Mathematical Methods 29:4761
Mathematical Methods I
29:4761
Fall 2022
Class Information
- Room: 618, Van Allen Hall
- Time: 9:30AM - 10:45AM
- Days: Tuesday, Thursday
- Text: Physical Mathematics,
Kevin Cahill, Cambridge University Press, 2019.
Instructor Information
- Instructor: Wayne Polyzou
- Office: 618 Van Allen Hall
- Office Hours:M 1:00-2:30, W 3:00-4:30
- Grader: Zoe Bellis
- E-mail: polyzou@uiowa.edu
- Phone: 319-335-1856
Grading Policy
Possible final grades are A+,A,B,C,D,F. The grade of A+ is for
performance that is a full grade above an A. Grades are based on
homework scores (15%), hour exam scores (25% x2), and the final exam
(35%). Exam dates will be determined by the instructor after
consultation with the students.
Homework assignments and important announcements will appear
on the web version of this syllabus
(http://www.physics.uiowa.edu/~wpolyzou/phys4761/). Homework
solutions, exam solutions, and lecture notes will be posted on
the class website
here. Homework will be due on Thursdays. The
lecture notes are for your benefit, but they are no substitute for
taking your own good notes during lectures. My lecture notes are
normally written during the evening before each lecture and posted on
the morning of the lecture. I do not have time to proofread the notes
so be warned that they may have errors. If you do not understand
something in the posted lecture notes, check with me before or after
class. I will try to correct errors as I go so expect changes in the
latter parts of the notes.
General Information
This is the first half of a two semester course on mathematical
methods in physics. The purpose of this course is to expose students
to the type of mathematics that is used in intermediate and advanced
physics classes. The main focus for the first semester will be
on complex analysis, linear algebra and analysis. If there is time
this semester lie groups will also be covered. These are topics from
chapter 6, chapter 1, chapter 5, and chapter 11 of the text.
These are both used extensively in the core graduate courses.
``Physical Mathematics'' by
Kevin Cahill. In the past I have used "Mathematics for Physicists",
Philippe Dennery and Andre Krzywicki which is an inexpensive alternative.
Lectures may also include supplementary topics not covered in the text.
In addition to the text there are a number of excellent
references on specific areas of mathematics that are used in physics.
The references listed below go deeper in many of the subjects that I
will cover in this class and cover some relevant areas of mathematics
that will not be covered in this class; I have chosen them because
they are the books that I have found to be useful both as a
student, teacher, and researcher.
- Functional Analysis,
Frigyes Reisz and Bela Sz.-Nagy, Dover, 1990(1950).
Readable treatment of functional analysis.
- Mathematics for Physicists,
Philippe Dennery and Andre Krzywicki, Dover, 1995(1967)
- Mathematics for Physicists,
Alexander Altland and Jan Von Delft, Cambridge University Press, 2109.
- Group Theory in a Nutshell for Physicists,
A. Zee, Princeton University Press, 2106.
- Orthogonal Polynomials
G. Szego, AMS Colloquium Publications, 1939.
- Non-Commutative Analysis
P. Jorgensen and F. Tian, World Scientific, 2017.
- Operators and Representation Theory
P. Jorgensen, Dover, 1988.
- Methods of Theoretical Physics, V1-2
P. Morse and H. Feshbach, Feshbach Publications, 1953.
- Functional Analysis and Semigroups,
Einar Hill, Ralph S. Phillips, AMS Colloquium Publications,
Vol. XXXI, 1957.
The best reference on analytic properties of resolvent and semigroups.
- Methods of Modern Mathematical Physics, Vol 1-IV,
Michael Reed and Barry Simon, Academic Press, (1972,1975,1978,1979).
This is a four volume set of books that cover almost all aspects
of functional analysis that are relevant for physics. Contains excellent
historical references.
- Real and Complex Analysis,
Walter Rudin, McGraw Hill, 1972.
Standard first year graduate reference on analysis.
- Real Analysis,
H. L. Royden, Mac Millan, 1968.
Main competitor to Rudin.
- Generalized Functions, V1-6,
I. M. Gelfand, G. E. Shilov (V1-3), I. M. Gelfand and N. Ya. Vilenkin (Vol 4),
I. M. Gelfand, M. I. Graev, N, Ya. Vilenkin (Vol 5),
Academic Press (1964,68,67,64,66).
