The builders of Britain’s ancient stone circles like Stonehenge were using Pythagoras' theorem 2,000 years before the Greek philosopher was born, experts have claimed.

I’ll be bookmarking the book described in this piece for later. The author doesn’t get into the specifics of the claim in the title enough for my taste. What is the actual evidence? Is there some other geometrical construct they’re using to come up with these figures that doesn’t involve Pythagoras?

]]>University of Florida mathematician Kevin Knudson and I are excited to announce our new math podcast: My Favorite Theorem. In each episode, logically enough, we invite a mathematician on to tell us about their favorite theorem. Because the best things in life are better together, we also ask our guests to pair their theorem with, well, anything: wine, beer, coffee, tea, ice cream flavors, cheese, favorite pieces of music, you name it. We hope you’ll enjoy learning the perfect pairings for some beautiful pieces of math. We’re very excited about the podcast and hope you will listen here, on the site’s page, or wherever you get your podcasts. New episodes will be published approximately every three weeks. We have a great lineup of guests so far and think you’ll enjoy hearing from mathematicians from different mathematical areas, geographic locations, and mathematical careers.]]>

Should LaTeXiT be categorized, it would be an equation editor. This is not the plain truth, since LaTeXiT is "simply" a graphical interface above a LaTeX engine. However, its large set of features is a reason to see it as an editor; this is the goal in fact.]]>

Andrew Jordan reviews Peter Woit's Quantum Theory, Groups and Representations and finds much to admire.

For the tourists, I’ve noted before that Peter maintains a free copy of his new textbook on his website.

I also don’t think I’ve ever come across the journal *Inference* before, but it looks quite nice in terms of content and editorial.

A study of 10,000 school districts shows how local norms help grow or shrink gender achievement gaps.]]>

The Harvey Mudd College mathematician tells us why he loves playing with Brouwer's fixed-point theorem

I need to remember to subscribe to this podcast…

]]>It is astounding to me that mathematics — of all school subjects — elicits such potent emotional reaction when “reform” is in the air…

An interesting take on the changes in math curriculum over the past few years. Takeaway, we need to think about the pedagogy we use with the public and parents as well.

]]>The Futility Closet people recently posted “A Square Circle“, in which they showed: 49² + 73² = 7730 77² + 30² = 6829 68² + 29² = 5465 54² + 65² = 7141 71² + 41² = 6722 67² + 22² = 4973 which is a nice little result. I like this sort of recreational maths, so I spent a little time w...

An interesting cyclic structure here.

]]>Special Issue: Mathematical Oncology

h/t to @ara_anderson

]]>Its finally out! Our mammoth special issue of the @SpringerNature Bulletin of Mathematical Biology on #mathonco Mathematical Oncology! Jointly edited with @OxUniMaths Philip Maini and this is the single biggest issue in the journals history! @MoffittNews https://t.co/K9GqAPTjy8 pic.twitter.com/tUDs1ACZCW

— Sandy Anderson (@ara_anderson) April 28, 2018

To be published by Cambridge University Press in April 2018.

Upon publication this book will be available for purchase through Cambridge University Press and other standard distribution channels. Please see the publisher's web page to pre-order the book or to obtain further details on its publication date.

A draft, pre-publication copy of the book can be found below. This draft copy is made available for personal use only and must not be sold or redistributed.

This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.

h/t to @michael_nielsen via Nuzzel

]]>John Watrous's excellent quantum book just came out. It's still available free on his webpage: https://t.co/D2rr5FTly6

— michael_nielsen (@michael_nielsen) April 28, 2018

]]>Just Skyped with a math student @UofR who has built (beta) an interactive glossary/encyclopedia for challenging technical/academic jargon that can be layered into textbooks. He wants to develop it as an #opensource resource for #OER. More soon, but the future is SO OPEN!

