Academic Year 2021-2022

FOUNDATIONS OF HIGHER GEOMETRY

Teachers

Pietro De Poi
Francesco Zucconi
Unit Credits
12
Teaching Period
Annuity
Course Type
Characterizing
Prerequisites. First two years of first-level degree in Mathematics
Teaching Methods. Lectures.
Verification of Learning. Written examination (3 hours) on exercises followed by an oral theoretical examination.
More Information. Oral examination by appointment also.

The course could be held in English, upon proposal of the competent teaching structure.

Objectives
The student will have to:

Knowledge and understanding: To know some basic concepts and results of complex analysis in one variable and of differential geometry. To recognize a real analytic or geometric problem that can be tackled with complex analysis methods. To recognize a geometric problem which is resoluble through differential geometry methods.

Ability to apply knowledge and understanding: To know how to deal with and solve some classical problems of complex analysis in one variable and of differential geometry. To find analytical and geometric applications of complex analysis in one variable and differential geometry

Independent thinking: To know how to find the most appropriate analytical or geometric techniques in solving assigned problems. To address the difficulty of specific problems both in complex analysis of one variable and in differential geometry.

Communication skills: To introduce, orally and in writing, a subject, or a mathematical theory, learned during the course. Being able to present to a non-specialist public the salient aspects of classical theory of analytic functions in one complex variable, of Riemann surfaces, and of curves and surfaces immersed in ordinary space.

Learning ability: to be able to read a graduate degree book in the fields covered by the course. To work independently in literature search. To address the proposed problems by selecting independently the most meaningful ones.

Contents
The aim of the course is twofold:

to introduce the basic concepts of complex analysis in one variable;

to introduce the basic concepts of differential geometry.

Texts
Lars Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, 3rd ed. International Series in pure and applied Mathematics. Düsseldorf etc.: McGraw-Hill Book Company. xiv+331 p., 1979.

Walter Rudin, Real and complex analysis, 3rd ed. New York, NY: McGraw-Hill. xiv+416p., 1987.

W. Ballmann, “Introduction to Geometry and Topology”

M.P.Do Carmo, “Differential geometry of curves and surfaces”

M.P. Do Carmo “Riemannian geometry”