Topics in Topology and Geometry
We introduce basic concepts of algebraic topology, such as the fundamental group (together with the Van Kampen theorem)and singular homology (together with the Mayer-Vietoris long exact sequence). We also review basic notions of homological algebra. Fiber bundles and connections on them are discussed, and we define the concept of curvature. As a starting point of Riemannian geometry, we define the Levi-Civita connection and the Riemannian curvature tensor
By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.
Regular homework, and presentation or final
real analysis, linear algebra