# Course Details

#### Name:

**Advanced Geometry / Geometria Superiore **

### Basic information

##### Credits:

*Master Degree in Mathematics:* 12 Ects (b)

##### Term:

*Module Advanced Geometry 1:* 1° semester

*Module Advanced Geometry 2:* 2° semester

##### Degree(s):

Compulsory 1^{st} year Master Degree in Mathematics curriculum Generale

##### Language:

English

### Course Objectives

- 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)
-The goal is to acquire a good knowledge of basic concepts about topological manifolds, CW-complexes and simplicial complexes (Fedeli).

**Module Advanced Geometry 2:** -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

- (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.
- (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.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- The student should have a basic knowledge of topological manifolds and complexes.
- The student should be able to use the acquired tools.
- The student should be able to understand and solve problems.
- The student should be able to present in a clear and rigorous way the acquired knowledge.
- The student should develop those learning skills necessary to deal with the subsequent
studies.

### Prerequisites and Learning Activities

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

**Module Advanced Geometry 2:** An introductory course on algebraic topology (fundamental group and singular homology)
Basics on smooth manifolds (in particular differential forms and de Rham cohomology).

### Teaching Methods

**Language**: English

Lectures (Nelli)
Lecture (Fedeli)

**Module Advanced Geometry 2:** Lectures

### Assessment Methods and Criteria

Oral exam (Nelli)
Oral exam (Fedeli)

**Module Advanced Geometry 2:** Oral exam

### Textbooks

- Farkas-Kra,
**Riemann Surfaces**. Springer-Verlag. * *

### Course page updates

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

*Course information last updated on: 17 marzo 2016, 08:23*