i. review of fundamental notions of analysis
a. set theory， definitions
1. sets
2. mappings
3. relations
4. orderings
b. algebraic structures， definitions
1. groups
2. rings
3. modules
4. algebras
5. linear spaces
c. topology
1. definitions
2. separation
3. base
4. convergence
5. covering and compactness
6. connectedness
7. continuous mappings
8. multiple connectedness
9. associated topologies
10. topology related to other structures
11. metric spaces
metric spaces
cauchy sequence; completeness
12. banach spaces
normed vector spaces
banach spaces
strong and weak topology; compactedness
13. hilbert spaces
d. integration
1. introduction
2. measures
3. measure spaces
4. measurable functions
5. lntegrable functions
6. integration on locally compact spaces
7. signed and complex measures
8. integration of vector valued functions
9. l1 space
10. l1 space
e. key theorems in linear functional analysis
1. bounded linear operators
2. compact operators
3. open mapping and closed graph theorems
problems and exercises
problem 1： clifford algebra; spin（4）
exercise 2： product topology
problem 3： strong and weak topologies in l2
exercise 4： htlder spaces
see problem vi 4： application to the schrtdinger equation
ii. differential calculus on banach spaces
a. foundations
1. definitions. taylor expansion
2. theorems
3. diffeomorphisms
4. the euler equation
5. the mean value theorem
6. higher order differentials
b. calculus of variations
1. necessary conditions for minima
2. sufficient conditions
3. lagrangian problems
c. implicit function theorem. inverse function theorem
1. contracting mapping theorems
2. inverse function theorem
3. implicit function theorem
4. global theorems
d. differential equations
1. first order differential equation
2. existence and uniqueness theorems for the lipschitzian case
problems and exercises
problem 1： banach spaces， first variation， linearized equation
problem 2： taylor expansion of the action; jacobi fields; the feynman-green function; the van vleck matrix; conjugate points; caustics
problem 3： euler-lagrange equation; the small disturbance equation; the soap bubble problem; jacobi fields
iii. differentiable manifolds， finite dimensional case
a. definitions
1. differentiable manifolds
2. diffeomorphisms
3. lie groups
b. vector fields; tensor fields
1. tangent vector space at a point
tangent vector as a derivation
tangent vector defined by transformation properties
tangent vector as an equivalence class of curves
images under differentiable mappings
2. fibre bundles
definition
bundle morphisms
tangent bundle
frame bundle
principal fibre bundle
3. vector fields
vector fields
moving frames
images under cliffeomorphisms
4. covariant vectors; cotangent bundles
dual of the tangent space
space of differentials
cotangent bundle
reciprocal images
5. tensors at a point
tensors at a point
tensor algebra
6. tensor bundles; tensor fields
c. groups of transformations
i. vector fields as generators of transformation groups
2. lie derivatives
3. invariant tensor fields
d. lie groups
1. definitions; notations
2. left and right translations; lie algebra; structure constants
3. one-parameter subgroups
4. exponential mapping; taylor expansion; canonical coordinates
5. lie groups of transformations; realization
6. adjoint representation
7. canonical form， maurer--cartan form
problems and exercises
problem 1： change of coordinates on a fiber bundle， configuration space， phase space
problem 2： lie algebras of lie groups
problem 3： the strain tensor
problem 4： exponential map; taylor expansion; adjoint map; left and right differentials; haar measure
problem 5： the group manifolds of soo） and su（2）
problem 6： the 2-sphere
iv. integration on manifolds
a. exterior differential forms
1. exterior algebra
exterior product
local coordinates; strict components
change of basis
2. exterior differentiation
3. reciprocal image of a form （pull back）
4. derivations and antiderivations
definitions
interior product
5. forms defined on a lie group
invariant forms
maurer--cartan structure equations
6. vector valued differential forms
b. integration
1. integration
orientation
odd forms
integration of n-forms in r"
partitions of unity
properties of integrals
2. stokes theorem
p-chains
integrals of p-forms on p-chains
boundaries
mappings of chains
