Ma 191 c: Quantum Statistical Mechanics and Noncommutative Geometry in Number Theory
Spring 2016, Caltech Math Department, Tuesday-Thursday 1:00pm-2:25pm SLN159,
Brief Course Description
This class will present methods of Quantum Statistical Mechanics,
Operator Algebras, and Noncommutative Geometry applied to the
study of zeta and L-functions in Number Theory.
Suggested readings: books
- M.Marcolli, "Arithmetic Noncommutative Geometry"
American Mathematical Society, 2005.
- A.Connes and M.Marcolli, "Noncommutative Geometry,
Quantum Fields, and Motives", American Mathematical Society, 2008.
- A.Consani, M.Marcolli (Eds.) "Noncommutative Geometry and
Number Theory" Vieweg Verlag, 2006.
- M. van Frankenhuijsen, "The Riemann Hypothesis for Function Fields",
Cambride University Press, 2014.
- S.J. Patterson, "An Introduction to the Theory of the Riemann
Zeta-Function", Cambridge University Press, 1995.
Suggested readings: papers
- Matilde Marcolli, Goncalo Tabuada, "Bost-Connes systems,
Categorification, Quantum statistical mechanics, and Weil numbers",
- Alain Connes, Caterina Consani, Matilde Marcolli, "Noncommutative
geometry and motives: the thermodynamics of endomotives",
- Alain Connes, Caterina Consani, Matilde Marcolli, "The Weil proof
and the geometry of the adeles class space", arXiv:math/0703392
- Caterina Consani, Matilde Marcolli, "Quantum statistical mechanics
over function fields", arXiv:math/0607363
- Alain Connes, Matilde Marcolli, Niranjan Ramachandran,
"KMS states and complex multiplication", arXiv:math/0501424
- Marcelo Laca, Nadia S. Larsen, Sergey Neshveyev, "On Bost-Connes
type systems for number fields", arXiv:0710.3452
- Matilde Marcolli, "Cyclotomy and endomotives", arXiv:0901.3167
- Alain Connes, Caterina Consani, Matilde Marcolli, "Fun with F1",
- Javier Lopez-Pena and Oliver Lorscheid, "Torified varieties
and their geometries over F1", arXiv:0903.2173
- Yuri I. Manin, Matilde Marcolli, "Moduli Operad over F1",
- Gunther Cornelissen, Matilde Marcolli, "Zeta functions that
hear the shape of a Riemann surface", arXiv:0708.0500
- Gunther Cornelissen, Matilde Marcolli, "Quantum Statistical
Mechanics, L-series and Anabelian Geometry", arXiv:1009.0736
- A.Carey, M.Marcolli, A.Rennie, "Modular Index Invariants
of Mumford Curves", arXiv:0905.3157
- Matilde Marcolli, "Solvmanifolds and noncommutative tori
with real multiplication", arXiv:0711.2036
- Yuri Manin, Matilde Marcolli, "Modular shadows and the
Levy-Mellin infinity-adic transform", arXiv:math/0703718
Suggested readings for student presentations
More will be added soon
- Gunther Cornelissen, "Curves, dynamical systems and weighted
point counting", arXiv:1209.5782
- Jeffrey C. Lagarias, Sergey Neshveyev,
"Ergodicity of the action of K* on A_K" arXiv:1211.3256
- Eugene Ha and Frederic Paugam, "Bost-Connes-Marcolli systems
for Shimura varieties. I. Definitions and formal analytic properties",
- Sergey Neshveyev, Simen Rustad, "Bost-Connes systems
associated with function fields", arXiv:1112.5826
- Benoit Jacob, "Bost-Connes type systems for function fields",
- Alain Connes, Caterina Consani, "On the arithmetic of the BC-system",
- Marcelo Laca, Nadia S. Larsen, Sergey Neshveyev,
"Hecke algebras of semidirect products and the finite part of
the Connes-Marcolli C*-algebra", arXiv:math/0609316
- Marcelo Laca, Nadia Larsen, Sergey Neshveyev,
"Phase transition in the Connes-Marcolli GL2-system",
- Joachim Cuntz, Xin Li,
"C*-algebras associated with integral domains and crossed
products by actions on adele spaces", arXiv:0906.4903
Additional reading material
- Matilde Marcolli, "Noncommutative Geometry and Arithmetic",
- Mark Greenfield, Matilde Marcolli, Kevin Teh, "Type III sigma-spectral
triples and quantum statistical mechanical systems", arXiv:1305.5492
- Zhaorong Jin, Matilde Marcolli, "Endomotives of toric varieties",
- Javier Lopez-Pena and Oliver Lorscheid, "Mapping F1-land"
- Catharine Wing Kwan Lo, Matilde Marcolli, "F-zeta geometry, Tate
motives, and the Habiro ring", arXiv:1310.2261
- Jake Marcinek, Matilde Marcolli, "KMS weights on higher rank buildings",
- Gunther Cornelissen, Matilde Marcolli, Kamran Reihani, Alina Vdovina
"Noncommutative geometry on trees and buildings", arXiv:math/0604114
- Matilde Marcolli, Yujie Xu, "Quantum statistical mechanics
in arithmetic topology", arXiv:1602.04890
- Dori Bejleri, Matilde Marcolli, "Quantum field theory over F1",
Notes of classes
Notes will be posted here.
The class times of Tuesday May 31 and Thursday June 2 will be
used for student prasentations.
- Tuesday March 29: Quantum statistical mechanical
systems and twisted spectral triples
- Thursday March 31: Quantum statistical mechanical systems:
observables, time evolution, KMS states, symmetries by automorphisms
- Tuesday April 5: The Bost-Connes system: main properties
- Thursday April 7: The Bost-Connes system: Hecke algebra, KMS states,
- Tuesday April 12: Riemann gas and twisted spectral triples: Bosonic
and Fermionic Fock spaces
- Thursday April 14: Quantum statistical mechanics of 1-dimensional
Q-lattices and the Bost-Connes system
- Tuesday April 19: Quantum statistical mechanics of 2-dimensional
Q-lattices: groupoid algebras and Hecke convolution algebras, time evolution
- Thursday April 21: Quantum statistical mechanics of 2-dimensional
Q-lattices: symmetries by automorphisms and endomorphism
- Tuesday April 26: The modular field and the arithmetic algebra, zero
temeperature KMS states and automorphisms of the modular field
- Thursday April 28: Classification of KMS states of the QSM of
- Tuesday May 3: Quantum statistical mechanics of imaginary
quadratic fields, abelian extensions and Galois theory
- Thursday May 5: Dynamical systems and class field theory:
dynamical system and quantum statistical mechanical system
- Tuesday May 10: Dynamical systems and class field theory: orbit
equivalence, conjugacy, and reciprocity isomorphism
- Thursday May 12: Dynamical systems and class field theory: reciprocity
isomorphism and L-function isomorphism
- Tuesday May 17:
- Thursday May 19:
- Tuesday May 24: CLASS CANCELLED (conference travel)
- Thursday May 26: Student presentations
- Tuesday May 31: Student presentations
- Thursday June 2: Student presentations
The course is offered Pass/Fail. Requirements: one oral
presentation on reading materials chosen from the list
provided above (first come first serve) and attendance
of (most) lectures.
Assigned student presentations:
Assigned persentations and presentation schedule will be listed here.
- Thursday May 26: Jim Tao, "Bost Connes systems for number fields"
- Tuesday May 31: John Clark, "Curves, dynamical systems, and weighted point counting"
- Thursday Jun 2: Marcus Bintz, "Endomotives of Toric Varieties"