Arithmetic Geometry and Noncommutative Geometry

An MRI Master Class 2009-2010

Subject

The past decade has witnessed a lot of interaction between number theory, arithmetic geometry and noncommutative geometry. To name but a few examples: Manin’s real multiplication programme, Consani and Marcolli’s recast of Arakelov geometry in terms of spectral triples and Connes and Marcolli’s reformulation of explicit class field theory in a noncommutative thermodynamical formalism. One observes how these examples each nicely hook noncommutative geometry to one of the significant trends in modern number theory and geometry.

At the same time, noncommutative geometry is further extending its (controversial?) scope in mathematical physics, more precisely in renormalisation (explaining the occurrence of such number theoretical gadgets as Tate motives therein) and the standard model.

The master class will lead students to the forefront of this research field. The structure is to offer courses on basic noncommutative geometry in connection and interaction with a variety of background material from `traditional’ mathematics.

The first semester has an introductory programme. The basics of C*-algebras are taught, with special focus on the examples that occur in number theory (such as crossed product algebras, Cuntz-Krieger algebras). There is a course on analysis of pseudo-differential operators on manifolds (with an eye towards Dirac operators). A student seminar is devoted to uniformisation of Riemann surfaces and Kleinean groups, and a basic course on ergodic theory culminates in Hopf’s theorem on
ergodicity of geodesic flow.

The second semester starts with a topics course in number theory, half of which is about p-adic numbers, and half of which is on the theory of zeta functions (broadly interpreted). There is a basic course on noncommutative geometry, dealing with some K-theory of operator algebras, spectral triples and noncommutative tori. Finally, there is a student seminar on spectral triples in which some of the original papers will be read. This is complemented by expert lectures (`master class within the master class’) by Consani, Marcolli, Robertson and van Suijlekom.

Programme

Some items have a working link to a more detailed website for the course.

FIRST SEMESTER

Courses start in week 37 (7 Sep) and end in week 51 (16 Dec). No lectures in week 45 (2 Nov). All courses at Utrecht University.

    > Tuesdays 11-13, Minnaert Building room 021 (week 37+38), Maths Building room 611AB (start in week 39)
    Seminar: Fuchsian Groups
    students; coordination: G. Cornelissen, B. Mesland, J. Plazas (Utrecht)

    > Tuesday 15-17, Maths Building room 611AB
    Ergodic Theory – Dynamics on Manifolds
    lecturer: K. Dajani (Utrecht)

    > Wednesday 10-13, BBL Building room 276
    Analysis on Manifolds
    lecturers: E. van den Ban, M. Crainic (Utrecht)
    Note: with a one week intensive reminder of differential geometry in week 2/3

    > Wednesday 14-17, BBL Building room 276
    C*-algebras
    lecturer: Michael Mueger (RU Nijmegen)

Another potentially interesting course, not part of the Masterclass:
   Monday, 13-15, BBL Building, room 516
   Geometry of PDE's
   lecturer: Sergei Igonin

SECOND SEMESTER

Courses are scheduled to start in week 6 (8 Feb), but a Master Class within the Master Class talk will be scheduled somewhere between weeks 2 and 5.

    > Topics in Number Theory: p-adic numbers and zeta functions
    lecturers: R. de Jeu (VU Amsterdam), J. Plazas (Utrecht)

    > Noncommutative Geometry
    lecturers: N.P. Landsman (RU Nijmegen)

    > Seminar: Spectral Triples
    students, coordination: G. Cornelissen, B. Mesland, J. Plazas (Utrecht)

    > Master Classes within the Master Class
        G. Robertson (Newcastle): 1-5 March: Noncommutative geometry of euclidean buildings and their boundaries
        C. Consani (Johns Hopkins): 15-17 March: Around the field with one element
        W. van Suijlekom (RU Nijmegen): 26-29 april: Noncommutative geometry and physics
        M. Marcolli (Caltech): 8-17 June: t.b.a.

Other potentially interesting course, not part of the Masterclass:
   Quantum mechanics for Mathematicians
   lecturer: Benoit Dehrin
   Algebraic Geometry
   lecturer: Eduard Looijenga
   Riemann Surfaces
   lecturer: Gerard van der Geer

Prerequisites

Applicants should have completed at least three years of undergraduate studies in pure mathematics or theoretical and mathematical physics. Students are assumed to know some basic functional analysis (Hilbert space). For students not familiar with the theory of differentiable manifolds, a crash course will be organized before the start of the master class.