# Course Details for A.Y. 2015/2016

#### Name:

**Geometria Superiore 2 / Advanced Geometry 2**

### Basic information

##### Credits:

*:* Master Degree in Mathematics 6 CFU (b)

##### Degree(s):

Master Degree in Mathematics 2^{nd} anno curriculum Generale Compulsory

##### Language:

English

### Course Objectives

-The goal is to acquire a good knowledge of basic concepts about topological
manifolds, CW-complexes and simplicial complexes (Fedeli).
- The student should learn the basic notions of the theory of Riemann surfaces necessary to establish some theorem
and to solve problems
about this subject (Nelli)

### Course Content

- (Fedeli) Locally euclidean spaces, topological manifolds:properties
and examples. Open and closed n-cells, examples. Cell
decompositions, characteristic maps. Cell complexes,
CW-complexes, examples. Properties of CW-complexes.
Subcomplexes, n-skeleton. Regular CW-complexes.
Inductive construction of CW-complexes. Regular
CW-decomposition of a 1-manifold. Simplices. Simplicial
complexes. Subcomplexes. Polyhedron of a simplicial
complex. Triangulations. Simplicial maps. Abstract simplicial
complexes.
- (Nelli)Definition and exsmples of Riemann surfaces. Topology of Riemann surfaces.
Differential forms and integration formulasBasic notions about Hilbert spaces theory. Weyl's Lemma.
Hilbert space of differential form with integrable square: L2(M).
Harmonic differential. Holomorphic and meromorphic differential.
Intersection theory on compact surfaces. Harmonic and analytic differential on compact surfaces.
Bilinear relations. Divisors and Riemann-Roch theorem.
Some applications of Riemann-Roch's theorem.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- The student should learn the basic notions of Riemann surfaces theory.

The student should have a basic knowledge on topological manifolds and complexes.

- The student should be able to solve small problems about the theory Riemann surfaces,
using notions and theorems of the course.

The student should be able to use the acquired tools.

- The student should understand how to apply the acquired notions of Riemann Surfaces theory to
the proposed problems.

The student should be able to understand and solve problems.

- The student should be able to explain the statements and the proofs of the theorems about
Riemann surfaces.

The student should be able to present in a clear and rigorous way the acquired
knowledge.

- The student should have acquired the ability of reading and understanding more
advanced result about Riemann surfaces.

The student should develop those learning skills necessary to deal with the subsequent
studies.

### Prerequisites and Learning Activities

- Familiarity with set-theory and general topology (Fedeli).
- Basic notions of differential geometry and complex analysis (Nelli)

### Assessment Methods and Criteria

Oral exam (Fedeli)
Oral exam (Nelli)

### Textbooks

- H. M. Farkas, I. Kra, Riemann Surfaces , Springer.
* *

### Course page updates

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 16 marzo 2016, 11:45*