While the internet is ablaze (relatively speaking) about the AI-generated solution to the unit distance problem, I must remind the general public that a totally different problem with the name “distance problem” is still open! Prelude Here is a simple question which can fit right in on the exam of an “introduction to probability” undergraduate … More The distance problem is still open
This is post #4 in the series about the “Useful Inequalities” cheat-sheet. Today’s inequality: The Lehmer mean inequality. After I finished the original three “useful inequalities” blogposts, I asked my friend and then-officemate Yotam Dikstein if there is any particular inequality he would like to be featured next. “Lehmer’s inequality”, he solemnly answered. “I know … More The useful(?) Lehmer inequality
Today is the anniversary of my arrival at Cambridge. I have been here for one year; the city, on the other hand, has been here for quite a bit longer, there having been some sort of permanent settlement on Castle hill since 70 AD. Even if you consider its official founding date to be that … More A year in Cambridge
When pressure builds in ancient depths,the earth trembles.At times just a tiny tremor,at others erupting mountains and flowing lava.This is a law of nature. When leaves fall dry and shrubs grow rampant,all it takes is a spark.At times no more than a meadow of parched grass,at others a whirlwind of consuming flame.How could it be … More Self organized criticality
There are three types of mathematicians: those who know how to count, and those who don’t. In this post, we will see that luckily, if you know how to integrate, you actually do not need to know how to count. Along the way we will see a beautiful interplay between combinatorics, probability, and topology. Combinatorics … More Now you are counting with Gaussians
Recently, I have been watching the Harry Potter movies on Netflix. After the first three, the program already learned what I’m doing, and started suggesting the next one in the series as soon as I logged in. Now, alas, my entire home screen is filled with Potter and Potter-related media, something which will undoubtedly take … More You may also like
A well-used door needs no oil on its hinges.A swift-flowing stream does not grow stagnant.A deer blends perfectly into the forest colors.Software rots if not used. These are great mysteries. The Tao of Programming, Geoffrey James And so it is with Google Play. Two years ago I published the incredibly successful app on the play … More G(n,p) does not decay
When I was a kid, I learned about the seasons. Autumn, winter, spring, and summer, repeating in an endless cycle. Autumn, I must admit, I didn’t much notice. But in winter it rained, and sometimes hailed, and we would watch the droplets water our tiny vegetable garden. In spring every square centimeter bloomed and flowered, … More The seasons
I’m happy to say that my postdoc advisor Asaf Nachmias, his student Guy Lachman and I have recently uploaded a paper to the arXiv under the title “A sharp lower bound on the eigenvalues of surfaces” (https://arxiv.org/abs/2407.21780). In this post, I’d like to give some background to the problem, as well as outline the main … More New paper on arXiv: A sharp lower bound on the eigenvalues of surfaces
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
Oh! The places you’ll go! You’ll be on your way up!You’ll be seeing great sights!You’ll join the high flierswho soar to high heights. You won’t lag behind, because you’ll have the speed.You’ll pass the whole gang and you’ll soon take the lead.Wherever you fly, you’ll be best of the best.Wherever you go, you will top … More You only need one
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
Sarcastic Resonance 