# Algebra and Geometry track

Contact person: Dimitri Wyss

Below we explain the structure of the Algebra and Geometry track. Not all courses are given each year. If you decide to focus on the Algebra and Geometry track, we suggest that you take MOST of the courses below during the third year of your bachelors and during your masters. You have the freedom how to interpret "most", but we suggest you interpret it based on your mathematical taste. That is, you take all courses that are in topics that you like the most. It is very important that you create a coherent set of courses for yourself. That is, you learn certain aspects of pure math deeply, in a way that the courses strengthen each other. In particular, you should be careful to take all prerequisite courses to the courses that you will want to take later on. You can have a rough idea of the dependencies of the courses by looking at the table below. The columns of this table are arranged based on "prerequisite channels" (the general courses, up to BA 5, are for all channels). We also remark that a big part of pure math is analysis, so for almost all of you it might be a good idea to complement the offering below with some courses from the analysis track.

The main areas present in the Algebra and Geometry track (in alphabetical order) are:

• algebraic geometry
• differential geometry
• group theory & representation theory (written only as group theory in the table below)
• number theory
• topology

Many of the courses of the track are interdisciplinary, so it is hard to assign them to only one of the above areas. So, the way couses are assigned in the following table to columns is only a guideline. Also, some of the courses listed in the table are not given every year. Such courses are typically not essential prerequisites for other courses. Discounting these discrepancies, the table below can be used well to understand a first approximation of the "prerequisite channels" of the track.

 algebraic geometry differential geometry group theory number theory topology BA 1 (joint) Structures algébriques Linear Algebra I BA 2 (joint) Linear algebra II BA 3 (joint) Théorie des groupes Espaces métriques et topologiques Géométrie différentielle BA 4 (joint) Anneaux et corps Topologie BA 5 (joint) Algebra IV - Rings and modules Algebra V - Galois theory BA5 Differential geometry II Smooth manifolds Number theory I.a Algebraic number theory Number theory I.b Analytic number theory Number theory 1.c Combinatorial number theory BA6 Algebraic geometry I Curves Differential Geometry III Riemannian geometry Representation theory Topology III Homology Topological groups MA1 (joint) Student seminar in pure mathematics Riemann surfaces MA1 Algebraic geometry II Schemes and sheaves Differential Geometry IV Introduction to general relativity Lie groups and algebras Ergodic theory Topology IV.a Algebraic K-theory MA2 Algebraic geometry III Selected topics Abstract analysis on groups Number theory II.a Modular forms Topology IV.b Homotopy theory Quantum (loop) groups and quivers Number theory II.b Cryptography

For more information please click below on the name of the research group that you are most interested in (names of the groups in alphabetical order):

 Name of the group & link Name of the professor Algebraic Geometry Prof. Zsolt Patakfalvi Analytic Number Theory Prof. Philippe Michel Arithmetic Geometry Prof. Dimitri Stelio Wyss Differential Geometry Prof. Marc Troyanov Ergodic Theory Prof. Florian Richter Ergodic and Geometric Group Theory Prof. Nicolas Monod Number Theory Prof. Maryna Viazovska Reductive Groups Prof. Donna Testermann Topology Prof. Kahtryn Hess Bellwald General Relativity Prof. Georgios Moschidis

General description :

A big part of today's theoretical mathematical research has some algebraic or geometric flavor to it. Examples are endless: number theory, representation theory, differential geometry, algebraic geometry, arithmetic geometry, homotopy theory, geometric group theory, low dimensional topology, symplectic geometry, algebraic combinatorics, combinatorial geometry, and others, including many different combinations of the above listed fields. This track gathers some of the topics that are important to study for most of these fields. However, it is important to stress that for many of the above areas it is similarly essential to take some of the analysis courses. For that please consult the links above where the research groups at EPFL list the courses suggested to take to be able to conduct research in the given group.

Related Minors :

• Analysis track
• Algorithmic and Dirscrete Mathematics track
• Probability track
• This wiki