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Alex Kontorovich Profile picture
Alex Kontorovich
@AlexKontorovich
Mathematician (Prof of #Math at @RutgersU). Here to learn about research, education, and community. Let’s build something together.
Jun 16, 2023 • 18 tweets • 6 min read
then they *would* solve professional-level mathematical problems. (This is not unrelated to issues of "alignment", which I don't have time or expertise to go into here.) Image And even if I’m completely wrong, and LLMs are indeed capable of producing, in natural language, something that reads like a perfect math paper, how can we ever trust it? We now have to go and referee these thousands or millions of papers that it can produce and determine which,
Jun 16, 2023 • 26 tweets • 6 min read
Notes on my lecture at the National Academies workshop on: AI to Assist Mathematical Reasoning. 🧵

I find it useful to work backwards from an end goal; what should be our “Holy Grail” here? Image Perhaps it’s that AI should solve the Riemann hypothesis. I see two ways this could go:

1. AI might give a million-line, dense, incomprehensible proof of RH, and I spend the rest of my life just trying to understand what it's saying and why. (Nightmare) Image
Mar 26, 2023 • 8 tweets • 5 min read
Awesome @MoMath1 presentation on the discovery of the Hat! A summary 🧵:

This is Dave Smith, a mathematical artist. He spends *a lot* of time just messing around, seeing what shapes he can tile in usual ways.

Nov 20, 2022, he emails @cs_kaplan to say: he can't figure out... 2/ how to get this shape to tile periodically. [By the way, Craig, I'd love to know more of the history predating this email -- how did he stumble onto it?]

4 days later: "now wouldn't that be a thing?" !!!

There are two problems: (i) does it really tile the whole plane? And...
Nov 14, 2021 • 10 tweets • 3 min read
Why I'm excited about @AMathRes: It's as a startup. It's an experiment. Whether it succeeds or fails will be a function of how good its ideas are, and how hard the people that get involved work. (That's why it's rather disheartening to see a coordinated campaign to ... 2/ intimidate founding members into resigning, based purely on ad hominem attacks...) The AMR, again, is only in an organizational phase; the membership (now being assembled) will decide what projects they wish to put their energies towards. That said, here are some of the ...
Jun 30, 2021 • 13 tweets • 4 min read
Ugh. Another media frenzy over another purported proof of RH. Usually I just delete such media requests without reply. For some reason (perhaps the fanfare with which this story is spreading), I felt it was my turn to have a quick look and debunk things. TL;DR No, RH isn't proved 2/ If you want to read the paper yourself (which all the press releases seem to not want to let you do; they want you to read the reports of their panels of "experts"- many of whom, if you actually look at the reports, say it's not a proof!...), it's here: researchgate.net/publication/32…
May 11, 2021 • 31 tweets • 11 min read
Thread: What is the Music of the Primes? (From last night's @MoMath1 lecture on the Riemann Hypothesis. TL;DR: watch this @QuantaMagazine video )

The tricky thing about prime numbers is that they’re defined by what they’re not. A composite number is a 2/ product of two other whole numbers; for example, 28 is composite (and hence not prime) since 4 × 7 = 28. For millennia, primes have fascinated people of all ages. Even babies can understand primes! Indeed, before my oldest could talk, he enjoyed playing with a set of 20 blocks
May 10, 2021 • 4 tweets • 1 min read
Steve (et al), I'm curious for your thoughts on what I think of as the "helix" model of education: introduce many ideas well before the student is ready, in tiny bites, moving on and circling back again and again, so when they get the "big reveal", the concepts are familiar? Eg 2/ "Algebra" should start in pre-K. "You want 6 M&Ms and I gave you 4. So 4+X=6. What is X?" What we do now is: before 6th grade, all symbols are numbers. Then out of nowhere, letters?!?

"Cartesian coordinates" could also start in pre-K: "Go six to the right and five up".
May 10, 2021 • 4 tweets • 1 min read
Another great question that's above my paygrade to answer. Help please?

My take is just to meet the kid wherever they are. Before starting anything, I find it important to map their understanding, specifically the boundary thereof. What's easy, and what's too hard? I also ... 2/ consider timing. Sometimes they're just not in the mood, leave them alone! I also start them very early. Once they can talk, they should count (learn the words in order). Then they should really count (often with treats/dessert): "how many M&Ms"? That way math is just a part..
Jan 21, 2021 • 4 tweets • 2 min read
I'm a big fan of @viktorblasjo's perceptive analysis, but I think on this note about the 5th postulate, we disagree. If Euclid was so concerned *here* with construction (I agree he was elsewhere!), why not simply define the 5th postulate in terms of the construction we all ... 2/ know of parallel lines? New 5th postulate: "Construct a perpendicular off a line, then construct another; the first and last lines don't meet". That would be much more constructive than "Suppose someone gives you two lines and a third line that crosses the two, and you measure
Dec 18, 2020 • 4 tweets • 1 min read
I just stumbled onto the paper that came out of my very first research job (Memories 🎶...).

jstor.org/stable/3211603

Summer after high school I worked for Christina Paxson (then at Princeton, now Brown prez), essentially writing code to numerically solve certain PDEs. The job.. 2/ was passed down to me from her previous research assistant (coincidentally, my brother).

