In complex analysis, I'm teaching my students about "the argument principle" now. It provides info about where the zeros of a function f(z) are. Here's a toy example, using the function w = f(z) = (z-a)(z-b), where the zeros a and b are located at the black points shown below. When we apply the function f to all points in the z plane, they get mapped into the w plane. In particular, the black dots get sent to the origin (since they are zeros, by construction!) and the blue, orange, and green circles get transformed as shown here:

As a teacher, I understand why it can be helpful to tell our students that "infinity is not a number." But can we discuss that statement a bit further? I may come to regret opening Pandora's box about this , but here goes: We all agree that infinity is not a number in the “real number" and “natural number" systems. But being too rigid about whether infinity can _ever_ be regarded as a number is problematic in its own ways. That's what worries me about saying "infinity is not a number."

Roger Penrose - Is Mathematics Invented or Discovered? via @YouTube Steven Weinberg - Is Mathematics Invented or Discovered?

Twisted architecture in the Meatpacking District Now we’re talking meatpacking.

Looking for accessible articles about deep ideas in topology? Check out @QuantaMagazine. For example, here are three fine recent explainer articles about (1) holes: quantamagazine.org/topology-101-h… (2) homology: quantamagazine.org/how-mathematic… (3) equivalence quantamagazine.org/in-topology-wh… quantamagazine.org/how-mathematic…

Peter Sarnak on the “best” rotation in the plane. Turns out there are 16 equally good ones, including one that uses the golden angle! Wonderful Math Encounters talk @momath1. More at golden.momath.org A theorem about the sense in which a golden rotation is “best”

Evolution of a manuscript: Three versions of the introduction to INFINITE POWERS, followed by the published version. Here's version 1: Version 2:

While my wife and I were binge watching #ThisIsUs tonight, we were shocked to hear some new characters discussing math concepts like the discrete cosine transform and the Karhunen-Loève transform. I was personally amazed at how accurate the math was! By the end of the show, it all made sense. The show was celebrating the real-life mathematician Nasir Ahmed who gave us the means of sending pictures and videos in compressed form. His work helped many folks stay connected during this crazy pandemic. cc @Dan_Fogelman

Lots of great tips in this thread from @BettySLai. My own technique is to use the dictation function on my iPhone. I dictate while I’m walking my dog. Just talk and talk. It gives the writing a conversational sound. Most of it is garbage, but occasionally something good comes out

https://twitter.com/BettySLai/status/1423637484408279046

Like many of the commenters in the thread, my own biggest weakness is a tendency to censor myself, and to keep editing the same sentence over and over. But when I dictate, the sentences flow. I pick one small topic and talk to myself about it. It generates a lot of raw material.

I’m thrilled about this preprint w Martin Kassabov and Alex Townsend. We prove that a network of identical Kuramoto oscillators synchronizes —regardless of the details of its wiring diagram — if every oscillator is connected to at least 75% of the others. arxiv.org/abs/2105.11406 This puzzle has fascinated a lot of us in my little corner of nonlinear dynamics since 2012. What’s the smallest level of connectivity that guarantees that a homogeneous Kuramoto model (the simplest kind of oscillator system) will always fall into sync?

Regarding the natural place to introduce e, my preference is to wait until calculus. Once you learn that the antiderivative of x^n is x^[n+1]/(n+1), it becomes fascinating to ask: what happens when n = -1? So define L(x) as the indefinite integral of 1/x and explore it. Doing it this way, you discover something truly amazing and beautiful: L(x) behaves like a logarithm! For instance, it obeys L(ax)=L(a)+L(x), as you can show by taking d/dx of L(ax). Once you know L is a log function, the natural question is: what is its base? Answer: call it e

I’ve spent the morning reading this preprint: arxiv.org/abs/1805.11556. It has a story behind it. A finance person named @MarcosCarreira does math for pleasure, inspired by @CutTheKnotMath. While playing with a classic problem, he finds something weird in a famous paper about it. It seems that Marcos discovered an error in that famous paper (by Gilbert and Mosteller) which nobody noticed until now. But I’m not an expert in probability, and it would be great if those of you who are would take a look at Marcos’s paper. It strikes as a neat piece of work.

I just received this new book, and at a glance, it looks terrific. Very creatively conceived, written, and illustrated. I came to that conclusion after reading two pages at random. Take a look at them below and see what you think: The teacher in me likes the question in the cloudy enclosure, and the gentle way it’s approached after that. The playful drawings help too. The question itself is really deep, and you can see the author appreciates that.

For those pointing out that the Fibonacci sequence originated in India, yes, I agree! I learned this from Manjul Bhargava, and discussed the matter in some tweets a few years ago:

https://twitter.com/stevenstrogatz/status/1080623259593465856?s=21

I'm teaching a course on asymptotics and perturbation methods, and thought it might be fun to share the lectures on @YouTube. Here's lecture 1, which introduces the idea of asymptotic expansions. (For more about the course, see the rest of this thread.) Asymptotic methods and perturbation theory are clever techniques for finding approximate analytical solutions to complicated problems by exploiting the presence of a large or small parameter. This course is an introduction to such methods.

In an hour, I'll meet my students in Math Explorations, a course we teach in an inquiry-based format. I LOVE this class! But like all teachers, I'm feeling the usual first-day jitters. To calm myself, I'm re-reading "The art of asking good questions": artofmathematics.org/blogs/vecke/th… The course materials are available at the website for the wonderful "Discovering the Art of Mathematics" project artofmathematics.org. They offer 11 free books on fascinating topics like math and music, patterns, dance, games and puzzles, knots, infinity, etc. A real feast!

My kids know I see math everywhere. One of them made this series of photos of Murray to amuse me (or maybe to tease me). (1/4) (2/4)

Today was my first day of class, and I zoomed with the 200+ students in my "multivariable calculus for engineers" course. We had an unexpectedly wonderful session, super interactive. As you can see, I'm still on a high from it! And I want to share what worked so well... 1/n I was really anxious about this first class, and about how dead it would be on Zoom, but somehow by making it warm and fun, the class and I established a rapport. Most of them are freshmen and very apprehensive. This warm welcome to @Cornell seems to have been the right call. 2/n

An interesting approach to calculus. Instead of using the limit-based definition of continuity, Olver defines a function to be continuous if the inverse image of every open set is open (as in topology). He says this is both more rigorous and more understandable for beginners.

https://twitter.com/academath/status/1298196681461243905

You can check out Olver’s approach here [PDF]: www-users.math.umn.edu/~olver/ln_/cc.… (ht @academath)

Regarding my earlier tweet about eigenvectors, some of you have asked for more explanation about page rank. Here’s a chapter from “The Joy of x“ where I tried to do that. 1/5 2/5

Jo Boaler just showed me what a typical Japanese textbook looks like. Check out the thickness… Not! In the next series of tweets, I’ll show you just how much thinking is expected from students. I love how conceptual and interesting these textbooks are. And this is for fifth grade!