December 23, 2004: Introducing co-insertion
and co-Lie derivative operators.
November 25, 2004: Proving exactness of the Spencer sequence.
November 18, 2004: Elementary facts of Rational Mechanics,
Lagrangian and Hamiltonian formalism, constrain submanifolds, elastic and
non-elastic shocks. Simplectic structure on T^{*}(M), conceptual
definition of the universal form, even on a generic vector bundle.
De Rham homology in T^{*}(M). Fundamental theorem of Mechanics.
Hamilton-Jacobi equations.
November 11, 2004: First properties of the horizontalization.
"Totalization" of vector fields and linear differential operators.
Defining the horizontalization operator on jets of smooth functions of
J^{k}(π). Properties and alternative definitions
of the functor D_{k}.
November 4, 2004: Understanding nonlinear differential
operators in the framework of jet spaces of submanifolds. Introducing the "smaps", i.e.
the morphisms in the category of Diffietis. Conceptual definition of the
horizontalization operator of differential forms on jet spaces and its
coordinate expression. Definition of the horizontal differential d_{0}.
October 29, 2004: Definitions and basic properties of
the Diff-Spencer sequence and its dual, the Jet-Spencer sequence. Introducing the Spencer
cohomologies associated with a linear differential operator. Higher
symmetries of PDEs are Spencer cohomologies associated with the
universal linearization operator l_{F}.
October 25, 2004: Symmetries of graded structures with respect
to left and right notations. Functors Df_{k} and their representative objects.
October 14, 2004: Introduction to the spectral
sequences arising in
J^{∞}(π).
Introducing the functor Df_{2} and its representative
object P_{2}.
April 30, 2004:
Introducing into the higher analogs of the de Rham complex.
April 15, 2004:
Conceptual definition of the insertion of a derivation into a form.
Natural differential operators between representable functors. Why is
d^{2}=0? Defining functor D_{n}.
April 8, 2004:
Isomorphism between Diff_{k}^{+}(P,Q) and Hom_{A}(P,Diff_{k}^{+}Q), properties of homomorphisms between modules and bimodules.
March 26, 2004:
Construction of the jet algebras and the jet modules. Multiplication
on the jet modules and the corresponding natural transformation of functors. Isomorphism between the jet module of P and the tensor product of the tensor algebra and P. Conceptual definition of the de Rham differential from 1-forms to 2-forms. Derivations of graded algebras. Linear algebra (linear maps, invariant subspaces, splitting operators, diagonalization).
March 19, 2004:
Representative object of D_{2}. Natural transformations
between representable functors.
Inclusion of D_{2} into D^{2} yields the skew multiplication of 1-forms. Jet
algebras and jet modules. Spencer's diff-sequences and jet-sequences.
March 12, 2004:
Multiple module structures of differential operators with values in a bimodule.
Defining functor D_{2}, composition operator c_{k,s}. Defining functor D_{k}, and the "evaluation on 1" operator _{k}.
March 5, 2004:
Introduction to the fundamental functors of differential calculus. Construction of the
representative object for the functor D. Representative object of D in the
category of geometric modules over a smooth algebra. Introducing the differential calculus over the Grassmann algebra.
January 23, 2004:
Surgical property of the cohomology. Bundles of pairs
manifold-submanifold, bundles of n-tuples of manifolds (flags), Thom's isomorphism. Tubular
neighbourhoods and intersection index of two submanifolds. Fixed point theorem, Hopf's
formula, Euler-Poincare's characteristic.
January 23, 2004:
Lattice structure within the first term of the spectral
sequence associated with the CW-complex of a smooth maifold. De Rham cohomologies with arbitrary coefficients. The degree of a map between connected and oriented manifolds is an integer number. Computing the differential d_{1} in a CW-complex.
January 9, 2004:
Constructing the first term of the spectral sequence
associated with a differential CW-complex by means of deformation retractions. Applications: computing cohomology spaces of compact surfaces.
December 30, 2003:
Historical facts about the Spectral Sequence: Chern
classes, homotopy groups, Pontriagin classes, Serre's bundles. Hints for further development of the spectral sequence: pseudo-bundles (maps that are not bundles themselves but give rise to cohomological bundles), symplectic manifolds, conctact structures.
December 19, 2003:
Introduction to CW-complexes: k-cells and k-skeletons,
spectral sequence associated with a CW-complex, differential d_{1}, term E_{2} surviving to infinity. Cellular decomposition of the Grassmannian manifold.
November 28, 2003:
Bundle of the orientations (principal bundle of Z_{2}). Defining integrals on
non-orientable manifolds: cohomological bundle associated with the bundle
of tangent n-vectors. Linear algebra (theorem of the sheaf).
November 21, 2003:
Theorems on the dimension of H_{c}^{n}(M), for M connected
smooth n-dimensional manifold.
November 14, 2003:
De Rham complex with compact support, theorem of suspension.
November 7, 2003:
Homotopy formula, relative cohomological bundle, theorem of Rene Thom.
Linear algebra (linear functionals, continuous functionals).
October 31, 2003:
Introduction to differential topology (double
coverings, differential
forms with compact support, degree of a map, intersection index, theory of
residuals, theorem of Newton-Leibnitz, Stokes' formula). Linear algebra (linear
dependency, generators).