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Alex Kontorovich Profile picture
Alex Kontorovich
@AlexKontorovich
Mathematician (Distinguished Professor of #Math at @RutgersU). Here to learn about research, education, and community. Let’s build something together.
Sep 28 • 36 tweets • 40 min read
“The Shape of Math To Come”

(with apologies to Ornette Coleman for the title…)

This thread contains slides from (a version of) my talk at last week’s CMSA conference at Harvard, discussing a vision for what research mathematics may look like in the age we now seem to be entering, of AI and formalization.

We will decompose our discussion into three broad categories: (1) the Discovery of new mathematics, (2) the Teaching of mathematics, and (3) the better Understanding of existing mathematics and Communication of it to others. The difference between Teaching and Communication is that the former is meant to apply to newcomers to our subject, whereas by the latter, I mean communication among professionals, perhaps even experts in the same field. I originally wanted to split Understanding and Communication into their own categories but could not find anything that meaningfully distinguished them: any improvement in understanding led to a corresponding improvement in communication, and vice versa.

The old adage “it’s hard to make predictions, especially about the future” (variously attributed to Yogi Berra, Niels Bohr, and others) is very apt here. So we begin by taking a step back.Image
Image At the turn of the millennium, some of the world’s most prominent mathematicians gathered in Tel Aviv for a conference to discuss what mathematics might look like in the 21st century. The proceedings were published as a special issue in GAFA called “Visions of Mathematics”, and I strongly recommend one read the whole issue. A particularly prescient contribution was given by Tim Gowers @wtgowers, titled “Rough Structure and Classification”.Image
Sep 19 • 5 tweets • 4 min read
Congratulations to Jesse, Jared @jdlichtman, and Christian @ChrSzegedy on this great result! (They told me and Terry about it weeks ago, but released it while I was giving a lecture series in Italy last week, followed by speaking at a conference this week at Harvard -- where I got to chat some more with Jared; so I’m only now getting around to perusing the blueprint+code.) What I’m impressed by:

(cont'd) • Generating 25K lines of Lean code that compiles! If I understand correctly, *none* of the Lean was touched by humans; the statements as well as the proofs are generated by Gauss. Anyone who’s tried to get 10 or 100 lines of AI-generated code in one shot, knows how many orders of magnitude more difficult that is!
The code is, in some places, highly nontrivial, and even looks nice! Compare their formalization of Borel-Caratheodory to a some places in PNT+ where Thomas Hubert and I collaborated with AlphaProof (which trained via massive RL, and so learned extremely … unorthodox strategies); the latter is completely incomprehensible.Image
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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:
Apr 11, 2020 • 13 tweets • 3 min read
This is just awful news. My introduction to "real" mathematics came in the fall of my freshman year, when I took Linear Algebra from Conway. We have 100 Conway stories from that one course alone. How do you learn vector spaces? Compute dimension of the space of 3x3 magic squares! 2/ Why is the Jordan normal form useful? To compute asymptotic growth rates of the Audioactive Decay sequence!

He explained simultaneous diagonalization as follows: brought in a wire hanger, penny, and umbrella. Pulled the hanger into a diamond shape and held it on his finger,
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 23, 2019 • 32 tweets • 8 min read
Collecting here a meta-thread of threads. Thread on Pi Day

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, ...