"Math Methods"
11 Mar 2024 21:21
Physics and engineering, and some related fields, have a well-established type of course and book which teaches "mathematical methods" for the field. This means it covers ideas, results and techniques of branches of advanced mathematics which are widely used in the field. (Here "advanced" means something like "beyond basic calculus and linear algebra.") These typical have no pretense of rigor, and lots of examples illustrating the intended applications. (This is not what is usually meant by "applied math", for complicated historical reasons.) I liked these courses as a physics student, and wish we had more of them in statistics. This notebook is for collecting relevant references.
What would go in a "mathematical methods for statistics" class? (Again, I exclude basic calculus and finite-dimensional, vectors-and-matrices linear algebra.) Combinatorics, of course, but what else? My off-the-cuff, unordered list for a course for beginning graduate students would be:
- Fourier transforms
- Convolutions (and their connection to Fourier transforms)
- Laplace transforms (?)
- Rudimentary functional analysis: Hilbert and Banach spaces; linear algebra in infinite-dimensional spaces
- Calculus in function spaces: functional derivatives, calculus of variations
- Measure theory (rudiments of)
- Asymptotic expansions for sums and integrals (Laplace's method, the various series expansions we use)
- Optimization
- Numerical approximation: function approximation, interpolation, approximations of definite integrals
- Information theory
- Graph theory?
I'm sure this list is open to many objections and corrections! If anyone can point me to a math-methods-for-stats book, I'd appreciate it; I very much do not want to write one. (I have too many unfinished book projects as it is.)
- Recommended (very misc.):
- V. I. Arnol'd, Mathematical Methods of Classical Mechanics
- Mary L. Boas, Mathematical Methods in the Physical Sciences
- Frederick W. Byron and Robert W. Fuller, Mathematics of Classical and Quantum Physics
- John C. Neu, course notes for Math 121 at Berkeley [These are from a version of the class almost twenty years after the one I took in the early 1990s, and so the content is somewhat different, but that class was genuinely one of the highlights of my college education]
- Bernard Schutz, Geometrical Methods of Mathematical Physics
- To read (with thanks to Lucy Keer and Daniel Weissman for suggestions):
- Walter Appel, Mathematics for Physics and Physicists
- Arfken and Weber, Mathematical Methods for Physicists
- Carl M. Bender and Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory
- Richard Courant and David Hilbert, Methods of Mathematical Physics [I did say that they're usually not very rigorous, not always]
- Henry Margenau and George Moseley Murphy, The Mathematics of Physics and Chemistry
- Mathews and Walker, Mathematical Methods of Physics
- Morse and Feshbach, Methods of Theoretical Physics
- Roel Snieder, Guided Tour of Mathematical Methods: For the Physical Sciences
