MM854: Graduate seminar

Students taking the course are expected to:

• Have a basic knowledge of topology and functional analysis, corresponding to the contents of the courses MM548 and MM549.
• Be able to use basic arguments from topology.
• Be able to work independently with linear algebra.
• Have a basic knowledge of the theory of groups and rings and be able to work comfortably with these objects, corresponding to the contents of the courses MM570 and MM567.
  

The learning objectives of the course is that the student demonstrates the ability to: 

• reproduce definitions and results, including their proofs, covered in the course.
• be able to use these results to analyse concrete examples.
• select and structure relevant mathematical material within a given topic and present the material to fellow students in a lecture.
• Communicate and explain mathematical insights to fellow students in a clear and well-structured fashion.
• Formulate and present definitions, proofs and calculations in a mathematically rigorous way.

The following main topics are contained in the course:
Introduction to one or more topics in analysis. This could, for example, be:

• Representation theory.
  
• Cohomology theory for groups and/or algebras.
• Introduction to K-theory.
• Important classes of discrete groups.
• Von Neumann algebras.
• Index theory.
• Unbounded operators and their spectral theory.
• Introduction to noncommutative geometry.
• Hilbert C*-modules.
• Quantum groups.
• Symplectic geometry.
• Quantum field theory and classical field theory.
• Algebraic geometry.
• Deformation quantization.
• Geometric quantization.
• Cluster algebras.
• Gauge theory.
• Moduli space theory.
• Homological mirror symmetry.
• Micro local sheaf theory.
  

• The students are expected to familiarize themselves with the material covered in the lectures.
• To acquire knowledge of selected topics independently.
• Select and structure mathematical material and prepare to present this material, in the form of lectures, to fellow students.