A very positive review of my ergodic theory book has now appeared in the American Math Monthly. See simonrs.com/ETbookreview... to read it!

Another nice problem about dice that has a slick generating function argument: it is not possible to place weights on the sides of two dice (numbered 1--6) in such a way that every possible sum from 2 to 12 occurs with probability 1/11.

So many good options here! But I'll just post one:

There's a very beautiful and short generating function proof of the following result: There is no nontrivial partition of the nonnegative integers into arithmetic progressions with distinct moduli.

Have you taught an introductory class on Lean? Would you be interested in teaching one at Euler Circle next summer? If so, let's chat!

My copies of the ergodic theory book just arrived!

I just learned a charming theorem, the Tait-Kneser Theorem, while editing my differential geometry notes for my summer class: If a smooth plane curve has strictly monotonic curvature, then the osculating circles are disjoint and nested.

Our reward for accepting them is that they all come together so spectacularly into one formula, thus showing us that we were right to accept them.

Many people find Euler's identity e^{iπ}=-1 or e^{iπ}+1=0 to be the most beautiful formula in mathematics. To me, this identity is a triumph of bravery: every ingredient in this formula was a controversial idea at one point: zero, negative numbers, irrational numbers, imaginary numbers.

But there is a positive takeaway, which is that it encourages students to accept that their world can grow. If you don't have an object with properties that you want, then you can just make one. Of course it might not have good properties elsewhere, but the idea may be worth exploring.

I believe that this is a harder pill for students to swallow than limits are, in that it requires a greater strain on their sense of reality.

Should we teach calculus without limits? The obvious approach is using the dual numbers, i.e. introducing an ε such that ε is nonzero, but ε^2=0. Then you can define the derivative of f to be (f(x+ε)-f(x))/ε, at least for polynomials or power series, with no limits needed.

I just learned that the Galois group of the nth truncation of the Taylor series for the exponential function is S_n if n is not a multiple of 4 and A_n if n is a multiple of 4. A very nice result of Schur.

Also, every book title has to start with "An Introduction to" or "A First Course in" or "Elementary."

What Could Gifted Education Look Like? What Could Gifted Education Look Like? Simon Rubinstein-Salzedo Introduction The purpose of this essay is to investigate what gifted education could look like, if we decided as a society that we car...

Last month, I hosted two dinner parties to discuss logistics for gifted education. At the request of one of the attendees, I wrote a report, based in part on what we discussed there. I don't know what I should do with it, but I'll start by posting it here, and I'd love to hear your feedback on it.

This is one reason you should never use an indefinite integral. Instead, always use a definite integral, possibly with variable bounds.

Ergodic Theory

Coming soon! bookstore.ams.org/view?Product...

Applications to teach at Kaleidoscope are due in one week! Apply to teach physics, AI, CS, or something else to amazing high-school students who aim to become professionals in your chosen area. Ideal candidates will be able to teach in Palo Alto weekly and will have a PhD.

www.kalcircles.com/hiring

Do you wish you could do my job? Do you want to teach physics, computer science, or artificial intelligence to outstanding kids? Apply to teach at Kaleidoscope Circles at www.kalcircles.com/hiring! Applications are due February 1.

Come back, both of you!

New paper up by Euler Circle alum and TA Andrew Lin: arxiv.org/pdf/2412.19730

Kaleidoscope Circles

🎓 Seeking teacher to develop & lead college-level classes at Kaleidoscope! Focus areas: AI/AI safety, CS, physics (other fields possible). Teach the classes you've always dreamed of to talented high school students. Like Euler Circle, but for your field. Apply by Feb 1 at kalcircles.com.

Recommend your favorite videos of math talks that you think I'll enjoy!

Sleep Sort in Python - GeeksforGeeks A Computer Science portal for geeks. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview...

I just learned about a hilarious sorting algorithm: sleep sort.

www.geeksforgeeks.org/sleep-sort-i...

New paper by Euler Circle student Anay Aggarwal: arxiv.org/pdf/2412.05857

The quote about standing on the shoulders of giants is most commonly attributed to Isaac Newton. But it goes back further, to Bernard of Chartres and William of Conches in the 12th century. en.wikipedia.org/wiki/Standin...

Kaleidoscope Circles

My new institute, Kaleidoscope, is now up and running! This is an extension of Euler Circle to subjects other than mathematics. We're looking to hire a director of a new circle, likely AI/AI safety, CS, or physics. Please encourage people you know to apply!

www.kalcircles.com

New paper by Euler Circle alumna and TA Emma Cardwell: arxiv.org/pdf/2412.00895.