proof of stokes theorem
3. global properties
homology and cohomology
o-forms and o-chains
betti numbers
poincar6 lemmas
de rham and poincare duality theorems
c exterior differential systems
1. exterior equations
2. single exterior equation
3. systems of exterior equations
ideal generated by a system of exterior equations
algebraic equivalence
solutions
examples
4. exterior differential equations
integral manifolds
associated pfaff systems
generic points
closure
5. mappings of manifolds
introduction
immersion
embedding
submersion
6. pfaff systems
complete integrability
frobenius theorem
integrability criterion
examples
dual form of the frobenius theorem
7. characteristic system
characteristic manifold
example： first order partial differential equations
complete integrability
construction of integral manifolds
cauchy problem
examples
8. invariants
invariant with respect to a pfaff system
integral invariants
9. example： integral invariants of classical dynamics
liouville theorem
canonical transformations
10. symplectic structures and hamiltonian systems
problems and exercises
problem 1： compound matrices
problem 2：poincar6 lemma， maxwell equations， wormholes
problem 3： integral manifolds
problem 4： first order partial differential equations， hamilton-jacobi
equations， lagrangian manifolds
problem 5： first order partial differential equations， catastrophes
problem 6： darboux theorem
problem 7： time dependent hamiltonians
see problem vi 11 paragraph c： electromagnetic shock waves
v. riemannian manifolds. kahlerian manifolds
a. the riemannian structure
1. preliminaries
metric tensor
hyperbolic manifold
2. geometry of submanifolds， induced metric
3. existence of a riemannian structure
proper structure
hyperbolic structure
euler-poincare characteristic
4. volume element. the star operator
volume element
star operator
5. isometries
b. linear connections
1. linear connections
covariant derivative
connection forms
parallel translation
affine geodesic
torsion and curvature
2. riemannian connection
definitions
locally flat manifolds
3. second fundamental form
4. differential operators
exterior derivative
operator
divergence
laplacian
c. geodesics
1. arc length
2. variations
euler equations
energy integral
3. exponential mapping
definition
normal coordinates
4. geodesics on a proper riemannian manifold
properties
geodesic completeness
5. geodesics on a hyperbolic manifold
d. almost complex and kahlerian manifolds
problems and exercises
problem 1 maxwell equation; gravitational radiation
problem 2： the schwarzschild solution
problem 3： geodetic motion; equation of geodetic deviation; exponentiation; conjugate points
problem 4： causal structures; conformal spaces; weyl tensor
vbis. connections on a principal fibre bundle
a. connections on a principal fibre bundle
1. definitions
2. local connection l-forms on the base manifold
existence theorems
section canonically associated with a trivialization
potentials
change of trivialization
examples
3. covariant derivative
associated bundles
parallel transport
covariant derivative
examples
4. curvature
definitions
cartan structural equation
local curvature on the base manifold
field strength
bianchi identities
5. linear connections
definition
soldering form， torsion form
torsion structural equation
standard horizontal （basic） vector field
curvature and torsion on the base manifold
bundle homomorphism
metric connection
b. hoionomy
1. reduction
2. holonomy groups
c. characteristic classes and invariant curvature integrals
1. characteristic classes
2. gauss-bonnet theorem and chern numbers
3. the atiyah-singer index theorem
problems and exercises
problem 1： the geometry of gauge fields
problem 2： charge quantization. monopoles
problem 3： instanton solution of euclidean su（2） yang-mills theory （connection on a non-trivial su（2） bundle over s4）
problem 4： spin structure; spinors; spin connections
vi. distributions
a. test functions
1. seminorms
definitions
hahn-banach theorem
topology defined by a family of seminorms
2. d-spaces
definitions
inductive limit topology
convergence in dm（u） and d（u）
examples of functions in
truncating sequences
density theorem
b. distributions
1. definitions
distributions
measures; dirac measures and leray forms
distribution of order p
support of a distribution
