Methods of Mathematical Physics, Part II


M393C # 59240
Fall 2025


Instructor: Thomas Chen
Email

Office: PMA 12.138

Lectures: TTH 12:30 - 2:00 PM.
Location: PMA 11.176

Office hours: TBA via Zoom
(Link will appear in Canvas)




Syllabus


The purpose of this graduate course is to provide an introduction to spectral and renormalization group (RG) methods in Quantum Mechanics and Quantum Field Theory, with connections and applications to neighboring research areas. Specific topics tentatively include stability of matter, RG in fluid equations and deep learning. No background in physics is required, but some preparation in Analysis/PDE is useful.
Familiarity with the material of Part I of this course is useful but not necessary.

There will be no HW and exams, but attendance is expected.

Updated course information will be posted here and on Canvas.

Topics


This list of topics is tentative and will be modified frequently.
  1. Quantum manybody systems.
    Spectral theory, Schrodinger equations, spectrum and dynamics.
    Ground states, kinetic energy estimates, Lieb-Thirring estimates.
    Stability of matter.

  2. Renormalization.
    Basics of Quantum Field Theory.
    Perturbative renormalization, Feynman graphs, renormalization group.
    Nonperturbative renormalization, renormalization group analysis of spectral problems in QFT.
    Renormalization group analysis of turbulence in fluids.

  3. Applications in Deep Learning
    Underparametrized DL networks, dynamical complexity reduction, renormalization.
    Overparametrized DL networks, sufficient conditions for reachability of zero loss.

Texts


These are some recommended texts.
  1. V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer.
  2. R. Abraham, J.E. Marsden, Foundations of Mechanics, AMS.
  3. T. Cazenave, Semilinear Schrodinger Equations (Courant Lecture Notes), AMS.
  4. L.C. Evans, Partial differential equations, AMS.
  5. S. Gustafson, I.M. Sigal, Mathematical concepts of Quantum Mechanics, Springer.
  6. J. Glimm, A. Jaffe, Quantum Physics from a functional integral point of view, Springer.
  7. R. Haag, Local quantum physics: Fields, particles, algebras. Springer.
  8. T. Kato, Perturbation Theory for Linear Operators, Springer.
  9. L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, Elsevier.
  10. E.H. Lieb, M. Loss, Analysis, AMS.
  11. E.H. Lieb, R. Seiringer, The stability of matter in Quantum Mechanics, Cambridge.
  12. J. Moser, E.J. Zehnder, Notes on Dynamical Systems (Courant Lecture Notes), AMS.
  13. C. Muscalu, W. Schlag, Classical and multilinear harmonic analysis, Vol. 1, Cambridge.
  14. M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vols. 1 - 4, Academic Press.
  15. E. Stein, Harmonic Analysis, Princeton University Press.
  16. T. Tao, Nonlinear dispersive equations, AMS.
  17. G. Teschl, Mathematical Methods in Quantum Mechanics, AMS.
  18. T. Wolff, Lectures on harmonic analysis, AMS.
Online resources:
  1. Lecture notes by T. Arbogast and J. Bona for Methods of Applied Mathematics.
  2. Lecture notes by W. Schlag on Harmonic Analysis.
  3. G. Teschl's book is available for download here.
  4. Lecture notes by T. Wolff on Harmonic Analysis.
If you would like to prepare ahead over the summer break, the following will be useful: Topics in Functional Analysis, including distributions, Hilbert spaces, spectral theory of selfadjoint operators (book by Teschl, lecture notes by Arbogast-Bona). Methods of Harmonic Analysis (lecture notes by Schlag, Wolff).

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities.
For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.