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.

# Tag: UCLA Extension

*(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.

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

## 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.

*Introduction to Category Theory*. As usual he passed out a short survey to accept ideas for the Fall and Winter quarters this coming year at UCLA Extension.

If you didn’t get a chance to weigh in, feel free to email him directly, or respond here with your suggestions (in order of preference) and I’ll pass them along.

I keep a list of his past offerings (going back to 2006, but he’s been doing this since 1973) on my site for reference. He’s often willing to repeat courses that have been previously offered, particularly if there’s keen interest in those topics.

Some of the suggestions on last night’s list included:

combinatorics

combinatorial group theory

number theory

game theory

group theory

ring theory

field theory

Galois theory

real analysis

point set topology

differential equations

differential geometry

Feel free to vote for any of these or suggest your own topics. Keep in mind that many of the topics in the past decade have come about specifically because of lobbying on behalf of students.

## 🔖 Introduction to Category Theory | UCLA Continuing Education

*(UCLA Continuing Education)*

This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra.

Category theory, since its development in the 1940s, has assumed an increasingly center-stage role in formalizing mathematics and providing tools to diverse scientific disciplines, most notably computer science. A category is fundamentally a family of mathematical obejcts (e.g., numbers, vector spaces, groups, topological spaces) along with “mappings” (so-called morphisms) between these objects that, in some defined sense, preserve structure. Taking it one step further, one can consider morphisms (so-called functors) between categories. This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra. Topics to be discussed include: isomorphism; products and coproducts; dual categories; covariant, contravariant, and adjoint functors; abelian and additive categories; and the Yoneda Lemma. The course should appeal to devotees of mathematical reasoning, computer scientists, and those wishing to gain basic insights into a hot area of mathematics.

January 8, 2019 - March 19, 2019

Tuesday 7:00PM - 10:00PM

Location: UCLA

Instructor: Michael Miller

Fee: $453.00

Oddly, it wasn’t listed in the published physical catalog, but it’s available online. I hope that those interested in mathematics will register as well as those who are interested in computer science.

## Gems And Astonishments of Mathematics: Past and Present—Lecture One

*Gems And Astonishments of Mathematics: Past and Present*class at UCLA Extension. There are a good 15 or so people in the class, so there’s still room (and time) to register if you’re interested. While Dr. Miller typically lectures on one broad topic for a quarter (or sometimes two) in which the treatment continually builds heavy complexity over time, this class will cover 1-2 much smaller particular mathematical problems each week. Thus week 11 won’t rely on knowing all the material from the prior weeks, which may make things easier for some who are overly busy. If you have the time on Tuesday nights and are interested in math or love solving problems, this is an excellent class to consider. If you’re unsure, stop by one of the first lectures on Tuesday nights from 7-10 to check them out before registering.

## Lecture notes

For those who may have missed last night’s first lecture, I’m linking to a Livescribe PDF document which includes the written notes as well as the accompanying audio from the lecture. If you view it in Acrobat Reader version X (or higher), you should be able to access the audio portion of the lecture and experience it in real time almost as if you had been present in person. (Instructions for using Livescribe PDF documents.)

We’ve covered the following topics:

- Class Introduction
- Erdős Discrepancy Problem
- n-cubes
- Hilbert’s Cube Lemma (1892)
- Schur (1916)
- Van der Waerden (1927)

- Sylvester’s Line Problem (partial coverage to be finished in the next lecture)
- Ramsey Theory
- Erdős (1943)
- Gallai (1944)
- Steinberg’s alternate (1944)
- DeBruijn and Erdős (1948)
- Motzkin (1951)
- Dirac (1951)
- Kelly & Moser (1958)
- Tao-Green Proof

- Homework 1 (homeworks are generally not graded)

Over the coming days and months, I’ll likely bookmark some related papers and research on these and other topics in the class using the class identifier MATHX451.44 as a tag in addition to topic specific tags.

