$\newcommand \E {\mathbb E}$ $\newcommand \I {\mathbb I}$

X-posted from the SIMO X-Men blog

Can you remember the proof of PIE off the top of your head? Most students I’ve taught know it as a complicated, black-boxed formula.

Over the summer, I gave weekly talks as part of a reading group. The first half of the talks were centered on information complexity, while the second half was about random kSAT threshold and methods from statistical physics. Here’s an attempt to transcibe (and revise) the “sales pitch” I gave for statistical physics.

(Credits. The first example is largely also a sales pitch for large deviations theory, which I took from Chatterjee’s introduction to this subject.)

Earlier today, I managed to get KaTeX working the way I wanted it to. I had been trying to do that for the last three years. Subsequently, I went into a frenzy porting over all my old content onto the blog.

As you might guess, most of the dates are awfully off - lots of these blogposts are 3-4 years old. But I hope to be more active now that it functions (mostly!) the way I want it to, and also to customize the layout more.

This is the coolest inequality I’ve seen in a while.

$\newcommand\p[1]{\left(#1\right)}$

(Dongyi Wei) Let $a_1,…,a_n\in (-1,1)$. Show that $$\prod_{1\le i,j\le n} \frac{1+a_ia_j}{1-a_ia_j}\ge 1$$ and determine all equality cases.