It’s 07:00 in the morning. Having just gotten out of bed, you already poured yourself some breakfast cereal into the bowl, when you come to the most horrifying realization: you’re fresh out of milk. Beads of sweat start rolling down your forehead, and your heart starts beating like it’s trying to attract a sandworm on … More How convenient!
This is post #3 in the series about the “Useful Inequalities” cheat-sheet. Today’s inequality: The Paley-Zygmund inequality. A large portion of probability theory involves showing that random variables behave “nicely”, which often means “show that a random variable is not too far away from its mean”, or at least “show that is not too large”. … More The useful Paley-Zygmund inequality
This is post #2 in the series about the “Useful Inequalities” cheat-sheet. Today’s useful inequality: Chebyshev’s sum inequality. Almost everyone I know is familiar with the famous Chebyshev’s inequality in probability theory, which states that most of the time, a random variable can’t be too far away from its mean: This inequality is very important … More The useful Chebyshev sum inequality
This is post #1 in the series about the “Useful Inequalities” cheat-sheet. Today’s useful inequality: The Weierstrass inequality. In its most basic form (simpler than what appears in the cheat-sheet), it reads as follows. Theorem (Weierstrass): Let be a sequence of numbers. Then Proof: Here is a simple proof by induction. For , we just … More The useful Weierstrass inequality
It has recently come to my attention that not every living being in the world knows the famous “Useful Inequalities” cheat-sheet, produced by László Kozma and available from his website (or here for a local copy). At the time of writing, it is a 3-page document, containing the gist and main formulation of some of … More The useful inequalities cheat-sheet
In my current research, I study hyperbolic surfaces; these are surfaces where every point locally has the geometry of the hyperbolic plane. In this post, we will talk a bit about hyperbolic geometry, investigate various hyperbolic surfaces, answer the question “what is the simplest hyperbolic doughnut”, and see why it is actually a pretzel. More … More The simplest hyperbolic doughnut
1. There is a very special level of hell. A level they reserve for people who write, “This type of result can be extended to higher dimensions, see e.g. [reference1]”, where [reference1] is a 300 page book. How difficult could it have been to cite a specific theorem? a page number? A section? a chapter? … More Four remarks on citations
Seasoned readers of this blog already know the ins and outs of the Erdős–Rényi random graph model. They have probably read the introductory blogpost, have doodled and ruminated about them in class, and have dreamt about them frequently, if not constantly. But until now, they have not had the pleasure of sharing their fresh-from-the-oven random … More New app on Play Store: G(n,p) random graphs!
Some mathematicians like probability, and some mathematicians like graphs, so it’s only natural that some mathematicians like probabilistic graphs. That is, they like to generate graphs at random, and then ask all sorts of questions about them: what are the features of a random graph? Will it be connected? Will it contain many triangles? Will … More Random graphs: The duplicube graph + new paper on arXiv
Humans are animals (in the sense that they are members of the biological kingdom “Animalia”), but in English, they can be animals in a different sense: “John’s a chicken, he won’t dare say something against me” is quite clear to a native speaker. In Hebrew, by the way, saying that someone is “a chicken” makes … More If you can be an animal, why can’t you be math?
Question: Alice decides to visit her good friend Bob. How much time will it take her to get there? Alas, in life, as in mathematics, we do not always know everything, and predicting exactly how much time it will take Alice to get to Bob can be a hard task. But we can try to … More Are we there yet?
Although I would never willingly be part of a club that would willingly accept me as a member, I have relatively recently entered (relative to the cosmic timescale, anyway) the fatherhood club. One perk of the job is that you get to completely indoctrinate and brainwash a child from a very early age. Literally no … More Stochastic Calculus for Babies
Over the past thousand years, I’ve been working on setting up a shared online wiki called the Boolean Zoo (https://booleanzoo.weizmann.ac.il/index.php/Main_Page). The point of the wiki is to collect lots of different examples of Boolean functions, together with pretty much all the properties that are known about them. Each function has a page containing a short … More The Boolean Zoo
In case the previous post concerning concentration on the Boolean hypercube wasn’t enough for you, you can now watch a short, 20-minute animated video that I made for the STOC 2020 conference in theoretical computer science: STOC is a large (by theoretical computer science standards anyway) annual conference, which was supposed to take place in … More STOC 2020 lecture available online
I’m happy to say that my advisor Ronen Eldan and I somewhat recently uploaded a paper to the arXiv under the title “Concentration on the Boolean hypercube via pathwise stochastic analysis” (https://arxiv.org/abs/1909.12067), wherein we prove inequalities on the Boolean hypercube using a cool continuous-time random process. In the previous post, I pretended that I had … More New paper on arXiv: Concentration on the Boolean hypercube via pathwise stochastic analysis
I’m happy to say that my advisor Ronen Eldan and I somewhat recently uploaded a paper to the arXiv under the title “Concentration on the Boolean hypercube via pathwise stochastic analysis” (https://arxiv.org/abs/1909.12067), wherein we prove inequalities on the Boolean hypercube using a cool continuous-time random process. That’s quite a mouthful, I know, and quite unfortunately, … More Catastrophic cubic crash course
Sarcastic Resonance 