Program Overview

Program Overview

The university program in question is "Ma 148: Topics in Mathematical Physics - Number Theory and Physics," offered by the Caltech Math Department during Winter 2024. The course takes place on Tuesdays and Thursdays from 9:00 to 10:30 am in Linde 187.

Brief Course Description

This class explores the interplay between Number Theory and Physics, including topics such as quantum statistical mechanics, number fields, modular curves, and modular forms. The course is graded pass/fail, with grades based on attendance, participation, and completion of assigned readings or projects.

Course Structure

• The course covers various aspects of 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

Lecture Summaries

A summary of each lecture is provided, covering topics such as:

• Introduction to modular forms
• Order of vanishing of modular forms and counting of zeros
• Dimension estimate for the space of SL(2,Z) modular forms of weight k
• Eisenstein series
• Discriminant, cusp forms, and continued fractions algorithm for GL(2,Z)
• Modular symbols and continued fractions
• Periods of cusp forms
• Irrational boundary of modular curves, bad quotients, and crossed product algebras
• Limiting modular symbols and continued fractions
• Fixed point theorem in positive cones in Banach spaces and invariant measure of the continued fraction expansion
• Limits of limiting modular symbols, mixmaster cosmology models, and continued fractions and geodesics on modular curves
• Modular complex and modular symbols, homology of modular curves
• Boundary crossed product algebra and K-theory, modular complex from K-theory of the boundary algebra
• Pimsner exact sequence for actions on trees
• Quantum statistical mechanics, observables, and time evolution, symmetries, states, and equilibrium states
• Gibbs states and KMS states, ground states, action of symmetries, algebra of the Bost-Connes system
• Time evolution and KMS states of the Bost-Connes system, quantum statistical mechanics, and the explicit class field theory problem
• Bost-Connes system and 1-dimensional Q-lattices, arithmetic algebra Eisenstein series, and trigonometric functions
• 2-dimensional Q-lattices, groupoids, quotients, and convolution algebra

Book References

Useful references for the course include:

• "Arithmetic Noncommutative Geometry" by Matilde Marcolli
• "Noncommutative Geometry, Quantum Fields and Motives" by Alain Connes and Matilde Marcolli
• "From Number Theory to Physics" by Michel Waldschmidt and Pierre Cartier
• "Physics and Number Theory" by Louise Nyssen
• "Frontiers in Number Theory, Physics, and Geometry" (2 volumes) by Pierre Cartier et al.

Other Reading Material

Additional suggested reading material includes:

• Elliptic modular forms
• Notes on modular forms
• Continued fractions, modular symbols, and noncommutative geometry
• Parabolic points and zeta functions of modular curves
• Limiting modular symbols and the Lyapunov spectrum
• Limiting modular symbols and their fractal geometry
• Period functions and the Selberg zeta function for the modular group
• Modular shadows and the Levy-Mellin infinity-adic transform
• Higher-weight limiting modular symbols
• Iterated integrals of modular forms and noncommutative modular symbols
• Remarks on modular symbols for Maass wave forms
• Iterated Shimura integrals
• Noncommutative generalized Dedekind symbols
• Modular forms of real weights and generalized Dedekind symbols
• Local zeta factors and geometries below Spec(Z)
• Zeta-polynomials for modular form periods
• Quantum statistical mechanics at the boundary of modular curves
• Quantum statistical mechanics of Q-lattices
• KMS states and complex multiplication
• Bost-Connes type systems and complex multiplication
• Bost-Connes-Marcolli system for the Siegel modular variety
• Bost-Connes type systems for number fields
• Arithmetic models and functoriality of Bost-Connes systems
• Noncommutative Geometry and Motives: The Thermodynamics of Endomotives
• The Weil Proof and the Geometry of the Adeles Class Space
• Bost-Connes systems, Categorification, Quantum Statistical Mechanics, and Weil Numbers
• Characterization of global fields by Dirichlet L-series
• Reconstructing global fields from dynamics in the abelianized Galois group
• q-deformations of statistical mechanical systems and motives over finite fields
• Endomotives of toric varieties
• Quantum statistical mechanics over function fields
• Prolate spheroidal operator and zeta
• The scaling Hamiltonian
• Weil positivity and trace formula - the archimedean place
• Quantum modular forms
• Mock theta functions and quantum modular forms
• Harmonic Maass forms, mock modular forms, and quantum modular forms
• Perspectives on mock modular forms
• (Mock) modular forms in string theory and moonshine
• Black holes and modular forms in string theory
• SIC-POVMs and the Stark conjectures
• Moment maps and Galois orbits in quantum information theory
• Generating Ray class fields of real quadratic fields via complex equiangular lines

Schedule of Presentations

The schedule for student presentations includes:

• Tuesday, March 12: Elizabeth (Endomotives and toric varieties), George (Lyapunov exponents)
• Thursday, March 14: Laura (SIC-POVMs and Stark conjectures), Khyathi (quintic periods and mock modular forms), Cameron (3-manifold quantum invariants and mock theta functions), Samuel (modular symbol and continued fractions in higher dimensions), Jonah (quantum modular forms and mock modular forms), Pedro (black holes and modular forms), Michael (TBA)