— Robin DeRosa (@actualham) April 27, 2018

Co-Founder and CEO at Cambridge Quantum Computing

Dear god, I wish Ilyas had a traditional blog with a true feed, but I’m willing to put up with the inconvenience of manually looking him up from time to time to see what he’s writing about quantum mechanics, quantum computing, category theory, and other areas of math.

]]>This short article is the result of various conversations over the course of the past year or so that arose on the back of two articles/blog pieces that I have previously written about Category Theory (here and here). One of my objectives with such articles, whether they be on aspects of quantum computing or about aspects of maths, is to try and de-mystify as much of the associated jargon as possible, and bring some of the stunning beauty and wonder of the subject to as wide an audience as possible. Whilst it is clearly not possible to become an expert overnight, and it is certainly not my objective to try and provide more than an introduction (hopefully stimulating further research and study), I remain convinced that with a little effort, non-specialists and even self confessed math-phobes can grasp some of the core concepts. In the case of my articles on Category Theory, I felt that even if I could generate one small gasp of excited comprehension where there was previously only confusion, then the articles were worth writing.

I just finished a course on Algebraic Geometry through UCLA Extension, which was geared toward non-traditional math students and professionals, and wish I had known about Smith’s textbook when I’d started. I did spend some time with Cox, Little, and O’Shea’s *Ideals, Varieties, and Algorithms* which is a pretty good introduction to the area, but written a bit more for computer scientists and engineers in mind rather than the pure mathematician, which might recommend it more toward your audience here as well. It’s certainly more accessible than Hartshorne for the faint-of-heart.

I’ve enjoyed your prior articles on category theory which have spurred me to delve deeper into the area. For others who are interested, I thought I’d also mention that physicist and information theorist John Carlos Baez at UCR has recently started an applied category theory online course which I suspect is a bit more accessible than most of the higher graduate level texts and courses currently out. For more details, I’d suggest starting here: https://johncarlosbaez.wordpress.com/2018/03/26/seven-sketches-in-compositionality/

]]>By making the first progress on the “chromatic number of the plane” problem in over 60 years, an anti-aging pundit has achieved mathematical immortality.]]>

This is a list of geometry topics, by Wikipedia page.

h/t to @mathematicsprof

]]>One misconception of the general public is that geometry is the kind of geometry the Greeks studied and nothing else. That’s like asking an engineer if engineering has progressed past the wheel. Here is a list of the many kinds of geometries. https://t.co/4gjGsCVqkX

— math prof (@mathematicsprof) April 19, 2018

For hundreds of years, x has been the go-to symbol for the unknown quantity in mathematical equations. So who started this practice?

]]>

Decades after physicists happened upon a stunning mathematical coincidence, researchers are getting close to understanding the link between two seemingly unrelated geometric universes.

An interesting story in that physicists found the connection first and mathematicians are tying the two areas together after the fact. More often it’s the case that mathematicians come up with the theory and then physicists are applying it to something. I’m not sure I like some of the naming conventions laid out, but it’ll be another decade or two after it’s all settled before things have more logical sounding names. I’m a bit curious if any category theorists are playing around in either of these areas.

After having spent the last couple of months working through some of the “rigidity” (not the best descriptor in the article as it shows some inherent bias in my opinion) of algebraic geometry, now I’m feeling like symplectic geometry could be fun.

]]>Cox-Zucker machine.What sounds like a high-tech device for oral sex is actually an algorithm used in the study of certain curves, including those that arise in cryptography. The story goes that David A. Cox co-authored a paper with fellow mathematician Steven Zucker, just so that the dirty-sounding term would enter the lexicon.

I always thought he was cool before (many of his students didn’t “get” him), but I’m now even more proud to have had Steven Zucker as my first math professor in college.

]]>This book is an invitation to discover advanced topics in category theory through concrete, real-world examples. It aims to give a tour: a gentle, quick introduction to guide later exploration. The tour takes place over seven sketches, each pairing an evocative application, such as databases, electric circuits, or dynamical systems, with the exploration of a categorical structure, such as adjoint functors, enriched categories, or toposes. No prior knowledge of category theory is assumed. [.pdf]

This is the textbook that John Carlos Baez is going to use for his online course in Applied Category Theory.

