Ma 191 c: Noncommutative Geometry, Part II
Spring 2010, Caltech Math Department
Instructor:
Matilde Marcolli
Brief Course Description
This will be a course with focus on
the interplay between Noncommutative
Geometry and Arithmetic Geometry,
with special emphasis on the use of
tools from quantum statistical mechanics
and quantum field theory in number
theoretic settings.
Prerequisites
Attending the previous term 140b will
be useful but not strictly necessary:
this course will use techniques from
noncommutative geometry that will be
covered in the previous class but a
good part of it will be self contained.
Syllabus
- The Bost-Connes system: quantum statistical
mechanics and class field theory, analytic and
algebraic endomotives, Q-lattices modulo
commensurability and scaling, the dual system
and the Riemann zeta function, the Bost-Connes
system and geometry over the "field with
one element".
- Noncommutative Geometry and modular forms:
limiting modular symbols and modular shadows,
Connes-Moscovici modular Hecke algebras,
2-dimensional Q-lattices, complex multiplication
and quantum statistical mechanics, function
field case.
- The noncommutative geometry approach to the
Riemann zeta function: the scaling Hamiltonian
and prolate spheroidal wave functions, the
Weil explicit formula as a trace formula, the
Weil proof and the noncommutative geometry of
the adele class space.