As a reminder for local Los Angeles area hobbyist mathematicians and physicists, Dr. Miller will be teaching Fundamentals of Point-Set Topology at UCLA starting September 24th. I hope some new folks will join our merry band for some math fun this fall. First time taking some math after high school/college? I’ve got some tips here.

# Category: Mathematics

*(UCLA Extension)*

Point-set topology is the branch of mathematics that deals with collections of points endowed with sufficient structure to make meaningful the notions of closeness, separation, and convergence. Beginning with familiar notions concerning open sets, closed sets, and convergence on the real number line and Euclidean plane, this course systematically develops the theory of arbitrary topological spaces. Topics include bases and subbases, separation axioms (Hausdorff, regular, and normal spaces), countability (first- and second-countable spaces), compactness and compactification, connectedness, and convergence (nets and filters). Instruction emphasizes examples and problem solving. The course appeals to those seeking a better understanding of the algebraic and geometric underpinnings of common mathematical constructs.

September 24 - December 3 on Tuesday 7:00PM - 10:00PM PT

Fee: $453.00

Location: UCLA, Math Sciences Building, Room 5127

As usual, there’s no recommended textbook (yet), and he generally provides his own excellent notes over a required textbook. I’d suspect that he’ll recommend an inexpensive Dover Publication text like those of Kahn, Baum, or Gamelin & Greene.

If you’re curious about what’s out there, I’ve already compiled a bibliography of the usual suspects in the space:

- Armstrong, M. A. Basic Topology. Undergraduate Texts in Mathematics, 3.0. Springer, 1983.
- Conover, Robert A. A First Course in Topology: An Introduction to Mathematical Thinking. Reprint. Dover Books on Mathematics. Mineola, N.Y: Dover Publications, Inc., 2014.
- Conway, John B. A Course in Point Set Topology. Undergraduate Texts in Mathematics. Springer, 2015.
- Crossley, Martin D. Essential Topology. Corrected printing. Springer Undergraduate Mathematics Series. 2005. Reprint, Springer, 2010.
- Gaal, Steven A. Point Set Topology. 1st ed. Pure & Applied Mathematics 16. Academic Press, 1964.
- Gamelin, Theodore W., and Robert Everist Greene. Introduction to Topology. 2nd ed. Dover Books on Mathematics. 1983. Reprint, Mineola, N.Y: Dover Publications, Inc., 1999.
- Kahn, Donald W. Topology: An Introduction to the Point-Set and Algebraic Areas. Dover Books on Mathematics. Mineola, N.Y: Dover Publications, Inc., 1995.
- Kasriel, Robert H. Undergraduate Topology. Dover Books on Mathematics. Mineola, N.Y: Dover Publications, Inc., 2009.
- López, Rafael. Point-Set Topology: A Working Textbook. 1st ed. Springer Undergraduate Mathematics Series. Springer, 2024.
- Mendelson, Bert. Introduction to Topology. 3rd ed. Dover Books on Mathematics. Dover Publications, Inc., 1990.
- Morris, Sidney A. Topology Without Tears, 2024. [.pdf]
- Munkres, James R., 1930-. Topology. 2nd ed. 1975. Reprint, Prentice-Hall, Inc., 1999.
- Shick, Paul L. Topology: Point-Set and Geometric. 1st ed. Wiley-Interscience, 2007.
- Sierpinski, Waclaw. General Topology. Translated by C. Cecilia Krieger. Repring. Dover Books on Mathematics. Mineola, N.Y: Dover Publications, Inc., 2020.
- Viru, O. Ya., O.A. Ivanov, N. Yu. Netsvetaev, and V.M. Kharlamov. Elementary Topology: Problem Textbook. American Mathematical Society, 2008.
- Waldmann, Stefan. Topology: An Introduction. Springer, 2014.
- Willard, Stephen. General Topology. Dover Books on Mathematics. Mineola, N.Y: Dover Publications, Inc., 2004.

AI generated featured photo courtesy of Glif Alpha

The text for the class will be *Rational Points on Elliptic Curves* (Springer, Undergraduate Texts in Mathematics) by Joseph H. Silverman and John T. Tate. He expects to follow and rely more on it versus handing out his own specific lecture notes.

He mentioned that while it would suggest a more geometric flavor, which it will certainly have, the class will carry an interesting algebraic component which those not familiar with the topic may not expect.

To register, look for the listing sometime in the coming month or so when the Winter catalog is released.

*(UCLA Extension)*

Topics to be discussed include the isomorphism theorems; ascending and descending chain conditions; prime and maximal ideals; free, simple, and semi-simple modules; the Jacobson radical; and the Wedderburn-Artin Theorem.