distributions with compact support
2. operations on distributions
sum
product by c function
direct product
derivations
examples
inverse derivative
3. topology on d
weak star topology
criterion of convergence
4. change of variables in rn
change of variables in rn
transformation of a distribution under a diffeomorphism
invariance
5. convolution
convolution algebra l1（rn）
convolution algebra d+ and d-
derivation and translation of a convolution product
regularization
support of a convolution
equations of convolution
differential equation with constant coefficients
systems of convolution equations
kernels
6. fourier transform
fourier transform of integrable functions
tempered distributions
fourier transform of tempered distributions
paley-wiener theorem
fourier transform of a convolution
7. distribution on a c∞ paracompact manifold
8. tensor distributions
c. sobolev spaces and partial differential equations
i. sobolev spaces
properties
density theorems
w? spaces
fourier transform
plancherel theorem
sobolevs inequalities
2. partial differential equations
definitions
cauchy-kovalevski theorem
classifications
3. elliptic equations; laplacians
elementary solution of laplaces equation
subharmonic distributions
potentials
energy integral， greens formula， unicity theorem
liouvilles theorem
boundary-value problems
green function
introduction to hilbertian methods; generalized dirichlet problem
hilbertian methods
example： neumann problem
4. parabolic equations
heat diffusion
5. hyperbolic equation; wave equations
elementary solution of the wave equation
cauchy problem
energy integral， unicity theorem
existence theorem
6. leray theory of hyperbolic systems
7. second order systems; propagators
problems and exercises
problem 1： bounded distributions
problem 2： laplacian of a discontinuous function
exercise 3： regularized functions
problem 4： application to the schrbdinger equation
exercise 5： convolution and linear continuous responses
problem 6： fourier transforms of exp （-x2） and exp （ix2）
problem 7： fourier transforms of heaviside functions and pr（l/x）
problem 8： dirac bitensors
problem 9： legendre condition
problem 10： hyperbolic equations; characteristics
problem 11： electromagnetic shock waves
problem 12： elementary solution of the wave equation
problem 13： elementary kernels of the harmonic oscillator
vii. differentiable manifolds， infinite dimensional case
a. infinite-dimensional manifolds
1. definitions and general properties
e-manifolds
differentiable functions
tangent vector
vector and tensor field
differential of a mapping
submanifold
immersion， embedding， submersion
flow of a vector field
differential forms
2. symplectic structures and hamiltonian systems
definitions
complex structures
canonical symplectic form
symplectic transformation
hamiltonian vector field
conservation of energy theorem
riemannian manifolds
b. theory of degree; leray-schauder theory
i. definition for finite dimensional manifolds
degree
integral formula for the degree of a function
continuous mappings
2. properties and applications
fundamental theorem
borsuks theorem
brouwers fixed point theorem
product theorem
3. leray-schauder theory
definitions
compact mappings
degree of a compact mapping
schauder fixed point theorem
leray-schauder theorem
c. morse theory
1. introduction
2. definitions and theorems
3. index of a critical point
4. critical neck theorem
d. cylindrical measures， wiener integral
1. introduction
2. promeasures and measures on a locally convex space
projective system
promeasures
image of a promeasure
integration with respect to a promeasure of a cylindrical
function
fourier transforms
3. gaussian promeasures
gaussian measures on rn
gaussian promeasures
gaussian promeasures on hilbert spaces
4. the wiener measure
wiener integral
sequential wiener integral
problems and exercises
problem a： the klein-gordon equation
problem b： application of the leray-schauder theorem
problem c1： the reeb theorem
problem c2： the method of stationary phase
problem d1： a metric on the space of paths with fixed end points
problem d2： measures invariant under translation
problem d3： cylindrical σ-field of c（[a， b]）
problem d4： generalized wiener integral of a cylindrical function
references
symbols
index
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