Homotopic and homological properties of holomorphic functions. Riemann representation theorem. Extension of biolomorphisms to the boundary, Rosay-Wong theorem for open sets in C. Non-compact groups of automorphisms. Introduction to Riemann surfaces. Curvature methods in geometric theory of functions. Uniformization of Riemann surface. Approximation and Runge theory. Construction of holomorphic functions: Mittag-Leffler and Weierstrass theorems, domains of holomorphy.
Notes provided by the Teacher.
Recommended reading:
W. Fischer - I. Lieb,A Course in Complex Analysis. From Basic Results to Advanced Topics,-Vieweg-Teubner(2011)
O. Forster, Lectures on Riemann Surfaces, Springer (1981)
L. V. Ahlfors, Complex Analysis, Third Edition, Mc Graw Hill 1979
J. B. Conway, Functions of One Complex Variabl, GTM Springer-Verlag, 1978
Learning Objectives
Knowledge acquired:
Review of the basic knowledge of the theory of functions of a complex variable.
Introduction to advanced topics of complex variable function theory
through the use of algebraic, geometric and analytical techniques.
Competence acquired:
Elements of complex function theory necessary for advanced topics in
Analysis, Geometry and Applied Mathematics
Skills acquired (at the end of the course):
Ability of using Complex Funcrion Theory in Analysis, Geometry and
Applied Mathematics.
Prerequisites
Courses required: All the required courses of the Laurea In Mathematics
(first three year cicle). In particular, it is expected familiarity with the elementary theory
of the theory of holomorphic functions as presented in the Analisi III course.
Courses recommended: All basic courses courses in Algebra, Analisi and Geometria of the Laurea In Mathematics (first three year cicle)
Teaching Methods
Total hours of the course (including the time spent in attending lectures,
seminars, private study, examinations, etc...): 225
Hours reserved to private study and other indivual formative activities: 153
Hours for lectures: 72
Further information
Attendance of lectures, practice and lab:
Not mandatory
Teaching tools:
All the necessary teaching material will be made available on the Moodle page of the course
For Office hours and contacts see institutional page:
https://www.unifi.it/p-doc2-2013-200010-P-3f2a3d2f34272c-0.html
Type of Assessment
Written and oral exam
Course program
Review of elementary properties of holomorphic functions.
Homotopy and integration along curves of holomorphic functions.
Simple connectivity and holomorphic primitives. Winding number. Homotopic Cauchy formula.
Existence of a holomorphic logarithm and of roots of non-zero holomorphic functions.
Local behavior of holomorphic functions, open mapping theorem.
Homological version of Cauchy theorem and formula.
Equivalent analytical properties on simply connected open sets.
Cauchy formula for C ^ 1 functions.
Inequalities of Cauchy version L ^ 1.
Weierstrass theorem, Montel theorem, normal families.
Review of Open Application Theorem and Maximum Module Principle.
Schwarz's lemma. Automorphisms of the unitary disk, Schwarz-Pick Lemma, Moebius distance.
Riemann sphere and elementary properties of the Moebius transformations.
Principle of the argument and Rouché's Theorem, Hurwitz theorem. Biolomorfismi
Definition of Riemann surface and examples: the Riemann sphere and complex projective space,
implicit function theorem and locus of zeros of holomorphic functions of two variables,
quotients and complex tori; holomorphic functions between Riemann surfaces.
Properties of holomorphic functions between Riemann surfaces.
Meromorphic functions on Riemann surfaces, in particular on the sphere.
Automorphisms of C and of the Riemann sphere.
Riemann representation theorem and classification of the simply connected open sets of the Riemann sphere.
Introduction to the problem extension of biolomorphisms to homeomorphisms of closures. Easily accessible boundary points.
Extension of biolomorphisms from a domain with accessible boundary points to the disc to homeomorphisms of the closures.
Groups of automorphisms of open of C and homogeneous open sets.
Peak functions and their existence in regular points of the open border of C. Rosay-Wong theorem for open sets of C.
Non-compact automorphisms groups and boundary orbit accumiulation points.
Conformal pseudomtrics. Curvature of conformal pseudometrics. Poincaré metric of the disk.
Lemma of Ahlfors. Non-existence of conformal metrics with strictly negative curvature on C, Liouville theorem.
Construction of conformal metric with strictly negative curvature on C minus two points.
Picard's Little Theorem, Landau's Theorem.
Spherical Metric: isometries and distance. Normal families of holomorphic applications with values in the Riemann sphere.
Spherical derivative. Marty's theorem. Montel theorem for holomorphic applications with values in the Riemann sphere.
Shottky's theorem, Picard's great theorem.
Non regular pseudometrics and supporting metrics, Ahlfors' Lemma in this case. Bloch theorem.
Uniformization of Riemann surfaces: analytical properties, potential theory and differential geometric properties.
Approximation and Runge theory. Construction of functions by classical methods and by d-bar tecniques:
Mittag-Leffler and Weierstrass theorems; domains of holomorphy.