Unit TOPOLOGY I

Course
Mathematics
Study-unit Code
55A00101
Curriculum
Didattico-generale
Teacher
Nicola Ciccoli
Teachers
  • Nicola Ciccoli
Hours
  • 42 ore - Nicola Ciccoli
CFU
6
Course Regulation
Coorte 2022
Offered
2022/23
Learning activities
Affine/integrativa
Area
Attività formative affini o integrative
Academic discipline
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.
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