Study-unit TOPOLOGY I
| Course name | Mathematics |
|---|---|
| Study-unit Code | 55A00101 |
| Curriculum | Didattico-generale |
| Lecturer | Nicola Ciccoli |
| Lecturers |
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| Hours |
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| CFU | 6 |
| Course Regulation | Coorte 2023 |
| Supplied | 2023/24 |
| Supplied other course regulation | |
| Learning activities | Affine/integrativa |
| Area | Attività formative affini o integrative |
| Sector | MAT/03 |
| Type of study-unit | Opzionale (Optional) |
| Type of learning activities | Attività formativa monodisciplinare |
| Language of instruction | Italian The course can be given in English if students should request it. |
| Contents | Algebraic Topology: basic of homotopy and singula rhomology theories. |
| Reference texts | To be decided according to students average skills |
| Educational objectives | Introducing both classical and more recent ideas in Algebrai Topology with an eye towards applications, |
| Prerequisites | Elements of General Topology (open and closed sets continuity, connection, compactness) |
| Teaching methods | Lectures |
| Learning verification modality | Oral exam (45' approx). It will be possible to agree on concentrating on specific parts of the program. The oral exam aim is to verify: 1. knowledge of the main techniques used in Algebraic Topology; 2. the level of comprehension of theoretical issues encountered in the course, their relation with different subfields of mathematics, the capabilty of establishing connections between different topics of the program. For these reasons it is required to be able to reproduce sufficiently complete proofs of the main theorems explained during lectures. 3. the students capability to expose correctly and with a suitably precise use of mathematical language the main topics of Algebraic Topology. Specific informations on support services with student with disabilities and/or DSA can be found at: http://www.unipg.it/disabilita-e-dsa |
| Extended program | Absolute and relative homotopy. Homotopy equivalence. Retractions and strong retractions. Computing homotopy groups. Applications to classical theorems in topology (e.g. Brouwer's fixed point theorem, the hairy ball theorem, the ham sandwich theorem). Introduction to singular homology. Eilenberg-Steenrod axioms. Computing singular homology groups. Topological and differentiable degree theory. Applications to classical theorems in topology (dimension theory, winding number, index of a vector field) Simplicial homology and applications to graph theory. Applications to Topological Data Analysis: the notion od persistence homology. |