Ma 148: Topics in Mathematical Physics - Number Theory and Physics
Winter 2024: Caltech Math Department, Tuesday-Thursday 9:00-10:30 am, Linde 187
Brief Course Description
This class will present various instances of the rich interplay between Number Theory and Physics, including quantum statistical mechanics and number fields, modular curves and modular forms, physics related aspects of the Riemann zeta function, mock and quantum modular forms in physics
The class is graded P/F, the grade is assigned on the basis of attendance/participation and completion of an assigned reading/presentation or project individually assigned by the instructor.
Notes
Notes of the lectures will be posted here
- pdf
Notes on modular forms
- pdf Notes on modular symbols
- pdf Notes on quantum statistical mechanics and the Bost-Connes system
Summary of Lectures
A short summary of each lecture will be posted here
- Thursday January 4: Introduction to modular forms, functions on lattices, modular functions, modular forms, Fourier expansion, fundamental domain and modular curve
- Tuesday January 9: order of vanishing of modular forms and counting of zeros
- Thursday January 11: dimension estimate for the space of SL(2,Z) modular forms
of weight k, Eisenstein series
- Tuesday January 16: disciminant, cusp forms, continued fractions algorithm for GL(2,Z)
- Thursday January 18: modular symbols and continued fraction, periods of cusp forms
- Tuesday January 23: irrational boundary of modular curves, bad quotients and crossed product algebras, limiting modular symbols and continued fractions
- Thursday January 25: Fixed point theorem in positive cones in Banach spaces and invariant measure of the continued fraction expansion, limits of limiting modular symbols
- Tuesday January 30: limits of limiting modular symbols, mixmaster cosmology models, continued fractions and geodesics on modular curves
- Thursday February 1: modular complex and modular symbols, homology of modular curves
- Tuesday Fabruary 6: boundary crossed product algebra and K-theory, modular complex from K-theory of the boundary algebra, Pimsner exact sequence for actions on trees
- Thursday February 8: Quantum statistical mechanics, observables and time evolution, symmetries (automorphisms and endomorphisms), states and equilibrium states
- Tuesday February 13: Gibbs states and KMS states, ground states, action of symmetries,
algebra of the Bost-Connes system
- Thursday February 15: time evolutiona and KMS states of the Bost-Connes system, quantum statistical mechanics and the explicit class field teory problem
- Tuesday February 20: Bost-Connes system and 1-dimensional Q-lattices, arithmetic algebra Eisentein series and trigonometric functions
- Thursday February 22: 2-dimensional Q-lattices groupoids quotients, convolution algebra
- Tuesday February 27:
- Thursday February 29:
- Tuesday March 5:
- Thursday March 7:
Some book references
There is no official textbook for this class, but the following books
will be useful references.
All these books should be available in the Caltech library.
If you have any difficulty locating them contact me.
- Matilde Marcolli, "Arithmetic Noncommutative Geometry"
- Alain Connes, Matilde Marcolli, "Noncommutative Geometry, Quantum Fields and Motives"
- Michel Waldschmidt, Pierre Cartier, "From Number Theory to Physics"
- Louise Nyssen, "Physics and Number Theory"
- Pierre Cartier et al. "Frontiers in Number Theory, Physics, and Geometry" (2 volumes)
Other Reading Material
Suggested reading material including both material covered in the
lectures and possible suggestions for student presentations (more will
be added as the class progresses)
- pdf Elliptic modular forms
- pdf Notes on modular forms
- pdf Continued fractions, modular symbols, and noncommutative geometry
- pdf Parabolic points and zeta functions of modular curves
- pdf Limiting modular symbols and the Lyapunov spectrum
- pdf Limiting modular symbols and their fractal geometry
- pdf Period functions and the Selberg zeta function for the
modular group
- pdf Modular shadows and the Levy-Mellin infinity-adic transform
- pdf Higher-weight limiting modular symbols
- pdf Iterated integrals of modular forms and noncommutative modular symbols
- pdf Remarks on modular symbols for Maass wave forms
- pdf Iterated Shimura integrals
- pdf Nonommutative generalized Dedekind symbols
- pdf Modular forms of real weights and generalized Dedekind symbols
- pdf Local zeta factors and geometries below Spec(Z)
- pdf zeta-polynomials for modular form periods
- pdf Quantum statistical mechanics at the boundary of modular curves
- pdf Quantum statistical mechanics of Q-lattices
- pdf KMS states and complex multiplication
- pdf Bost-Connes type systems and complex multiplication
- pdf Bost-Connes-Marcolli system for the Siegel modular variety
- pdf Bost-Connes type systems for number fields
- pdf Arithmetic models and functoriality of Bost-Connes systems
- pdf Noncommutative Geometry and Motives: The Thermodynamics of Endomotives
- pdf The Weil Proof and the Geometry of the Adeles Class Space
- pdf Bost-Connes systems, Categorification, Quantum Statistical Mechanics, and Weil Numbers
- pdf Characterization of global fields by Dirichlet L-series
- pdf Reconstructing global fields from dynamics in the abelianized Galois group
- pdf q-deformations of statistical mechanical systems and motives over finite fields
- pdf Endomotives of toric varieties
- pdf Quantum statistical mechanics over function fields
- pdf Prolate spheroidal operator and zeta
- pdf The scaling Hamiltonian
- pdf Weil positivity and trace formula - the archimedean place
- pdf Quantum modular forms
- pdf Mock theta functions and quantum modular forms
- pdf Harmonic Maass forms, mock modular forms and quantum modular forms
- pdf Perspectives on mock modular forms
- pdf (Mock) modular forms in string theory and moonshine
- pdf Black holes and modular forms in string theory
- pdf SIC-POVMs and the Stark conjectures
- pdf Moment maps and Galois orbits in quantum information theory
- pdf Generating Ray class fields of real quadratic fields via complex equiangular lines
Schedule of Presentations:
- Tuesday March 12: Elizabeth, Endomotives and toric varieties
- Tuesday March 12: George, Lyapunov exponents
- Thursday March 14: Laura, SIC-POVMs and Stark conjectures
- Thursday March 14: Khyathi, quintic periods and mock modular forms
- Thursday March 14: Cameron, 3 manifold quantum invariants and mock theta functions
- Thursday March 14: Samuel, modular symbol and continued fractions in higher dimensions
- Thursday March 14: Jonah, quantum modular forms and mock modular forms
- Thursday March 14: Pedro, black holes and modular forms
- Thursday March 14: Michael, TBA
Links to Papers of Assigned Student Presentations:
- pdf What are Lyapunov exponents and why are they interesting?