]]>Some awesome news just as I’ve wrapped up a class on Algebraic Geometry and was actively looking to delve into some category theory over the summer. John Carlos Baez announced that he’s going to offer an online course in applied category theory. He’s also already posted some videos and details!

]]>Maybe yet another hint that working on the Langlands program over the summer might be a fun diversion?

]]>G. W. Peck is a pseudonymous attribution used as the author or co-author of a number of published mathematics academic papers. Peck is sometimes humorously identified with George Wilbur Peck, a former governor of the US state of Wisconsin. Peck first appeared as the official author of a 1979 paper entitled "Maximum antichains of rectangular arrays". The name "G. W. Peck" is derived from the initials of the actual writers of this paper: Ronald Graham, Douglas West, George B. Purdy, Paul Erdős, Fan Chung, and Daniel Kleitman. The paper initially listed Peck's affiliation as Xanadu, but the editor of the journal objected, so Ron Graham gave him a job at Bell Labs. Since then, Peck's name has appeared on some sixteen publications, primarily as a pseudonym of Daniel Kleitman.

I’d known about Bourbaki, but this one is a new one on me.

]]>A physicist and best-selling author, Dr. Hawking did not allow his physical limitations to hinder his quest to answer “the big question: Where did the universe come from?”

Some sad news after getting back from Algebraic Geometry class tonight. RIP Stephen Hawking.

]]>There's no need to go out tonight for a movie. There are 100s of math videos on every conceivable 'math' topic' at --> https://www.pinterest.com/mathematicsprof/]]>

Three new books on the challenge of drawing confident conclusions from an uncertain world.

Not sure how I missed this when it came out two weeks ago, but glad it popped up in my reader today.

This has some nice overview material for the general public on probability theory and science, but given the state of research, I’d even recommend this and some of the references to working scientists.

I remember bookmarking one of the texts back in November. This is a good reminder to circle back and read it.

]]>Simple math shows how widespread vaccination can disrupt the exponential spread of disease and prevent epidemics.

This is a very clear and lucid article with some very basic math that shows the value of vaccines. I highly recommend it to everyone.

]]>Mayonnaise: 20 parts oil: 1 part liquid: 1 part yolk

Hollandaise: 5 parts butter: 1 part liquid: 1 part yolk

Vinaigrette: 3 parts oil: 1 part vinegar

Rule of thumb: You probably don’t need as much yolk as you thought you did.

I like that he provides the simple ratios with some general advice up front and then includes some ideas about variations before throwing in a smattering of specific recipes that one could use. For my own part, most of these chapters could be cut down to two pages and then perhaps even then cut the book down to a single sheet for actual use in the kitchen.

Part 4: Fat-Based Sauces

But what greatly helps the oil and water to remain separate is, among other things, a molecule in the yolk called lecithin, which, McGee explains, is part water soluble and part fat soluble.

Highlight (yellow) – Mayonnaise > Page 168

Added on Sunday, February 4, 2018

The traditional ratio, not by weight, is excellent and works beautifully: Hollandaise = 1 pound butter: 6 yolks. This ratio seems to have originated with Escoffier. Some cookbooks call for considerably less butter per yok, as little as 3 and some even closer to 2 to 1, but then you’re creeping into sabayon territory; whats more, I believe it’s a cook’s moral obligation to add more butter given the chance.

Highlight (yellow) – Hollandaise> Page 185

more butter given the chance! Reminiscent of the Paula Deen phrase: “Mo’e butta is mo’e betta.”

Added on Sunday, February 4, 2018

]]>

How do nonsensical counting-out rhymes like these enter the lexicon?

I’d read this a year or two ago for a specific purpose and revisited it again today just for entertainment. There’s some interesting history hiding in this sort of exercise.

