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

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

Last night was the first lecture of Dr. Miller’s 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.

Register now

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.

The newly renovated classroom space in UCLA’s Math Sciences Building

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!

Gems And Astonishments of Mathematics: Past and Present—Lecture One was originally published on Chris Aldrich

🔖 Gems And Astonishments of Mathematics: Past and Present | Dr. Mike Miller at UCLA Extension

🔖 Gems And Astonishments of Mathematics: Past and Present | Dr. Mike Miller at UCLA Extension

I’ve been waiting with bated breath to see what Dr. Miller would be offering in the evenings at UCLA Extension this Fall and Winter quarters. The wait is over, though it’ll be a few days before we can register.

If you’re interested in math at all, I hope you’ll come join the 20+ other students who follow everything that Mike teaches. Once you’ve taken one course from him, you’ll be addicted.

🔖 Gems And Astonishments of Mathematics: Past and Present | Dr. Mike Miller at UCLA Extension was originally published on Chris Aldrich

🔖 actualham tweet about interactive glossary/encyclopedia for challenging technical/academic jargon that can be layered into textbooks

https://platform.twitter.com/widgets.js

🔖 actualham tweet about interactive glossary/encyclopedia for challenging technical/academic jargon that can be layered into textbooks was originally published on Chris Aldrich

🔖 Nonlinear Dynamics 1 & 2: Geometry of Chaos by Predrag Cvitanovic

🔖 Nonlinear Dynamics 1 & 2: Geometry of Chaos by Predrag Cvitanovic
Nonlinear Dynamics 1 & 2: Geometry of Chaos by Predrag CvitanovicPredrag Cvitanovic (Georgia Institute of Technology)

The theory developed here (that you will not find in any other course 🙂 has much in common with (and complements) statistical mechanics and field theory courses; partition functions and transfer operators are applied to computation of observables and spectra of chaotic systems.

Nonlinear dynamics 1: Geometry of chaos (see syllabus)
Topology of flows – how to enumerate orbits, Smale horseshoes
Dynamics, quantitative – periodic orbits, local stability
Role of symmetries in dynamics
Nonlinear dynamics 2: Chaos rules (see syllabus)
Transfer operators – statistical distributions in dynamics
Spectroscopy of chaotic systems
Dynamical zeta functions
Dynamical theory of turbulence
The course, which covers the same material and the same exercises as the Georgia Tech course PHYS 7224, is in part an advanced seminar in nonlinear dynamics, aimed at PhD students, postdoctoral fellows and advanced undergraduates in physics, mathematics, chemistry and engineering.

An interesting looking online course that appears to be on a white-labeled Coursera platform.

I’ve come across Predrag Cvitanovic’s work on Group Theory and Lie Groups before, so this portends some interesting work. I’ll have to see if I can carve out some time to sample some of it.

🔖 Nonlinear Dynamics 1 & 2: Geometry of Chaos by Predrag Cvitanovic was originally published on Chris Aldrich