Mathematics

KIERAN GREGORY O'GRADY
KIERAN GREGORY O'GRADY

General objectives: to acquire basic knowledge in general topology, with an introduction to
algebraic topology and differential geometry.

Specific objectives:

Knowledge and understanding: At the end of the course the student will have acquired the concepts and the results
basic general topology, with various possible approaches to the notions of topological space,
continuous application, homeomorphism; then constructions of topologies on subspaces, products and
quotients, topological properties of separation, numerability, compactness, and connection
connection for arches. The student will also have acquired the notion of fundamental group and the its use together with the relevant calculation techniques, and the fundamental elements of the theory of topological coatings. Finally, the student will have acquired the basics of geometry differential of curves and surfaces in three-dimensional Euclidean space.

Apply knowledge and understanding: At the end of the course the student will be able to solve
simple topology problems, even with the use of elementary algebraic topology. He will also know use the notion of curvature in the contexts of the differential geometry of the curves and of the surfaces.

Critical and judgmental skills: The student will have the basis for analyzing the similarities and relations between
the topics covered and the fundamental notions of the theory of continuity and differentiability,
also with tools that have historically led to the solution of classical problems.

Communication skills: Ability to expose the contents in the oral part of the verification and in the any theoretical questions present in the written test.

Learning ability: The acquired knowledge will allow a study, individual or given in a subsequent three-year or master's degree course, related to more advanced aspects of geometry.

Students will be introduced to basic notions and results of General Topologiy and to some basic results of Algebraic Topology.

Geometria I. Analisi Reale.

Topological spaces and continuous maps between topological spaces. Bases, products, subspaces, Hausdorff topological spaces. Connected omponents. Compactness. Topological manifolds. Quotient spaces . Countability axioms. Homotopy. Fundamental group and homotopy groups. The fundamental group of a circle. Van Kampen's Theorem. Coverings and fundamental group. Differentiable manifolds and differenziable maps. Tangent space and differential of a differentiable map. Sard.'s Theorem. Mod 2 degree of a differentiable map. Partitions of 1. Whitney's approximation Theorems. Brouwer 's Fixed point Theorems and Brouwer 's Invariance of domain Theorem.

Allen Hatcher: Algebraic topology. CUP, Cambridge, 2002, https://pi.math.cornell.edu/~hatcher/AT/AT.pdf

John W. Milnor,: Topology from the differentiable viewpoint. University Press of Virginia, Charlottesville, VA, 1965.

John M. Lee: Introduction to smooth manifolds.
Springer GTM 218, 2013.

The teacher explains, with the aid of the blackboard, trying to stimulating active participation by the students.

Not compulsory.

Midterms. Written and oral exam.

Prove that the product of two compact topological spaces is compact.