GEOMETRY III
Academic year and teacher
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- CINZIA BISI
- Credits
- 10
- Didactic period
- Annualità Singola
- SSD
- MAT/03
Training objectives
- 1st PART: this is the first course on the theory of functions in Complex Analysis and Riemann Surfaces.
The main goal of the course consists in providing the basic concepts ans results of the theory of complex functions of one variable.
The main acquired knowledge will be:
basic knowledge of a complex function , also of multi-valued functions;
basic knowledge to tackle the study of complex functions, sequences and series of functions, series representations, integrals of continuous complex functions along varios types of paths of integration;
basic knowledge of analytic function and their power series expansions. Gentle Introduction to Riemann Surfaces.
The basic acquired abilities will be:
analysis of the behavior of complex functions;
study of the analitic and algebraic properties of functions of a complex variable and the geometric aspects of the mapping defined by them;
use of the properties of complex functions to solve some problems in the real field.
2nd PART: further steps in the study of algebraic methods in geometry. Prerequisites
- 1st PART: We assume the student to be familiar with the real and complex number system, as well as with the properties of the real functions of one and several real variables , as developed in the usual courses in calculus and in real variables.
We assume also some acquaintance with the fundamental concepts and propositions of abstract algebra, modern geometry and topology.
2nd PART: the following courses: Geometria 1, Algebra, Geometria 2. Course programme
- The course is strctured in two parts, the first one is of 50 hours the second one is of 30 hours.
1st PART is complex analysis:
Functions of one complex variable: basic definitions, elmentary functions; analytic functions; equations of Cauchy-Riemann (5 hours); sequences and series of complex functions; convergences; power series (6 hours); conformal mappings; elementary transcendental functions; complex integration; Cauchy's theorem for star regions and integral formula for the disc ; properties of analytic functions: the Taylor series (5 hours); Liouville's theorem, Riemann identity theorem, Riemann prolongation theorem (4 hours); the open mapping theorem; isolated singularities of complex functions, meromorphic functions; Cauchy's theorem for the annulus (10 hours) ; Laurent's theorem; the expansion in Laurent series for holomorphic functions in the annulus; Index of a path towards a point; regions homologically simple connected; general Cauchy's theorem; the calculus of residues:the residue theorem and its applications in the evaluation of real integrals (10 hours) . Gentle Introduction to Riemann Surfaces. (10 hours).
2nd PART:
Complements of linear algebra: The theorem of Cayley-Hamilton. Minimal polynomial. Characteristic subspaces, the 'Killer lemma'. Frobenius canonical form and applications. Jordan normal form. (aprrox 20 hours).
Algebraic geometry: algebraic plane curves, first properties. Singularities of plane curves. Linear systems. (approx. 10 hours). Didactic methods
- 1st PART: Lectures on all the course's topics : written exposition of the theory with performed exercises carried out as illustrative examples of the treated theory.
2nd PART: this will be some kind of 'reading course'. Very detailed notes written by the teacher will be given to the students. These notes will contain the theoretical part and exercises.
The student should study by himself the theoretical part and solve the exercises. During the lessons the teacher will quickly review the theoretical part and answer questions. Then the students will expose their solutions of the exercises that will be discussed by all the participants.
The lessons of both the two parts will be classroom lessons in presence or distance lessons according to the pandemic (Covid 19) conditions. Learning assessment procedures
- 1st PART: at the end, the student has to pass a written test with exercises to solve online. The text must be scanned and sent by email to the teacher.
2nd PART: Written examination. The participation to the correction of the exercises during the course will be part of the final mark.
If pandemic (Covid 19) conditions will permit, the examinations could also be in presence.
The final exam score will be the mean of the arithmetic partial score. Reference texts
- 1st PART:
Specific topics can be further developed in the following tests
A.Del Centina - Teoria delle funzioni di una variabile complessa - Aracne editrice
M.O.Gonzalez - Classical complex Analysis - Dekker
Beardon-A primer on Riemannian Surfaces
Foster-Lectures on Riemannian Surfaces
2nd PART:
Notes by the teacher.
