AI and the Riemann Hypothesis

Just a couple of months ago, major progress was made on the Riemann Hypothesis, one of the great unsolved problems in mathematics (and one of the seven Clay Millenial Problems worth $1 million to the first person who solves them). What’s especially noteworthy is that this progress was made by AI. From the opening paragraphs of an article in Forbes:

Number theorists spent 37 years moving one number by less than a single percentage point. The number is the proven share of the Riemann zeta function’s zeros that sit on the critical line, the place where one of the most famous unsolved problems in mathematics says all of them belong. Decades of human refinement had carried it to 41.6%. On August 10, Anthropic published a result from an unreleased research version of Claude that moved it to 67.2% in roughly a day and a half.

The Riemann hypothesis has been open since 1859. It predicts the hidden structure behind how prime numbers are distributed, and it carries a million-dollar Clay Institute bounty. The hypothesis itself remains open. What Claude established is that at least two-thirds of the zeros behave the way it predicts, the largest single advance in the record’s history…

The tempting story is that an AI out-thought the mathematicians. The evidence points somewhere more specific. Number theory is organized into specializations deep enough that a career fits inside one of them, and the researchers behind the two lines Claude combined publish in the same journals without needing each other’s machinery. Claude read both lines at once and found that they fit. The advance was sitting in the published literature, waiting for a reader with no specialization.

The Anonymous Math Geek Who Quit Anthropic – and Became the Face of AI Safety

This week’s post is a kind of counterpoint to last week’s post about the job prospects of math majors; competition between AI and Wall Street have driven up wages for math majors who are seeking non-academic jobs.

The Wall Street Journal ran a profile of Jacob Coxon, a former International Mathematics Olympiad participant (representing the UK) and Cambridge math major who worked in commodity training and then for OpenAI and Anthropic before dramatically quitting over fears about the lack of guardrails on AI’s development.

The Million-Dollar Talent Wars for 20-Something Math Geniuses

That was the eye-catching headline of a recent Wall Street Journal article, with the subheading “Flush with cash, Wall Street and Silicon Valley are vying to hire the same tiny pool of genius graduates”.

From the article’s opening paragraphs:

There has never been a better time to be a math nerd.

New college graduates and Ph.D.s are now securing million-dollar pay deals from elite trading firms seeking to secure the best and brightest amid fierce competition from artificial-intelligence companies.

So-called quant firms, which use sophisticated mathematical models to come up with trades, have for years wooed top young talent with lucrative compensation that big banks struggle to match. Now, the AI boom is pushing those numbers even higher.

Just a few years ago, early-career packages pushing seven figures were anomalies, said Matt Stabile, founder of New York-based recruitment firm Stabile Search. “But a million dollars is something people don’t even bat an eye at anymore.”

“The delineation is pre-OpenAI and post-OpenAI,” Stabile added, “that’s when you saw competition really take off.”

The skills required to train large language models have always overlapped with quantitative finance, but the connection has deepened as trading firms have pivoted toward machine learning and AI to power their trades in recent years. Now, AI labs such as OpenAI and Anthropic are vying for the exact same tiny pool of genius math majors and Ph.D.s as Wall Street.

The competition poses a challenge for quant firms, which have sought to sweep up the next generation of math whizzes to retain the trading edge that has made them a force in finance.

Ken Ono feared AI. Now he trains it

Physics Today recently published an interesting and thought-provoking Q&A with eminent mathematician Ken Ono, who recently took a leave of absence from his tenure-track position at the University of Virgina to join Axiom Math, an company developing AI tools to verify mathematical proofs. From the article’s opening paragraphs:

“Am I doing a better service to science by writing 20 more papers that only people in my field will read or by participating in this movement?” Ken Ono asked himself last year. At the time, Ono was an endowed chair at the University of Virginia. In the following months, he’d take a leave of absence from the university, move to Silicon Valley, and become the founding mathematician at an AI startup, Axiom Math.

Axiom is one of several companies focused on developing AI tools to verify mathematical proofs. In the future, similar tools could be applied to check AI-generated computer code, a potentially lucrative service now that AI tools can write lines of code faster than any human can debug them. Startups Harmonic and Math Inc as well as Google’s DeepMind are targeting similar AI-powered verification tools.

Axiom has attracted investor attention, reaching a valuation of $1.6 billion earlier this year. (Ono says that he has shares in the company and that he receives a salary that is similar to his former compensation at Virginia.) His interests, he says, are in helping researchers use AI to think in more-creative ways and across more disciplines.

This Musical’s Secret Weapon Is A Math Major

The Wall Street Journal recently published an interesting article (well, I found it interesting) about how “applied mathematics [is being] used to move around six different audiences in a new version of The Phantom of the Opera.” From the article’s opening paragraphs:

Simon Broucke didn’t expect to use his mathematics degree for his music assistant job at “Masquerade,” the new interactive revival of “The Phantom of the Opera” in New York.

But when “Masquerade” producers were struggling to figure out how to make it so the actors and the audience moved through the show on schedule, Broucke, 26, volunteered his background. He has dual degrees from Columbia University in music and mathematics, which uses spreadsheets and modeling to solve real-world problems. 

“Masquerade” follows the point of view of the Phantom, a disfigured musical genius who lives below the Paris Opera House and is obsessed with singer Christine Daaé. Producer Randy Weiner and his wife, the director Diane Paulus, needed to get the audience in lockstep from the start to the finish of the show in just under two hours. They also needed to get all 42 actors to their spots to perform at the right time throughout the space.

This would be simple if there was only one audience. Instead there are six. 

I thought this was a fun application of mathematics in an utterly unexpected context.

News flash: ChatGPT sometimes asserts false statements are true (Part 2)

I feel like I’ve done my good deed for the day by uncovering another instance when ChatGPT claimed a “fact” from the secondary mathematics curriculum that is simply incorrect. This one’s a doozy: it claimed that the binomial coefficent \displaystyle {23 \choose 6} = 100,947 is equal to \displaystyle {22 \choose 5} = 26,334. (The first identity in the first line below is correct; the second one is not.)

Lesson: ChatGPT is a nice tool but you get what you pay for.

Predicate Logic and Popular Culture (Part 297): Megamind

Let p be the statement “This is an Easter Bunny,” let q be the statement “There is a tooth fairy,” and let r be the statement “There is a Queen of England.” Translate the logical statement

\sim p \land \sim q \land \sim r.

This matches a line from the movie “Megamind.”

green line

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 296): Kanye West

Let H be the set of all things, K(x) be the proposition “x kills you,” and let S(x) be the proposition “x makes you stronger.” Translate the logical statement

\forall x \in H( \sim K(x) Rightarrow S(x) ).

This matches the first line of “Stronger” by Kanye West.

green line

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.