My most striking contribution to geometry is, no doubt, my problem on the number of distinct distances. This can be found in many of my papers on combinatorial and geometric problems.
-Paul Erdős, On Some of My Favorite Theorems, 1996.
Erdős is one of the greatest mathematicians from history, and he considered this problem one of his greatest contributions. Let’s dive into what it is and what is known.
There are few popular problems in math that can be explained easily and have current research across many disciplines.
Something like the Hodge Conjecture cannot be explained easily, but there is…
When people think about the health benefits of giving up alcohol (or any drug), they tend to be long-term and difficult to quantify: lower blood pressure, heart health improvement, better sleep, lower cancer risk, better liver function, and so on.
I certainly had no idea when I stopped drinking that it would be the most important factor in a very concrete diagnosis for me. I have no idea how long I would have lived with Lyme disease if I hadn’t stopped, but it almost certainly would have developed into a chronic problem with serious complications.
Lyme disease is a tick-borne…
Before we begin, you should know that I’m not actually going to present a proof of the Riemann Hypothesis.
This article is about a fictional object known as the field with one element, sometimes denoted Fᵤₙ. You can probably guess why: F is for field and “un” is the French word for 1.
When I first heard about this in grad school, I thought it was a joke. The object is “Fun” and it doesn’t really exist. How could people actually take this seriously?
But a lot of great mathematicians have done quite a bit of work on it: Jacques…
There’s a narrative in our culture about the good kids. It goes like this. The straight-A, rule-following (or religiously devout) kids get to college or move out of the house and then all that repression comes out in a flash of rebellion.
The good kids are the ones that go wild once they no longer have to follow the rules. The idea is that we (I count myself among them) secretly desired to be breaking all the rules the whole time. We just needed an outlet to do it.
Another way people think of this narrative is that strict parenting…
Our brains have been hijacked. We binge watch Netflix or browse social media for hours. We use substances to ease the stress. It may seem like these behaviors aren’t related, but the underlying brain chemistry is very similar.
The modern world hyperstimulates our brains, and this makes it harder and harder to do anything that isn’t these addictive behaviors and substances. We need practices in our lives that calm things down and return our serotonin levels to a normal baseline.
People talk about “digital detoxes” and meditation as ways to do it. …
Recently I’ve been reading Elaine N. Aron’s famous book The Highly Sensitive Person: How to Thrive When the World Overwhelms You. It came out in 1997 and I know a lot of people who say it changed their life.
I’ve always been a bit skeptical about how much use it could have, which is why I’m so late to the party.
Well, I have to say I’m shocked. …
Sometimes I glance through the current Annals of Mathematics to see what types of new results are being recognized as important. If you’re unaware, Annals is widely considered to be the best journal of math research (obviously, others are in contention and I mean no disrespect to them).
If your research gets accepted there, it has something important to it: new ideas, old problems, widespread interest, etc.
I ran across an article entitled “Enumerating Number Fields” by Jean-Marc Couveignes this year. …
I’ve put off writing about this topic for some time because there’s no way to say it without sounding entitled and egotistical.
And the fact that this is something I have to come to terms with means that in my head, I somehow believed otherwise.
Why would I think that I’d have some grand impact on the world? Why would I think I’d be one of the handfuls to be written about in the history books?
Well, I am egotistical and entitled and that’s something I’ve been working really hard on since coming to sobriety.
I won’t bore…
There’s this meme out there to make people who care about artificial intelligence safety look crazy. It goes like this. If AI ever starts doing something that might destroy humanity, we’ll just shut it off.
AI requires power and computers to function. So, if machines start building nuclear weapons of their own free will or turning everything into paper clips, we’ll have no problem. We’ll just pull the plug. Problem solved.
Some prominent voices on the topic try to refute this argument by saying we don’t know the types of extremely strong arguments and persuasion a hyper-evolved machine can produce…
I always say I don’t have a rock bottom. I stick to that, but what I did have was a lightbulb moment when I realized I had a problem.
This is the first time I’m sharing this moment.
To set the scene, I was training for my fifth half-marathon. The first few I had done were to just experience it and get through it.
This was the first one where I specifically wanted to train for speed. I had picked a flat course in cool October to increase the chance of getting a personal record (no hills and no heat).