MATH-506: Topology IV.a - Cohomology ringsSingular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative algebra. We study an algebraic version, namely group coho ...
MATH-323: Topology III - HomologyHomology is one of the most important tools to study topological spaces and it plays an important role in many fields of mathematics. The aim of this course is to introduce this notion, understand its properties and learn how to compute it. There will be m ...
MATH-410: Riemann surfacesThis course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic ...
MATH-563: Student seminar in pure mathematicsThe goal of the seminar is to learn the main aspects of class field theory, which aims at understanding abelian extensions of global fields in terms of the arithmetic of the field. The theory can also be seenas a first instance of the Langlands corresponde ...
MATH-497: Topology IV.b - homotopy theoryWe propose an introduction to homotopy theory for topological spaces. We define higher homotopy groups and relate them to homology groups. We introduce (co)fibration sequences, loop spaces, and suspensions. We study long exact sequences. We construct Eilen ...
MATH-473: Complex manifoldsThe goal of this course is to help students learn the basic theory of complex manifolds and Hodge theory. ...
MATH-488: Topology IV.b - Algebraic K-theoryAlgebraic K-theory, which to any ring R associates a sequence of groups, can be viewed as a theory of linear algebra over an arbitrary ring. We will study in detail the first two of these groups and some of their applications to other areas of mathematics. ...
MATH-494: Topics in arithmetic geometryP-adic numbers are a number theoretic analogue of the real numbers, which interpolate between arithmetics, analysis and geometry. In this course we study their basic properties and give various applications, notably we will prove rationality of the Weil Ze ...
MATH-310: AlgebraThis is an introduction to modern algebra: groups, rings and fields. ...
MATH-645: Young Topologists Meeting Mini-CoursesWe expect these mini-courses to equip junior researchers with new tools, techniques, and perspectives for attacking a broad range of questions in their own areas of research while also inspiring students to learn more about ground-breaking advances in pure ...
MATH-679: Group schemesThis is a course about group schemes, with an emphasis on structural theorems for algebraic groups (e.g. Barsotti--Chevalley's theorem). All the basics will be covered towards the proof of such theorem, with an estress on the modern presentation using sche ...
