Mathematics

General aims of the course: acquire basic notions of algebraic topology. Detailed aims of the course: Notions and comprehension: in this course the student will acquire notions and basic results about homology and cohomology and about the theory of characteristic classes of vector bundles. Applying acquired notions: in this course the student will learn how to solve simple exercises using techniques from homotopy theory, homology and cohomology and algebraic structures related to them; such ability will be verified in the written test. Critical thinking abilities: in this course the student will discover analogies between the treated topics and other topics in topology (seen in Geometria 1-2), differential geometry (seen in Geometria 2, Geometira differenziale, Istituzioni di geometria superiore). Moreover, the student will learn tools that lead to the solution of some classical problems. Communication abilities: the student will learn how to comunicate the acquired mathematical content; such abilities will be verified during the oral examination and possibly in some of the theoretical questions in the written test. Learning abilities: the acquired knowledge will allow the student to pursue a deeper investigation of the topology of differentiable manifolds and algebraic varieties.