Readable and well written - the definitive reference on distribution theory,
harmonic analysis, infinite dimensional integration. One of my favorite
references.
- Linear Operators (Parts I,II and III),
N. Dunford and J. Schwartz, Wiley, (1957,1963,1971).
Comprehensive three volume work on linear operators.
- Methods of Mathematical Physics, Vol I and II,
R. Courant and D. Hilbert,
Wiley, 1989(1937)(v1) , 1962 (v2).
Comprehensive reference written by leading mathematical physicists.
- Functional Analysis,
K. Yoshida,
Springer, 1980.
Has useful material on semigroups of operators and material that is
important in quantum mechanics.
- Trace Ideals and Their Applications,
Barry Simon,
AMS Mathematical Surveys and Monographs, V120, 2005.
Has unique material that in important in statistical physics,
quantum mechanics, and quantum field theory.
- An Introduction to Probability Theory and its Application, V1,2,
W. Feller, Wiley, 1950.
Standard reference on probability theory, proof of the law of large numbers
and central limit theorem.
- Differential Equations, Dynamical Systems, and Linear Algebra,
M. Hisrch and S. Smale, Academic Press, 1974.
Modern treatment of linear algebra and differential equations -
nice emphasis on qualitative methods that are important for dynamical
systems.
- Ordinary Differential Equations,
V. I. Arnold, MIT Press, 1981.
Clear and concise treatment of differential equations from a modern point of
view.
- Transversal Mappings and Flows,
R. Abraham and J. Robbin, Benjamin Cummings, 1967.
Nice treatment of generic properties of dynamical systems that is
hard to find elsewhere.
- Singular Integral Equations,
N. I. Muskhelishvili, Dover, 1992 (1953).
One of the earliest references in treating a class of integral equation
that are important in scattering theory and imaging.
- Perturbation Theory for Linear Operators,
T. Kato, Springer, 1966.
Contains important material on time dependent scattering theory,
also illustrates many important concepts using finite-dimensional examples.
- Scattering Theory by the Enss Method,
P. Perry, Harwood, 1983.
The first section gives a beautiful introduction to functional analysis, and
uses Weiner's theorem to give a neat geometrical characterization
of spectral properties of linear operators that have applications in scattering.
- Foundations of Modern Analysis,
J. Dieudonne, Academic Press, 1969.
Elegant work on analysis by one of the Bourbaki
contributors; formulates many theorems of elementary calculus in a
geometric manner that applies equally to infinite and finite dimensional
spaces.
- Complex Variables,
R. Redheffer and N. Levinson,
Holden Day, 1970.
Clear elementary reference directed at physicists, mathematicians and
engineers
- The Theory of Functions,
E. C. Titchmarch,
Oxford, 1932.
Classic reference, contains much of what is now called
mathematical physics. Easy to read.
- The Theory of Functions,
E. C. Titchmarch,
Oxford, 1932.
Classic reference, contains much of what is now called
mathematical physics. Easy to read.
- Infinite Dimensional Analysis,
Palle Jorgensen and James Tian,
World Scientific, 2021.
U.I. Math Faculty
- Non-Commutative Analysis,
Palle Jorgensen and Feng Tian,
World Scientific, 2017.
U.I. Math Faculty
- Operators and Representation Theory,
Palle Jorgensen,
Dover, 2017.
U.I. Math Faculty
- Applied Analysis,
C. Lanczos, Dover, 1988(1956).
Contains practical material relevant to mathematical physics.
- Generalized Functions, Vol 1-6
I. M. Gel′fand, M. I. Graev, I. I. Pyatetskii-Shapiro, G. E. Shilov, N. Ya. Vilenkin,
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Clear treatments of distribution theory, harmonic analysis.
- Mathematical Physics,
Robert Geroch, University of Chicago Press, 1985.
A unique abstract treatment of mathematical physics that starts from
category theory. Good job of motivating why certain abstract mathematical
structures are important.
- A Survey of Modern Algebra,
G. Birkhoff and S. Maclaine,
Mac Millan, 1965(1941).
Standard introductory reference on algebra. Written by two
excellent mathematicians.
- Algebra
S. Lang, Springer, (1965).
Standard graduate text on algebra.