The summer after that, I was living in NYC (Misty water-colored memories 🎶...), by day, doing research at NYU for Neil Chriss, by night, sitting until 4 am at Smalls (their motto ...
Oct 11, 2020 • 15 tweets • 4 min read
Triple product L-functions arise naturally in (at least) the following two ways:

1) (Arithmetic) Quantum Unique Ergodicity (AQUE)
2) Langlands functoriality for GL(2)xGL(2)

Let's first talk about AQUE (for which Lindenstrauss won a Fields). The simplest setting is: ... Image 2/ you have a surface M and look at the Laplacian acting on L^2(M) (really the unit tangent bundle of M, where geodesic flow lives, but nevermind...). Each eigenfunction phi gives rise to a probability measure |phi|^2 dx, and a basic question in quantum chaos is: what happens ... Image
Aug 12, 2020 • 9 tweets • 3 min read
Hey everybody! Gather round; I just discovered this incredible ancient way for multiplying! It's so simple; here's what you do: Say you want 3x4. So you make 3 horizontal lines, and 4 vertical lines, and count the intersections! I can't believe nobody ever showed me this!(cont'd) 2/ Actually I'm not just trolling, I do have a point. People keep getting excited by "new algorithms" like this because adults, while fully capable of executing the "standard" algorithm, actually have zero understanding of *why* it works. Let's look at it together:
Jul 20, 2020 • 14 tweets • 3 min read
Ok, take the quadratic polynomial

f(n) = n^2-n+41,

and notice that f(1)=41, f(2)=43, ... f(40)=1601 are *all* prime. There's a rather deep reason for this. (Thread)

The discriminant of the quadratic is

D = B^2-4AC = (-1)^2 - 4*41 = -163.

Here's a crazy fact about this number 2/ Let chi(n) = 0 if n divides D, 1 if n is a square mod D, and -1 otherwise (this is the "quadratic Dirichlet character" mod D). The first many primes are all non-squares mod D! Just like Riemann zeta, the L-function of chi

L(chi,s) = sum chi(n)/n^s

has an Euler product:
Mar 6, 2020 • 4 tweets • 3 min read
With Pi Day just around the corner, let’s remember what Pi is all about.

After washing your hands thoroughly, cut the crust off a pizza pie and lay it across four others. You’ll see that the crust spans a little more than 3 pies. That’s Pi ≈ 3.14.

But that’s not all! (Cont’d) 2/ Remember that

Area = Pi r^2

formula for a circle? That’s not really the formula. It should be:

Area = radius x (half circumference),

where circumf=Pi*diameter, so half of that is Pi*radius.

Proof: cut your pizza into small slices and arrange into an r x (C/2) rectangle...
Jan 3, 2020 • 17 tweets • 5 min read
Forgot to record this. Adding to meta-thread (so I can find it again later...)

Nothing to look at here. Just recording these to meta-thread so I can find them later as needed. (Is there a better way?...)





Sep 14, 2019 • 17 tweets • 4 min read
Unpopular opinion: The 3x+1 Conjecture might be False!

Here's why I think this may be the case. (thread)

My first paper, with Y. Sinai in 2002, arxiv.org/abs/math/06016… proves that 3x+1 paths are a geometric Brownian motion (in a precise asymptotic sense), with drift log 3/4 < 0 2/ This suggests typical trajectories decay, and can be used to recover the fact (proved before us) that almost every seed eventually reaches a value below itself (but this cannot be iterated, since the paths could fall into a very sparse divergent trajectory). For a long time,
Sep 10, 2019 • 13 tweets • 3 min read
As often happens in mathematics, Fourier was trying to do something completely unrelated when he stumbled on Fourier series. What was it? (thread)

He was studying the propagation of heat in a uniform medium. To keep things as simple as possible, say you have a 1-d pipe, (cont'd) 2/ with position represented by 0<x<1, and let u(x,t) be the temperature at x at time t. He reasoned that heat at the next instant changed by taking a "local average" of nearby temperatures. For reasons I won't get into here, this local average is modeled by the Laplacian, ...
Aug 27, 2019 • 5 tweets • 2 min read
Most real analysis books start by saying: we have to develop Lebesgue integration, and before that measure theory, because Riemann int is insufficient. Why not? You can't integrate f(x)=1 on rationals and 0 on irrationals, or some other family of concocted functions. (cont'd) 2/ I always found this rather disingenuous. Who cares about these weird functions? Is that really why people decided Riemann was insufficient? I don't think so, but I'm not a historian. The Hubbards have a book where they prove the main thms of Real Analysis w.o. measure theory,
Jun 11, 2019 • 15 tweets • 4 min read
With summer coming soon, I was asked to explain the mathematics of the seasons.

"The Mathematics of the Seasons, an Ode to Trigonometry"

Many creatures on Earth experience seasons, here transitioning from spring towards summer. Why does this actually happen? And how do we know? (2/n) The first giant leap in our understanding of this phenomenon came once humanity accepted heliocentrism, that the Earth revolves around the Sun, and not, as one might naturally guess, the other way around. Today we could find this out by sending rockets into space...
Mar 12, 2019 • 24 tweets • 6 min read
Some people have asked about #PiDay, so here are some thoughts (warning: long-ish thread).

The number π ≈ 3.14, that is, the ratio of circumference to diameter of any circle, has continually captivated mankind since at least the ancient Babylonians four millennia ago. It even appears in the Hebrew Bible, where King Solomon’s Temple is described as containing a perfectly round ritual basin (“Molten Sea”) measuring exactly 10 cubits across and exactly 30 cubits around. This may suggest that π should be 30/10 = 3 identically,