## Course Description

Mathematics has evolved over the centuries not only by building on the work of past generations, but also through unforeseen discoveries or conjectures that continue to tantalize, bewilder, and engage academics and the public alike. This course, the first in a two-quarter sequence, is a survey of about two dozen problems—some dating back 400 years, but all readily stated and understood—that either remain unsolved or have been settled in fairly recent times. Each of them, aside from presenting its own intrigue, has led to the development of novel mathematical approaches to problem solving. Topics to be discussed include (Google away!): Conway’s Look and Say Sequences, Kepler’s Conjecture, Szilassi’s Polyhedron, the ABC Conjecture, Benford’s Law, Hadamard’s Conjecture, Parrondo’s Paradox, and the Collatz Conjecture. The course should appeal to devotees of mathematical reasoning and those wishing to keep abreast of recent and continuing mathematical developments.

### Suggested Prerequisites

Some exposure to advanced mathematical methods, particularly those pertaining to number theory and matrix theory. Most in the class are taking the course for “fun” and the enjoyment of learning, so there is a huge breadth of mathematical abilities represented–don’t not take the course because you feel you’ll get lost.

I’ve written some general thoughts, hints, and tips on these courses in the past.

## Renovated Classrooms

I’d complained to the UCLA administration before about how dirty the windows were in the Math Sciences Building, but they went even further than I expected in fixing the problem. Not only did they clean the windows they put in new flooring, brand new modern chairs, wood paneling on the walls, new projection, and new white boards! I particularly love the new swivel chairs, and it’s nice to have such a lovely new environment in which to study math.

## Category Theory for Winter 2019

As I mentioned the other day, Dr. Miller has also announced (and reiterated last night) that he’ll be teaching a course on the topic of Category Theory for the Winter quarter coming up. Thus if you’re interested in abstract mathematics or areas of computer programming that use it, start getting ready!

## Reply to Stephanie Hurlburt on Twitter

Separately but related, “adults” looking for a varied advanced math outlet in the Los Angeles area are welcome to join Dr. Mike Miller’s classes at UCLA Extension on Tuesday nights from 7-10pm. We’re working on Algebraic Geometry this quarter. For those who might need notes to play catch up, I’ve got copies, with full audio recordings, that I’m happy to share.

## RSVP to MATH X 451.43 Introduction to Algebraic Geometry: The Sequel | UCLA Extension

Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.

## MATH X 451.43 Introduction to Algebraic Geometry: The Sequel | UCLA Extension

*(UCLA Extension)*

Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.

Don’t forget to use the coupon code EARLY to save 10% with an early registration–time is limited!

## Algebraic Geometry Lecture 1

**Algebraic Geometry-Lecture 1 notes [.pdf file with embedded and linked audio]**

I’ve previously written some notes about how to best access and use these types of notes in the past. Of particular note, one must download the .pdf file and open in a recent version of Adobe Acrobat to take advantage of the linked/embedded audio file. (Trust me, it’s worth doing as it will be like you were there with the 20 of us who showed up last night!)

For those who prefer just the audio files separately, they can be listened to here, or downloaded.

### Lecture 1 – Part 1

### Lecture 1 – Part 2

Again, the recommended text is *Elementary Algebraic Geometry* by Klaus Hulek (AMS, 2003) ISBN: 0-8218-2952-1.

For those new to Dr. Miller’s classes, I’ve written up some hints/tips about them in the past as well.

## 🔖 Elementary Algebraic Geometry by Klaus Hulek

*(American Mathematical Society)*

This is a genuine introduction to algebraic geometry. The author makes no assumption that readers know more than can be expected of a good undergraduate. He introduces fundamental concepts in a way that enables students to move on to a more advanced book or course that relies more heavily on commutative algebra. The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory. The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary. It is also an excellent text for those working in neighboring fields (algebraic topology, algebra, Lie groups, etc.) who need to know the basics of algebraic geometry.

*Elementary Algebraic Geometry*by Klaus Hulek (AMS, 2003) ISBN: 0-8218-2952-1.

Sadly, I totally blew the prediction of which text he’d use. I was so far off that this book wasn’t even on my list to review! I must be slipping…