Learning objectives
The object of the course is to familiarize the students with the basic language and of some fundamental theorems in General Topology and in the theory of holomorphic finctions of one complex variable.
In particular,
1) the notions of topological spaces, subspaces, product and quotient topology, compact, connected and path-connected spaces, metric spaces and topological groups di spazio metrico will be provided;
2) the basic properties and facts of holomorphic finctions of one complex variable, as Cauchy-Riemann equations, Cauchy Theorem and Cauchy Formula, analytic continuation, Laurent series and residue theorem will be discussed.
Prerequisites
Geometry 1, Analysis 1, Algebra.
Course unit content
Topology and introduction to algebraic topology.
Full programme
Topological spaces. Metric spaces. Continuos maps, homeomrphisms. Closure, interior adn boundary of a set. Neighbourhoods and basis of neighbourhoods. Countable and separation axioms. Topological subspaces. Product topology. Quotient topology. Compact, connected and path-connected spaces. Tychonoff Theorem. Locally connected spaces. Metric spaces. Baire's Lemma. Extension of continuos map. Topological groups. Matrices groups.
Homotopy. Fundamental group of a topological space. Fundamental group and continuos maps. Fundamental group of the circle. Seifert and Van Kampen Theorems.
Spazi topologici. Spazi metrici. Applicazioni continue, omeomorfismi. Chiusura, parte interna, frontiera. Basi di aperti, sistema fondamentale di intorni, assiomi di numerabilita'. Sottospazi. Topologia prodotto. Topologia quoziente. Assiomi di separazione. Quasi compattezza e compattezza. Connessione e connessione per archi. Spazi localmente connessi. Spazi metrici completi. Estensioni di funzioni continue. Metrica della convergenza uniforme. Gruppi topologici. Gruppi di matrici.
Funzioni olomorfe di una variabile complessa. Le condizioni di Cauchy-Riemann. Il Teorema di Cauchy. La formula di Cauchy. Disuguaglianze di Cauchy. Serie di potenze. Analiticita' delle funzioni olomorfe. Il teorema di Liouville. Il teorema fondamentale dell'algebra. Sviluppo in serie di Laurent. Il teorema dei residui.
Bibliography
V. Checcucci, A. Tognoli, E. Vesentini, Lezioni di topologia generale, ed. - Milano, Feltrinelli, 1977. 242 pp.
E. Sernesi, Geometria 2, Torino Bollati Boringhieri, 1994.
V. Villani, Elementi di topologia generale
Teaching methods
Lectures and classroom exercises.
During lectures in traditional mode, the
topics will be formally and rigorously presented. The course will give particular emphasis to application and computation aspects, while not letting out
the theoretical aspect. The classroom exercises are aimed at showing how and where the abstract results can be applied to make the students understand better the relevance of what they are studying.
During the lectures the basic results of calculus for functions of one variable will be analyzed and discussed.Students will be provided also with the basic results on sequences of real numbers and numerical series. The classroom exercises are aimed at showing how and where the abstract results can be applied to make the students understand better the relevance of what they are studying.
Assessment methods and criteria
The exam consists of a written part and an oral part in different dates.
Other information
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2030 agenda goals for sustainable development
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