached to manifolds or bundles over manifolds, so called characteristic classes. This method of defining characteristic classes is called Chern-Weil theory.

Very interesting condition is when the curvature of a connection vanishes, usually expressed as so-called Maurer-Cartan equation. Then, we say that the connection is flat. Integrability condition in the theory of Hamiltonian systems is a special case. Vector bundle with a flat connection is called a local system. Every local system provides a representation of the fundamental group. In particular we assign endomorphisms of a fiber over the base point to any based loop: the monodromy. Monodromic behaviour is very important not only in the theory of integrable systems, but also in general theory of ordinary differential equations. Other way around, it is interesting to study (and sometimes even define) the fundamental group as the group of automorphism of a fiber in universal covering space; these automorphisms are precisly the monodromies for the corresponding local system. This leads to Grothendieck approach to Galois theory and Tannakian formalism.

On the other hand, a vector bundle with a connection determines a module over a sheaf of regular differential operators, so-called D-module. The simplest case is the case of the Weyl algebra of regular differential operators on the affine n-space over a field. This algebra is also of importance in quantum mechanics and the general study of quantization.

In a way, D-modules are more general than vector bundles with flat connection (local systems); D-modules coming from vector bundles are rather special. D-modules can be defined again in many setups, including D-modules on algebraic varieties and schemes (“algebraic D-modules”), on complex analytic varieties, on rigid analytic varieties, on differentiable manifolds, in synthetic differential geometry, on ind-schemes and so on. More geometric “nonlinear” version, D-schemes, can be also defined.

D-modules can be viewed as spaces of solutions of a linear differential operator. D-schemes are similarly related to nonlinear differential operators. This geometric language can be combined with symplectic geometry to give a proper treatment in the theory of pseudodifferential operators and microlocal analysis as well.

Universal enveloping algebra of the tangent Lie algebra of a Lie group GG is the algebra of left invariant differential operators on GG. Hence it is not surprising that differential operators can be usefully used to define many interesting representations of Lie groups. There is a fine statement, called Beilinson-Bernstein localization theorem relating the algebraic D-modules and representations. Sometimes in representation theory, a similar object is used instead of D-modules, so called perverse sheaves; the reason is that some deep geometric results are known for perverse sheaves without strong analogues in D-module theory (BBDG decomposition theorem).

Geometric version of the theory of vertex algebras, so called chiral algebras are formulated (by Alexander Beilinson and Vladimir Drinfel’d) also in terms of D-module theory.

In positive characteristics, regular differential operators are studied using special devices like divided powers and Frobenius automorphism. A convenient language is the language of so called crystals which are in a way dual to D-modules. For conceptual reasons like relationship with descent theory, infinitesimals and so on, this is a good approach to compare and understand better the D-module theory as well.

Literature

  • S. C. Coutinho, A primer of algebraic D-modules, London Math. Soc. Stud. Texts 33, Cambridge University Press, 1995. xii+207 pp.

  • D-modules: Joseph Bernstein‘s notes ps; P. Schneiders’ notes; D. Miličić‘s notes D-modules, Localization and representation theory of reductive Lie groups; Victor Ginzburg’s 1998 Chicago notes pdf; A. Braverman-T. Chmutova, Lectures on algebraic D-modules, pdf; R. Bezrukavnikov, MIT course notes on D-modules, pdf (handwritten)

  • Donu Arapura?, Notes on D-modules and connection with Hodge theory, pdf

  • Lurie’s notes on crystals, exposition in Gaitsgory’s seminar pdf

  • D. Gieseker: Flat vector bundles and the fundamental group in non-zero characteristics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 1, 1–31, numdam

  • A. Beĭlinson, J. Bernstein, A proof of Jantzen’s conjectures, I. M. Gelʹfand Seminar, 1–50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc. 1993, pdf MR95a:22022

  • R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory, Progress in Mathematics 236, Birkhäuser

  • A. Borel et al., Algebraic D-modules, Perspectives in Mathematics, Academic Press, 1987.

  • P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ.P. 1978. vi+243, ISBN0-691-08218-9

  • Masaki Kashiwara?, D-modules and microlocal calculus, gBooks

  • Secret Blogging Seminar Musings on D-modules, Musings on D-modules, part 2

  • The Everything Seminar D-module Basics I, D-Module Basics II.

  • P. Maisonobe, C. Sabbah, D-modules cohérents et holonomes, Travaux en cours, Hermann, Paris 1993. (collection of lecture notes)

  • R. Bott, L. W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer 1982. xiv+331 pp.

  • V. Guillemin, S. Sternberg, Supersymmetry and equivariant de Rham theory, Springer 1999.

  • Ieke Moerdijk, Gonzalo Reyes, Models for Smooth Infinitesimal Analysis

  • V. Guillemin, S. Sternberg, Geometric asymptotics, AMS 1977, online

  • J. J. Duistermaat, Fourier integral operators, Progress in Mathematics 130, Birkhäuser, Boston 1996. x+142 pp.

  • David Ben-Zvi, David Nadler, Loop spaces and representations, arxiv/1004.5120, Loop spaces and connections, arxiv/1002.3636, The character theory of a complex group, arxiv/0904.1247, Perverse bundles and Calogero-Moser spaces, math.AG/0610097

  • Zoghman Mebkhout, Le formalisme des six opérations de Grothendieck pour les 𝒟 X\mathcal{D}_X-modules cohérents, With supplementary material by the author and L. Narváez Macarro. Travaux en Cours 35, Hermann, Paris, 1989. x+254 pp.

Last revised on February 24, 2011 at 16:07:56. See the history of this page for a list of all contributions to it.