- Algebra
T. Hungerford, Springer, 1974.
Graduate text on algebra, clear presentation of many topics.
- Lie Algebras in Particle Physics,
H .Georgi, Benjamin Cummings, 1982.
Excellent and readable treatment of group theory
for physicists.
- Representations of the Rotation and Lorentz Groups and their Applications,
I. M. Gelfand, Martino Publications, 1963.
- The Theory of Groups,
H. J. Zassenhaus, Dover, 1999(1958).
Clear and compact reference that focuses on group theory.
- Theory of Group Representations,
M. A. Naimark and A. I. Stern, Springer, 1982.
Nice treatment of group representation theory.
- The Theory of Lie Groups,
C. Chevalley, Princeton, 1999(1946).
This is a classic and well written reference. It deals
with some of the fundamental properties of Lie Groups.
- The Classical Groups - Their Invariants and Representations,
H. Weyl, Princeton, 1939.
A classic reference which includes material on the classification of
groups.
- Representation Theory of Semisimple Groups,
A. Knapp, Princeton, 1986.
Readable and useful.
- Group Theory and its Application to Physical Problems,
M. Hammermesh, Dover, 1989(1962).
One of the earlier references on group theory written by a physicist.
- Gian-Carlo Rota on Combinatorics,
G. C. Rota, Birkhauser, 1995 .
Collection of Rota's papers.
Has useful material on Mobius functions, Zeta functions and partial
orderings that is important in theories involving many degrees of
freedom.
- General Topology,
J. L. Kelly, D. Van Nostrand, 1955.
Clear treatment of the branch
mathematics that is used to properly formulate convergence.
- Topological Groups,
L. S. Pontryagin, Gordon and Breach, 1966,
Stands out as one of the most readable book on advanced mathematics -
contains excellent treatment of foundation material for topics that are
important in Lie groups.
- Differential Geometry, Lie Groups, and Symmetric Spaces,
S. Helgason, Academic Press, 1978.
Best treatment of symmetric spaces, pretty good for differential geometry
and Lie groups as well.
- A Comprehensive Introduction to Differential Geometry (V1-5),
M. Spivak, Publish or Perish, .
Readable treatment of differential geometry
that is relevant for general relativity and gauge theories - contains
some fun historical material.
- The Topology of Fiber Bundles,
N. Steenrod, Princeton University Press, 1951.
One of two treatments of the formal mathematics behind gauge symmetries
- Fiber Bundles,
Dale Husemoller, Springer, 1966.
Covers same material as Steenrod.
- Measure Theory
P. Halmos, Springer, 1974(1950).
The subject is becoming more important in physics, especially for
problems in dynamical systems, statistical physics, and quantum
field theory.
- Foundations of Differentiable Manifolds and Lie Groups,
M. Warner, Scott Foresman, 1970.
Nice compact treatment of Differentiable Manifolds and
Lie groups. Now available in a Dover edition.
- PCT Spin Statistics and All That,
R. F. Streater and A. S. Wightman, Benjamin Cummings, 1964.
Clear treatment of SL(2,C), finite dimensional representations of the
Lorentz group, representation analytic functions of several variables,
distribution theory with applications to quantum field theory.
- Algebraic Topology,
E. H. Spanier, Springer, 1989(1966).
The first comprehensive textbook on the subject. Most
contemporary mathematicians learned the subject from this
text.
- Gravitation and Cosmology,
Steven Weinberg, Wiley, 1971.
Nice treatment of Reimannian Geometry in Chapter 6, nice treatment of
the Weyl Tensor.
- The Quantum Theory of Fields, VI
Steven Weinberg,
Cambridge, 1995.
Chapter 2 contains a nice treatment of the
representation theory of the Poincare group, projective representation
and central extensions of groups,
- Operator Algebras and Quantum Statistical Mechanics V1,2,
O. Bratelli and D. Robinson, Springer 1979,1981.
Mathematics of systems of and infinite number of degrees of freedom.
Homework Assignments and Calendar
- Week 6
- Tuesday, September 27
-
Lecture 11:
- Reading: Chapter 1 text
- Thursday, September 29
- No Class today - I am a Kent State for a departmental review. I plan to make up this class by having an out of class exam.
- Final Exam: Monday, Dec 12, 5:30-7:30 , 618 Van
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