Mathematics

MARCO MANETTI

Knowledge and understanding: at the end of the course the student will have acquired the notions and the results
basics related to singular homology, to the study of differentiable varieties
and a fair knowledge of Riemann's theory of surfaces.

Apply knowledge and understanding: at the end of the course the student will be able to solve
even complex problems that require the use of techniques related to de Rham's cohomology,
the Hurwitz theorem and the Riemann Roch theorem for compact Riemann surfaces;
will be able to determine the kind of a Riemann Surface and the size of the linear systems
variety cohomology groups.

Critical and judgmental skills: the student will have the bases to analyze the analogies and the relationships between
topics that vary between algebraic topology, differential geometry, complex geometry
and also algebraic geometry.

Communication skills: ability to expose the contents in the oral part of the verification and in the
any theoretical questions present in the written test.

Learning skills: the knowledge acquired will allow you to devote yourself to more specialized aspects of geometry.

Knowledge and understanding: at the end of the course the student will have acquired the notions and the results
basics related to singular homology, to the study of differentiable varieties
and a fair knowledge of Riemann's theory of surfaces.

Apply knowledge and understanding: at the end of the course the student will be able to solve
even complex problems that require the use of techniques related to de Rham's cohomology,
the Hurwitz theorem and the Riemann Roch theorem for compact Riemann surfaces;
will be able to determine the kind of a Riemann Surface and the size of the linear systems
variety cohomology groups.

Critical and judgmental skills: the student will have the bases to analyze the analogies and the relationships between
topics that vary between algebraic topology, differential geometry, complex geometry
and also algebraic geometry.

Communication skills: ability to expose the contents in the oral part of the verification and in the
any theoretical questions present in the written test.

Learning skills: the knowledge acquired will allow you to devote yourself to more specialized aspects of geometry.

he course requires familiarity with the topics of the fundamental courses in Geometry, Analysis and Algebra of the Bachelor's Degree. This knowledge is indispensable. There are no preliminary steps.

Notes on functions of a complex variable: holomorphic functions. residue theorem, holomorphic functions as applications
Differential and analytical varieties: definitions and examples. Algebraic varieties, Zariski topology, algebraic plane curves.
Singular Homology, Mayer- Vietoris's Theorem . Calculus on differentiable varieties: differential forms. Stokes theorem,
de Rham's cohomology . The consequences of Stokes and de Rham theorems for Riemann surfaces. Meromorphic functions on
Compact Riemann Surfaces. Notes on Hodge's Theorem for Compact Riemann Surfaces. Riemann-Roch theorem.

Arbarello- Salvati Manni Notes free access on the web
Springer : Riemann Surfaces
Miranda: Algebraic Curves and Riemann Surfaces - Mathematics

Traditional lectures in the classroom

stongly encouraged

The exam aims to evaluate learning through a written test (consisting of the resolution of
problems of the same type as those carried out in the exercises) and an oral test (consisting of
discussion of the most relevant topics illustrated in the course). The written test will last about three
hours and can be replaced by two intermediate tests, both lasting two hours, the first one
of which will take place in the middle of the course and the second immediately at the end of the course.

To pass the exam, a grade of not less than 18/30 must be obtained. The student must
to demonstrate to have acquired a sufficient knowledge of the arguments of both
parts of the program and to be able to
carry out at least the simplest among the assigned exercises.
To achieve a score of 30/30 cum laude, the student must instead demonstrate
have acquired an excellent knowledge of all the topics covered during the course and
be able to link them in a logical and consistent way.

What is a meromorphic function?