Tag: number theory

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!

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👓 Kaisa Matomäki Dreams of Primes | Quanta Magazine

Read Kaisa Matomäki Dreams of Primes by Kevin Hartnett (Quanta Magazine)
Kaisa Matomäki has proved that properties of prime numbers over long intervals hold over short intervals as well. The techniques she uses have transformed the study of these elusive numbers.
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Primes as a Service on Twitter

Our friend Andrew Eckford has spent some time over the holiday improving his Twitter bot Primes as a Service. He launched it in late Spring of 2016, but has added some new functionality over the holidays. It can be relatively handy if you need a quick answer during a class, taking an exam(?!), to settle a bet at a mathematics tea, while livetweeting a conference, or are hacking into your favorite cryptosystems.

General Instructions

Tweet a positive 9-digit (or smaller) integer at @PrimesAsAService. It will reply via Twitter to tell you if the number prime or not.

Some of the usable commands one can tweet to the bot for answers follow. (Hint: Click on the buttons with the tweet text to auto-generate the relevant Tweet.)

If you ask about a prime number with a twin prime, it should provide the twin.

Pro tip: You should be able to drag and drop any of the buttons above to your bookmark bar for easy access/use in the future.

Happy prime tweeting!

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The L-functions and modular forms database

Bookmarked The L-functions and modular forms database by Tim Gowers (Gowers's Weblog)
...I rejoice that a major new database was launched today. It’s not in my area, so I won’t be using it, but I am nevertheless very excited that it exists. It is called the L-functions and modular forms database. The thinking behind the site is that lots of number theorists have privately done lots of difficult calculations concerning L-functions, modular forms, and related objects. Presumably up to now there has been a great deal of duplication, because by no means all these calculations make it into papers, and even if they do it may be hard to find the right paper. But now there is a big database of these objects, with a large amount of information about each one, as well as a great big graph of connections between them. I will be very curious to know whether it speeds up research in number theory: I hope it will become a completely standard tool in the area and inspire people in other areas to create databases of their own.

…I rejoice that a major new database was launched today. It’s not in my area, so I won’t be using it, but I am nevertheless very excited that it exists. It is called the L-functions and modular forms database. The thinking behind the site is that lots of number theorists have privately done lots of difficult calculations concerning L-functions, modular forms, and related objects. Presumably up to now there has been a great deal of duplication, because by no means all these calculations make it into papers, and even if they do it may be hard to find the right paper. But now there is a big database of these objects, with a large amount of information about each one, as well as a great big graph of connections between them. I will be very curious to know whether it speeds up research in number theory: I hope it will become a completely standard tool in the area and inspire people in other areas to create databases of their own.

–Tim Gowers

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Devastating News: Sol Golomb has apparently passed away on Sunday

The world has certainly lost one of its greatest thinkers, and many of us have lost a dear friend, colleague, and mentor.

I was getting concerned that I hadn’t heard back from Sol for a while, particularly after emailing him late last week, and then I ran across this notice through ITSOC & the IEEE:

Solomon W. Golomb (May 30, 1932 – May 1, 2016)

Shannon Award winner and long-time ITSOC member Solomon W. Golomb passed away on May 1, 2016.
Solomon W. Golomb was the Andrew Viterbi Chair in Electrical Engineering at the University of Southern California (USC) and was at USC since 1963, rising to the rank of University and Distinguished Professor. He was a member of the National Academies of Engineering and Science, and was awarded the National Medal of Science, the Shannon Award, the Hamming Medal, and numerous other accolades. As USC Dean Yiannis C. Yortsos wrote, “With unparalleled scholarly contributions and distinction to the field of engineering and mathematics, Sol’s impact has been extraordinary, transformative and impossible to measure. His academic and scholarly work on the theory of communications built the pillars upon which our modern technological life rests.”

In addition to his many contributions to coding and information theory, Professor Golomb was one of the great innovators in recreational mathematics, contributing many articles to Scientific American and other publications. More recent Information Theory Society members may be most familiar with his mathematics puzzles that appeared in the Society Newsletter, which will publish a full remembrance later.

A quick search a moment later revealed this sad confirmation along with some great photos from an award Sol received just a week ago:

As is common in academia, I’m sure it will take a few days for the news to drip out, but the world has certainly lost one of its greatest thinkers, and many of us have lost a dear friend, colleague, and mentor.

I’ll try touch base with his family and pass along what information sniff I can. I’ll post forthcoming obituaries as I see them, and will surely post some additional thoughts and reminiscences of my own in the coming days.

Golomb and national medal of science
President Barack Obama presents Solomon Golomb with the National Medal of Science at an awards ceremony held at the White House in 2013.
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What An Actual Handwaving Argument in Mathematics Looks Like

I’m sure we’ve all heard them many times, but this is what an actual handwaving argument looks like in a mathematical setting.

Handwaving during Algebraic Number Theory

Instagram filter used: Normal

Photo taken at: UCLA Math Sciences Building

Handwaving during Algebraic Number Theory

A photo posted by Chris Aldrich (@chrisaldrich) on

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Shinichi Mochizuki and the impenetrable proof of the abc conjecture

Read The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof of the ABC Conjecture (Nature News & Comment)
A Japanese mathematician claims to have solved one of the most important problems in his field. The trouble is, hardly anyone can work out whether he's right.

The biggest mystery in mathematics

This article in Nature is just wonderful. Everyone will find it interesting, but those in the Algebraic Number Theory class this fall will be particularly interested in the topic – by the way, it’s not too late to join the class. After spending some time over the summer looking at Category Theory, I’m tempted to tackle Mochizuki’s proof as I’m intrigued at new methods in mathematical thinking (and explaining.)

The abc conjecture refers to numerical expressions of the type a + b = c. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities a, b and c. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers — those that cannot be further factored out into smaller whole numbers: for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7. In principle, the prime factors of a and b have no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c.

Thanks to Rama for bringing this to my attention!

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