Mokshay Madiman

Algebraic numbers in the complex plane. Colors indicate the coefficient of the leading term: Red = 1 (algebraic *integer*), Green = 2, Blue = 3, Yellow = 4. Large dots mean fewer terms and smaller coefficients. Image from Wikipedia (made by Stephen J. Brooks).

Algebraic integers in the complex plane. Each red dot is the root of a monic polynomial of degree ≤ 7 with coefficients from {0, ±1, ±2, ±3, ±4, ±5}.

Algebraic integers are roots of monic polynomials. There are lots of fun visuals of these in the complex plane (just Google it!) Here are two, from Wikipedia.

Click the alt text for the descriptions!

2/19

It may help to also keep in mind that Homo sapiens is less than 10^6 years old and complex human civilization less than 10^4 years old; so we have 10^11 times the current age of recognizable human civilization before that last red dwarf winks out.

The joy balance at any given time appears to be oscillatory, with frequency that depends on the semester in question and amplitude that depends on how zen one is feeling.

Would be happy to be added, thanks!

Sometimes it feels like the machine is turning coffee into committee reports and emails and homework sets and graded papers, and the theorem production doesn’t need as much fuel since it is pure joy.

I’m enjoying catching up on several of the historical gems you unearthed! As regards this particular observation, I first saw it described as Arnold’s principle: www.math.fsu.edu/~wxm/Arnold....

Since V.I.Arnold does not say when M.Berry said this, it is hard to say if this predates Stigler.

Tricks Wiki article: The tensor power trick As many readers may already know, my good friend and fellow mathematical blogger Tim Gowers, having wrapped up work on the Princeton Companion to Mathematics (which I believe is now in press), has …

Other more sophisticated versions of this idea (going to higher dimensions) include situations where one goes from the real line to the complex plane (allowing the use of the powerful tools of complex analysis), and the tensorization trick: see, for example: terrytao.wordpress.com/2008/08/25/t...

Just to be clear, this does have a geometric visualization: instead of studying one object on a line (the uniform distribution on n points), we can study an essentially equivalent object in the plane (the uniform distribution on n^2 points in a rectangular grid).

Another instance of such a doubling trick is the way we calculate the integral of a Gaussian function… essentially by doubling and using Tonelli/Fubini.

This is a manifestation of the fact that for IID random variables X and Y, the variance of X is half of
E[(X-Y)^2] just by linearity of the expectation and the IID assumption. It is an instance of what one might call the “doubling trick”… where taking two copies of something creates simplification.

It seems like The Royal Society of London for Improving Natural Knowledge (its official name!) made a category error in the first place. Isn’t fellowship meant for scientists and scholars with direct contributions to human knowledge? When did they start recognizing the bankrollers?

underlies analysis, since the way we develop the notion of the (Lebesgue) integral with respect to a measure is to build up from sets to functions via this equivalence.

My favorite equivalence of this type is between sets and their indicator functions (for finite sets, equivalently between the power set of a set with n elements, and binary strings of length n). Thanks to the reflection principle, this underlies many combinatorial identities. But it also…

Yes! More generally, when we have two objects that are isomorphic, just realizing that we can think of it as one object with two descriptions is so useful. And beautiful— like looking at the same painting closing one eye at a time, and you get complementary perspectives.

Beyond the entropy power inequality, via rearrangements A lower bound on the Rényi differential entropy of a sum of independent random vectors is demonstrated in terms of rearrangements. For the special case of Boltzmann-Shannon entropy, this lower bound i...

Rearrangement inequalities (from Hardy-Littlewood at the simplest end to Rogers-Brascamp-Lieb-Luttinger) are such a useful tool! Classically much used as a step in proving functional inequalities, we (with Liyao Wang) found them useful in an entropy context here: arxiv.org/abs/1307.6018

YouTube video by Sony Pictures Entertainment India Pavitr Prabhakar sipping on Chai-Tea | Spider-Man: Across the Spider-Verse | Rent Now

I’m sure it’s delicious, but your tea store doesn’t seem familiar with the etymology, or with the Spiderverse for that matter: youtu.be/3_OZoPhHpiY?...

Cute! Also trivial but instructive for students, especially since it motivates the sample variance:
2 Var(X)= E [(X-Y)^2]
for IID X, Y.

Thanks for this! Anything written by Tom is luminously clear.

True. Often when reading a paper (especially on something I don’t really have expertise in), that part of the mental translation process is key. “What is the folklore I don’t know that makes X obvious to the author?”

I liked the ideas over incrementals, but wasn’t a huge fan of the 1-page abstract. Extra work, too concise to be meaningful for most readers, and the practical need of having printed proceedings of manageable length hasn’t applied for a while.

It would only be a good thing IMHO. “Prestige” would transfer over time in case of those for-profit journals that have accumulated it. There are plenty of good journals to go around in most (mathematical) fields without being forced to resort to a for-profit publisher.

Specifically, the mixed second derivative being nonnegative means the sum of the function values at any 2 points in the plane is at most the sum of the function values at their coordinate-wise minimum and maximum.

In particular, for a function of 2 variables, one can draw a nice picture involving rectangles to illustrate what the local condition of a positive mixed second derivative means globally.

Sumset estimates in convex geometry Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entro...

Just as the sign of the diagonal elements of the Hessian tell us about convexity or concavity, the sign of the off-diagonal elements is connected to submodularity or supermodularity. (One can explain this without invoking the Hessian as such.) See for example arxiv.org/abs/2206.01565

Hi Eugenia! We’ve never met in person, but I have to tell you that I *loved* reading “The Joy of Abstraction” and have given it as a gift multiple times. Your talent for exposition is something else! Glad to meet you here.

Wow this is a lot more comprehensive than the one I put together for myself! Wouldn’t mind being added to this too.

Hi Yiannis! 😊

Browsing a used book store (or one that sells both used and new books) beats any other bookstore activity hands down. My favorites are The Midtown Scholar in Harrisburg (Pennsylvania), and Blossom Book House in Bengaluru (India). Both have incredibly diverse selections with many hidden gems.

What is your favorite manifestation of mathematical beauty? #MathSky

Particularly when one is feeling weighed down by the vagaries of life or the existence of suffering or the state of humanity and the planet, that glimmer of the transcendental is deeply comforting and meaningful. Some may see it almost as a spiritual thing (as Einstein did), but one doesn’t need to.