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B018799 - COMPLEX VARIABLE
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Further information
Type of Assessment
Course program
Academic Year 2017-18
Coorte 2016 - Second Cycle Degree in MATHEMATICS
Course year
Second year - Second Semester
Belonging Department
Mathematics and Computer Science "Ulisse Dini" (DIMAI)
Course Type
Single education field course
Scientific Area
MAT/03 - GEOMETRY
Credits
9
Teaching Hours
72
Teaching Term
26/02/2018 ⇒ 15/06/2018
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Teaching Language
Italian
Course Content
Geometric interpretation of analytic properties of holomorphic functions. The Riemann sphere.
Schwarz and Schwarz Pick Lemmas. Poincare' metric and distance and associated geodesics..
Automorfphisms of the unitary disk. Automorphismsi of the complex plane and the Riemann sphere.
Linear fractional rasformations. Normal families.
Weierstrass Theorem. Hurwitz Theorem . Montel. Theorem. Uniformization (Riemann) Theorem.. Riemann surfaces and their classification.
Schwarz and Schwarz Pick Lemmas. Poincare' metric and distance and associated geodesics..
Automorfphisms of the unitary disk. Automorphismsi of the complex plane and the Riemann sphere.
Linear fractional rasformations. Normal families.
Weierstrass Theorem. Hurwitz Theorem . Montel. Theorem. Uniformization (Riemann) Theorem.. Riemann surfaces and their classification.
Suggested readings (Search our library's catalogue)
L. V. Ahlfors, Complex Analysis, Third Edition, Mc Graw Hill 1979
J. B. Conway, Functions of One Complex Variabl, GTM Springer-Verlag, 1978
J. Milnor, Dynamics n One Complex Variable, Third Edition, Annals of Mathematics Studies, 2006
E. Vesentini, Capitoli scelti della teoria delle funzioni olomorfe. UMI 1980
J. B. Conway, Functions of One Complex Variabl, GTM Springer-Verlag, 1978
J. Milnor, Dynamics n One Complex Variable, Third Edition, Annals of Mathematics Studies, 2006
E. Vesentini, Capitoli scelti della teoria delle funzioni olomorfe. UMI 1980
Learning Objectives
Knowledge acquired:
Basic Knoweledge of the the theory of a complex variable.
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
Aplied Mathematics.
Basic Knoweledge of the the theory of a complex variable.
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
Aplied Mathematics.
Prerequisites
Courses required: All the required courses of the Laurea In Mathematics
(first three year cicle)
Courses recommended: All basic courses courses in Algebra, Analisi and Geometria of the Laurea In Mathematics (first three year cicle)
IIn particular basic notions of Algebraic Topology (Fundamental Group and Covering Spaces) and
of Complex Analytic Functions (namely Cauchy Formula) are supposed to be familiar.
(first three year cicle)
Courses recommended: All basic courses courses in Algebra, Analisi and Geometria of the Laurea In Mathematics (first three year cicle)
IIn particular basic notions of Algebraic Topology (Fundamental Group and Covering Spaces) and
of Complex Analytic Functions (namely Cauchy Formula) are supposed to be familiar.
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
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:
http://web.math.unifi.it/users/patrizio/DidaI/
Office hours:
Monday 14:30 and by appointment
Contact:
Viale Morgagni, 67/a - 50134 Firenze
Phone: 055 4237109
Fax: 055 4237165
E-mail: giorgio.patrizio@math.unifi.it
Web: http://web.math.unifi.it/users/patrizio/
Not mandatory
Teaching tools:
http://web.math.unifi.it/users/patrizio/DidaI/
Office hours:
Monday 14:30 and by appointment
Contact:
Viale Morgagni, 67/a - 50134 Firenze
Phone: 055 4237109
Fax: 055 4237165
E-mail: giorgio.patrizio@math.unifi.it
Web: http://web.math.unifi.it/users/patrizio/
Type of Assessment
Written and oral exam
Course program
Complex numbers and topology of the Riemann sphere. Holomorphic functions and conformality. Power series.
Schwarz and Schwarz Pick Lemmas. Automprphisms of the unit disc, of the Riemann sphere and of the complex plane. Free and properly discontinuous actions of automorphiisms.
Normal families and Montel Theorem. Riemann Uniformizartion Theore,
Introduction of Riemann surfaces and their classification..
Ramified coverings and Riemann Hurwitz’ theorem.
Complex tori and their moduli space..Weierstrass function associated with a lattice in the complex plane.
Some connections between Riemann surfaces and algebraic curves,
Schwarz and Schwarz Pick Lemmas. Automprphisms of the unit disc, of the Riemann sphere and of the complex plane. Free and properly discontinuous actions of automorphiisms.
Normal families and Montel Theorem. Riemann Uniformizartion Theore,
Introduction of Riemann surfaces and their classification..
Ramified coverings and Riemann Hurwitz’ theorem.
Complex tori and their moduli space..Weierstrass function associated with a lattice in the complex plane.
Some connections between Riemann surfaces and algebraic curves,