I also considered these rhymes as simple counting games, but the’re not really used to count up as if they were ordinals. Most people couldn’t even come close to saying how many things they’d have counted if they sang such a song. I also find that while watching children sing these while “counting” they typically do so with a choice for each syllable, but this often fails in the very young so that they can make their own “mental” choice known while still making things seem random. For older kids, with a little forethought and some basic division one can make something seemingly random and turn it into a specific choice as well.

So what are these really and what purpose did they originally serve?

]]>Okay so right now I go to coffee shops to solve math problems alone, it's peaceful, I like it But someone mentioned they do cute tea parties with their girl squad & I said wow I want something like that but we all bring math textbooks & solve problems next to each other (1/2)

It’s not specifically femme yet does involve tea, but I’ve noticed something informal like this at the Starbucks just two blocks West of CalTech in Pasadena.

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.

]]>There is a proof for Brouwer's Fixed Point Theorem that uses a bridge - or portal - between geometry and algebra. Analogous to the relationship between geometry and algebra, there is a mathematical “portal” from a looser version of geometry -- topology -- to a more “sophisticated” version of algebra. This portal can take problems that are very difficult to solve topologically, and recast them in an algebraic light, where the answers may become easier. Written and Hosted by Tai-Danae Bradley; Produced by Rusty Ward; Graphics by Ray Lux; Assistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington; Made by Kornhaber Brown (www.kornhaberbrown.com)

I had originally started following Tai-Danae Bradley on Instagram having found her account via the #math tag. Turns out she’s burning up the world explaining some incredibly deep and complex mathematics in relatively simple terms. If you’re into math and not following her work already, get with the program. She’s awesome!

Personal Website: http://www.math3ma.com/

Twitter: @math3ma

Instagram: @math3ma

YouTube series: PBS Infinite Series

While this particular video leaves out a masters degree’s worth of detail, it does show some incredibly powerful mathematics by analogy. The overall presentation and descriptions are quite solid for leaving out as much as they do. This may be some of the best math-based science communication I’ve seen in quite a while.

I must say that I have to love and laugh at the depth and breadth of the comments on the video too. At best, this particular video, which seems to me to be geared toward high school or early college viewers and math generalists, aims to introduce come general topics and outline an incredibly complex proof in under 9 minutes. People are taking it to task for omitting “too much”! To completely understand and encapsulate the entirety of the topics at hand one would need coursework including a year’s worth of algebra, a year’s worth of topology including some algebraic topology, and a minimum of a few months worth of category theory. Even with all of these, to fill in all the particular details, I could easily see a professor spending an hour at the chalkboard filling in the remainder without any significant handwaving. The beauty of what she’s done is to give a very motivating high level perspective on the topic to get people more interested in these areas and what can be done with them. For the spirit of the piece, one might take her to task a bit for not giving more credit to the role Category Theory is playing in the picture, but then anyone interested is going to spend some time on her blog to fill in a lot of those holes. I’d challenge any of the comments out there to attempt to do what she’s done in under 9 minutes and do it better.

]]>Lecture one of six in an introductory set of lectures on category theory.

Take Away from the lecture: Morphisms are just as important as the objects that they morph. Many different types of mathematical constructions are best described using morphisms instead of elements. (This isn’t how things are typically taught however.)

Would have been nice to have some more discussion of the required background for those new to the broader concept. There were a tremendous number of examples from many areas of higher math that many viewers wouldn’t have previously had. I think it’s important for them to know that if they don’t understand a particular example, they can move on without much loss as long as they can attempt to apply the ideas to an area of math they are familiar with. Having at least a background in linear algebra and/or group theory are a reasonable start here.

In some of the intro examples it would have been nice to have seen at least one more fully fleshed out to better demonstrate the point before heading on to the multiple others which encourage the viewer to prove some of the others on their own.

Thanks for these Steven, I hope you keep making more! There’s such a dearth of good advanced math lectures on the web, I hope these encourage others to make some of their own as well.

