The seminar is dedicated to a topic that connects Galois theory, simple (non-commutative) algebras and algebraic geometry. A central simple algebra over a field k is a finite-dimensional associative algebra over k with the center equal to k and that has no non-trivial two-sided ideals. Examples of these are Hamilton quaternions over real numbers and matrix algebras over any base field. One can use tensor multiplication over k to define product of such algebras and form a monoid. After imposing what's known as Morita equivalence one obtains the Brauer group of k. A Severi-Brauer variety over k is a smooth projective variety X such that after base change to an algebraic closure of k it becomes isomorphic to a projective space. For example, a conic (i.e. a smooth projective curve of degree 2 in the projective plane) is the main of example of a Severi-Brauer variety of dimension 1. And finally from the point of view of Galois theory we will be interested in second Galois cohomology, i.e. the cohomology of the absolute Galois group of the base field k. Note that when studying this, one forgets the field k itself and works just with the absolute Galois group. It will be our goal to understand that there is a one-to-one (if properly explained) correspondence between objects defined above. This opens up a possibility of using methods of one area to the other: for example, of understanding conics via quaternion algebras, or of using cohomological techniques for a better understanding of algebraic geometry of certain varieties. Despite broad scope of the seminar, it should be accessible for students who have basic knowledge of algebra, Galois theory and some acquaintance with algebraic geometry. At least in the beginning of the seminar we will follow the book by Gille and Szamuely (see the list of the references). Moreover, the schedule could be slightly adapted along the way depending on the prerequisites of the students.

The focus of this course will be on the homotopy category of topological spaces and various tools that allow to study it. In particular, we will introduce higher homotopy groups (where higher refers to the fundamental group being the first homotopy group) and 2-categorical tools that allow to apply (2-)categorical constructions for the study of the homotopy world. The abstract nonsense part of the course will be complemented by the geometric introduction into vector bundles and Serre spectral sequence. The latter is an effective tool for the computation of various (co)homology groups.

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