00540 - Fundamentals of Higher Geometry

Course contents

Formal power series
– Ring of polynomials with coefficients in a ring. Ring of formal series with coefficients in a ring.
– Metric and topology in the ring of formal series. Contractions. The geometric series.
– Characterization of invertible elements of the ring of formal power series.
– Differential equations in the ring of formal power series. Existence and uniqueness for the Cauchy problem.
– Exponential, logarithmic and binomial series. Vandermonde formula.
– The functional equation for the exponential series and the Newton binomial theorem.

Endomorphisms of the ring of formal power series.
– The monoidal structure of the formal series of positive order with respect to composition and his anti-isomorphism with the monoid of the endomorphisms of the ring of formal series.
– Invertible elements of the monoid of formal series of positive order for the replacement operation. - Couples of reciprocal series: exp / log , tan / atan , sin / asin .

Formal Laurent series
– Formal Laurent series.
– The Laurent series are the quotient field of the ring of formal power series (if the ring of coefficients is a field)
– The Laurent expansion of rational functions.

Formal Derivations
– Extensions of a derivation to the polynomial ring.
– Extension of a derivation from the polynomials ring to the power series ring.
– Extension of a derivation to the quotient field.

Holomorphic functions.
– The complex derivative. The ring of the holomorphic functions. Composition of holomorphic functions . The conjugate function is not holomorphic .
– The Cauchy- Riemann . The complex exponential function in terms of the real exponential function and trigonometric functions.
– A function holomorphic with the real / imaginary part / constant module is constant .

Mean and maximum principles
– Mean invariance on concentric circles . Mean property for holomorphic functions.
– A continuous function holomorphic except at a point is holomorphic.
– Strong maximum principle for continuous functions stisfying the mean property.
– The derivative of a holomorphic function is holomorphic.
– An entire holomorphic function which tends to zero at infinity is identically zero.
– The fundamental theorem of algebra.
– The Liouville theorem.
– The Schwarz Lemma.
– The convergence theorem of Weierstrass.

Series expansions
– Zeros of holomorphic functions of multiplicity m and their characterization in terms of derivatives.
– Existence and uniqueness of the Taylor polynomial of order d associated to an holomorphic function at a point of the domain.
– Every holomorphic function on a disc vanishing together with all derivatives in the center of the disc is identically zero.
– The zeros of infinite multiplicity of a holomorphic function form an open and closed set.
– Principle of analytic continuation for holomorphic functions.
– The zeros of a holomorphic function are isolated.
– The holomorphic functions on a domain form an integral domain.
– Convergent power series.
– Radius of convergence of a power series. Characterization of analytic functions on a disk.
– Analytic functions . A function is analytic if , and only if , it is holomorphic.
– Cauchy Inequality.
– Cauchy Turin Theorem.

Meromorphic functions.
– Isolated singularities of a holomorphic function and their classification : apparent,  polar and essential singularities.
– The Riemann extension theorem .
– Meromorphic functions.
– The meromorphic functions on a domain form a field.
– The Casorati - Weierstrass theorem.
– Laurent expansion of meromorphic functions.
– Elementary Properties esponential and trigonometric functions in the complex field .

The complex logarithm and  the index of a path.
– The logarithm and the argument of a complex number.
– Extension and holomorphy of the logarithm on the right half-plane.
– Local determination of the logarithm and determinations of the logarithm along a continuous function.
– Principal determination of the  logarithm and argument.
– Homotopy, contractible spaces , simply connected spaces. The star-shaped open domains are contractible.
– Existence of determinations of the logarithm along a path.
– Existence of determinations of the logarithm along homotopies.
– Index of a closed path with respect to a point.
– Homotopy invariance of the index.
– Two paths are homotopic in C-{z_0} if , and only if , they have the same index with respect to z_0.
– Existence of determinations of the logarithm on simply connected domains.

Covering spaces.
– The notion of covering. Admissible open set.
– Sections and liftings.
– Existence of lifting along paths and homotopies.
– Existence of sections and lifting of  closed paths.
– Existence of lifting of functions defined on simply connected spaces.

Primitives of analytic functions.
– Existence of primitives of a holomorphic function on a disk.
– The covering associated with the primitives of a holomorphic function.
– Existence of primitives of a holomorphic function on simply connected domains.
– Determinations of the arctan and the arcsin .

Differential forms
– Definition of differential form. The differential of a function.
– Integration of a differential form along a piecewise smooth path.
– Exact and locally exact differential forms.
– Forma complex derivatives complex. exact forms and primitive of a holomorphic function.
– A continuous form is exact on a disk if, and only if, the integral along the boundary of each rectangle contained in the disk vanishes.
– The covering associated to a locally exact differential form.
– Primitives (local)  of a form as (local) sections of the associated covering.
– Primitive along a function of a locally exact form as lifting along the associated covering.
– Integration along a continuous path of a locally exact differential form.
– Invariance of the integral of a locally exact differential form along homotopic paths.
– The integral of a locally exact differential form along a closed path is zero if, and only if, the lift of the path along the associated covering is a closed path.
– A differential form on a domain is locally exact if, and only if, the integral on the boundary of each rectangle contained in the domain vanishes.
– Equivalence between the existence of a global primitive form and the vanishing of the integrals along closed paths.
– A locally exact differential form on a simply connected domains is exact.

The residues theory.
– The Morera theorem.
– The Schwarz reflection principle.
– Integral representation of the index of a closed path.
– Cauchy Integral representation formula.
– Decomposition of a function holomorphic on an annulus.
– Laurent expansion of a holomorphic function in an annulus.
– Classification of isolated singularities by the Laurent expansion.
– The residues theorem.
– The logaritmic derivative. The logarithmic indicator theorem.
– The open map theorem.
– The inverse map theorem for holomorphic functions.
– The existence of the  complex derivativeimplies the holomorphy of a function.
– The expansion of the cotangent function an the origin.

Readings/Bibliography

H. Cartan "Théorie élémentaire des fonctions analytiques d' une ou plusieurs variables complexes", Hermann Ed.

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Lectures

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