Ring theory is a branch of abstract algebra that deals with sets—for example, the collection of all integers—that admit both additive and multiplicative operations. Modules generalize the notion of vector spaces, but with scalars drawn from a ring rather than a field. Beginning with a survey of the basic notions of rings and ideals, the course explores some of the elegant algebraic structuring that defines the behavior of rings—both commutative and non-commutative—and their associated modules. Topics to be discussed include the isomorphism theorems; ascending and descending chain conditions; prime and maximal ideals; free, simple, and semi-simple modules; the Jacobson radical; and the Wedderburn-Artin Theorem. Theory will be motivated by numerous examples drawn from familiar realms of number theory, linear algebra, and real analysis.

We’re mounting a study group on quantum mechanics based on Peter Woit‘s Introduction to Quantum Mechanics course from 2022. We’ll be using his textbook *Quantum Theory, Groups and Representations:An Introduction* (free, downloadable .pdf) and his lectures from YouTube.

Shortly, we’ll arrange a schedule and some zoom video calls to discuss the material. If you’d like to join us, send me your email or leave a comment so we can arrange meetings (likely via Zoom or similar video conferencing).

Our goal is to be informal, have some fun, but learn something along the way. The suggested mathematical background is some multi-variable calculus and linear algebra. Many of us already have some background in Lie groups, algebras, and representation theory and can hopefully provide some help for those who are interested in expanding their math and physics backgrounds.

Everyone is welcome!

## Theory and Applications of Continued Fractions MATH X 451.50 | Fall 2022

For almost 300 years, continued fractions—that is, numbers representable as the sum of an integer and a fraction whose denominator is itself such a sum—have fascinated mathematicians with both their remarkable properties and their myriad applications in such fields as number theory, differential equations, and computer algorithms. They have been applied to piano tuning, baseball batting averages, rational tangles, paper folding, and plant growth … the list goes on. This course is a rigorous introduction to the theory and mathematical applications of continued fractions. Topics to be discussed include quadratic irrationals, approximation of real numbers, Liouville’s Theorem, linear recurrence relations and Pell’s equation, Hurwitz’ Theorem, measure theory, and Ramanujan identities.

Mike is recommending the *Continued Fractions* text by Aleksandr Yakovlevich Khinchin. I found a downloadable digital copy of the 1964 edition (which should be ostensibly the same as the current Dover edition and all the other English editions) at the Internet Archive at Based on my notes, it looks like he’s following the Khinchin presentation fairly closely so far.

If you’re interested, do join us on Tuesday nights this fall. (We’ve already discovered that going 11 for 37 is the smallest number of at bats that will produce a 0.297 batting average.)

If you’re considering it and are completely new, I’ve previously written up some pointers on how Dr. Miller’s classes proceed: Dr. Michael Miller Math Class Hints and Tips | UCLA Extension

*The First Astronomers: How Indigenous Elders Read the Stars*by Duane Hamacher, Ghillar Michael Anderson, Ron Day, Segar Passi, Alo Tapim, David Bosun and John Barsa (Allen & Unwin, 2022) to arrive in the US, so I immediately downloaded a copy of the e-book version.

@AllenAndUnwin @AboriginalAstro

https://www.uclaextension.edu/sciences-math/math-statistics/course/introduction-differential-topology-math-x-45148

*YouTube*

The Collatz Conjecture is the simplest math problem no one can solve — it is easy enough for almost anyone to understand but notoriously difficult to solve.

*(Penguin Press)*

You don’t make a bagel by first baking a bialy and then punching out the center. No—you roll out a snake of dough and join the ends together to form the bagel. If you denied that a bagel has a hole, you’d be laughed out of New York City, Montreal, and any self-respecting deli worldwide. I consider this final.

## Differential Topology—Two quarter sequence at UCLA Extension for Fall/Winter 2021

*Differential Topology: An Introduction*by David B. Gauld (M. Dekker, 1982 or Dover, 1996 (reprint)).

The offering is naturally dependent on potential public health measures in September, which may also create a class limit on the number of attendees, so be sure to register as soon as it’s announced. For those who are interested in mathematics, but have never attended any of Dr. Miller’s lectures, I’ve previously written some details about his stye of presentation, prerequisites (usually very minimal despite the advanced level of the topics), and other details.

A few of us have already planned weekly Thursday night topology study sessions through the end of Spring and into Summer for those interested in attending. Just leave a comment with your contact information and I’ll be in touch with details.

I hope to see everyone in the fall.