]]>The main purpose of this blog is to share updates about the open-access, open-source textbook Understanding Linear Algebra. Though work is continuing on this project, the HTML version of the text is now freely available, the forthcoming PDF version will also be free, and low-cost print options will be provided. The PreTeXt source code will be posted on GitHub as well.

h/t Robert Talbert

]]>My awesome colleague @davidaustinm is unveiling his new, open-source linear algebra text at the JMM, but you can access it NOW at his (new!) blog, the aptly named "More Linear Algebra": https://t.co/AAreqGk8DW

— Robert Talbert (@RobertTalbert) January 9, 2018

I'd like to embark on yet another mini-series here on the blog. The topic this time? Limits and colimits in category theory! But even if you're not familiar with category theory, I do hope you'll keep reading. Today's post is just an informal, non-technical introduction. And regardless of your categorical background, you've certainly come across many examples of limits and colimits, perhaps without knowing it! They appear everywhere - in topology, set theory, group theory, ring theory, linear algebra, differential geometry, number theory, algebraic geometry. The list goes on. But before diving in, I'd like to start off by answering a few basic questions.

A great little introduction to category theory! Can’t wait to see what the future installments bring.

Interestingly I came across this on Instagram. It may be one of the first times I’ve seen math at this level explained in pictorial form via Instagram.

]]>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.

]]>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.

I’m definitely attending the Winter Quarter!

]]>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.

Alright math nerds, it’s that time again! Be sure to register for Mike Miller’s excellent follow-on course for Algebraic Geometry.

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

]]>The injury was to the professor’s hand, but I’m pretty sure it wasn’t due to excessive hand-waiving…

]]>In the sixteenth and seventeenth centuries, gamblers and mathematicians transformed the idea of chance from a mystery into the discipline of probability, setting the stage for a series of breakthroughs that enabled or transformed innumerable fields, from gambling, mathematics, statistics, economics, and finance to physics and computer science. This book tells the story of ten great ideas about chance and the thinkers who developed them, tracing the philosophical implications of these ideas as well as their mathematical impact. Persi Diaconis and Brian Skyrms begin with Gerolamo Cardano, a sixteenth-century physician, mathematician, and professional gambler who helped develop the idea that chance actually can be measured. They describe how later thinkers showed how the judgment of chance also can be measured, how frequency is related to chance, and how chance, judgment, and frequency could be unified. Diaconis and Skyrms explain how Thomas Bayes laid the foundation of modern statistics, and they explore David Hume’s problem of induction, Andrey Kolmogorov’s general mathematical framework for probability, the application of computability to chance, and why chance is essential to modern physics. A final idea―that we are psychologically predisposed to error when judging chance―is taken up through the work of Daniel Kahneman and Amos Tversky. Complete with a brief probability refresher, Ten Great Ideas about Chance is certain to be a hit with anyone who wants to understand the secrets of probability and how they were discovered.]]>

]]>Simpson's Paradox Part 2. This video is about how to tell whether or not university admissions are biased using statistics: aka, it's about Simpson's Paradox again!

REFERENCES:

Original Berkeley Grad Admissions Paper

Interactive Simpson’s Paradox Explainer

No Lawsuit, But Yes, Berkeley Study on Gender Bias

Statistics on college majors by gender:

https://nces.ed.gov/programs/digest/2016menu\_tables.asp

http://www.npr.org/sections/money/2014/10/28/359419934/who-studies-what-men-women-and-college-majors

http://www.randalolson.com/2014/06/14/percentage-of-bachelors-degrees-conferred-to-women-by-major-1970-2012/

Earnings by college major

Wall Street Journal Article on Simpson’s Paradox

The Workshop on Applied Category Theory 2018 takes place in May 2018. A principal goal of this workshop is to bring early career researchers into the applied category theory community. Towards this goal, we are organising the Adjoint School. The Adjoint School will run from January to April 2018.

There’s still some time left to apply. And if nothing else, this looks like it’s got some interesting resources.

h/t John Carlos Baez

]]>