Study-unit TOPOLOGY I
| Course name | Mathematics |
|---|---|
| Study-unit Code | 55A00101 |
| Curriculum | Didattico-generale |
| Lecturer | Nicola Ciccoli |
| Lecturers |
|
| Hours |
|
| CFU | 6 |
| Course Regulation | Coorte 2022 |
| Supplied | 2022/23 |
| 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 |
| Contents | Homotopy and homology theory |
| Reference texts | 1. Singer-Thorpe Elements of General Topology and Differential Geometry 2. Hatcher, Algebraic Topology 3. Dubrovin-Novikov-Fomenko, Contemporary Geometry I-II-III (empohasis on vol. 3) More details will be clarified during lectures. |
| Educational objectives | Understanding Category Theory as a unifying language of mathematics. Basic topics in General Topology and learning techniques of general homotopy and homology theories. |
| Prerequisites | The content of previous courses in Algebra and Geometry is required, in particular for what concerns the Topology part. (Definition of topology, continuous functions, product and quotient topology. pull-back and push-forward topology, separation axioms, connectedness) |
| Teaching methods | Lectures on all required materials. Following lectures is not compulsory but strongly advised. Students are encouraged to actively participate in classes. On occassions it will be possible to use flipped teaching techniques, proposing case studies to students and only afterwards developing the thoeries needed to understand them. |
| Other information | Office hours will be fixed at the start of lectures. Disabled students and/or students with special needs are encouraged to contact me to discuss further issues. See also: http://www.unipg.it/disabilita-e-dsa |
| Learning verification modality | Oral discussion on the program (less than 90 minutes). The student is expected to be able to prove in details all the therems that were met in classes, filling in all details that were left to the audience. In evaluting the discussion also clairty of the exposition, precision in notations and property of lmathematical terms will play a role. Also the student is expected to be able to connect the material here presented with basic material studied before and to evaluta at the light of new concepts the content of previous courses. |
| Extended program | Categories. Functors and natural functors. Representable functors and Yoneda's lemma. Equivalence of categories. Adjoint functors. The category of (pointed) topological spaces. Compact spaces, Stone-Cech compactification. Locall compactness. Basic homotopy theory. Fundamental groupoid and group. Computation techniques. Coverings. Simplicial and singular homology. Computation techniques. Major topological results that can be easily be proven with homology. |