# Course Details for A.Y. 2014/2015

#### Name:

**Istituzioni Di Geometria Superiore I / Foundations of Advanced Geometry I**

### Basic information

##### Credits:

*:* Bachelor Degree in Mathematics 9 CFU (b)

##### Degree(s):

Bachelor Degree in Mathematics 3^{rd} anno curriculum Generale Compulsory

##### Language:

Italian

### Course Objectives

The goal of this course is to provide the motivations, definitions and techniques for the translation of topological problems into algebraic ones, hopefully easier to deal with.
On successful completion of this module, the student should understand the fundamental concepts of algebraic geometry and should be aware of potential applications of algebraic topological invariants in other fields as theoretical physics , including the computational fluid mechanics and electrodynamics.

### Course Content

- General Topology: Topological spaces and continuous maps, induced, quotient and product topology, Metric spaces, Hausdorff spaces, Compact spaces, connected spaces, Paths and path connected spaces
- Manifolds and surfaces: The pancake problems, n-dimensional manifolds, surfaces and classification of surfaces.
- Homotopy: Retracts and contractible spaces, paths and multiplication, the fundamental group, the fundamental group of the circle.
- Covering spaces: The fundamental group of a covering space, the fundamental group of a orbit space, lifting theory and existence theorems, the Borsuk-Ulam theorem, the Seifert-Van Kampen theorem, the fundamental group of a surface
- Introduction to singular homology : standard and simplicial simplexes.
Introduction to singular homology : standard and simplicial simplexes.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- have profound knowledge of basic techniques in Homotopy Theory,
- have knowledge and understanding of geometric and topological arguments,
- understand the fundamental concepts of Topology, algebra and their connections and be aware of potential applications in other fields,
- demonstrate skill in mathematical reasoning and ability to conceive a proof,
- understand and explain the meaning of complex statements using mathematical notation and language;

### Prerequisites and Learning Activities

The student must know the basic notions of General Topology and Geometry contained in the exams Geometry A, B and Algebra

### Assessment Methods and Criteria

Oral Exam

### Textbooks

- Czes Kosniowski, A first course in algebraic topology , Cambridge University Press. 1980.
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- Czes Kosniowski, Introduzione alla Topologia Algebrica , Zanichelli Ed.. 2010.
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### Course page updates

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

*Course information last updated on: 08 marzo 2015, 18:35*