Million Dollar Problems of Mathematics

Ken Ono, a renowned mathematics genius at the University of Virginia and former vice president of the American Mathematical Society, made the improbable choice at age 57 to leave academia and join a Silicon Valley startup called Axiom Math. He is leaving to work alongside 24-year-old math prodigy and former student Carina Hong. Hong dropped out of her joint law and math studies at Stanford after raising $64 million to build the first truly capable machine mathematician, with the goal of teaching artificial intelligence to prove theorems rather than just predicting words. In the race toward mathematical superintelligence, Axiom Math competes against three main approaches: formal verification languages like Lean, natural language models that excel at analogies, and hybrid architectures that attempt to combine both intuition and ironclad logic. This shift in artificial intelligence was underscored by the 2025 International Mathematical Olympiad, where experimental reasoning systems from OpenAI and Google DeepMind achieved scores meeting the gold medal threshold by solving five out of six problems under strict, unassisted conditions. Recognizing how rapidly these models are advancing, Ono relocated across the country to serve as Axiom's founding mathematician, designing complex problems to test and push these systems to their limits.

Calculus, independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, has powered our understanding of science and driven technological progress from the industrial revolution to modern artificial intelligence. In this episode, we explore the jagged function that broke calculus, examining how mathematicians for two centuries relied on the intuitive dogma that continuous, unbroken functions must be smooth and differentiable almost everywhere, as claimed in André-Marie Ampère's 1806 proof. We examine how Karl Weierstrass shattered this comfortable world in 1872 by presenting a function constructed from an infinite series of cosine waves that is continuous everywhere yet differentiable nowhere. Finally, we investigate the profound shock this "deplorable evil" caused among contemporaries like Henri Poincaré and Charles Hermite, forcing mathematicians to abandon flawed intuition and rebuild mathematical analysis from the ground up.

A 125-year-old math challenge, originally proposed by David Hilbert as his sixth problem at the 1900 International Congress of Mathematicians in Paris, sought to find a rigorous logical bridge connecting the microscopic world of Newtonian particles to the macroscopic flow of continuous fluids. In this episode, we explore how mathematicians finally linked these worlds to ground physics in law, addressing the deep paradox of time's arrow where reversible microscopic laws give rise to the irreversible, one-way flow of macroscopic systems. We examine the three-rung ladder of reality—from Newton's laws to Ludwig Boltzmann's mesoscopic statistical equation and the Navier-Stokes equations—alongside Oscar Lanford's 1975 landmark proof which was restricted to very short timescales before modern breakthroughs solved the exploding history of particle interactions.

Knots are found throughout our daily lives in curling smoke, matted yarn, braided hair, and tangled earbud cords, while statistical mechanics indicates that a long, agitated string has a 100 percent probability of forming a knot. In this episode, we explore the periodic table of molecular knots, examining how knots form naturally in about one percent of all known proteins and in tightly packed DNA coils, where they can disrupt cellular functions unless untangled by topoisomerase enzymes. We examine the mathematical and chemical definition of a true closed-loop knot characterized by a mechanical bond, minimum crossing numbers, and algebraic knot invariants like the Alexander and Jones polynomials. Finally, we investigate how chemists use directed self-assembly around metal ion scaffolds and analytical techniques like X-ray crystallography and NMR spectroscopy to synthesize and verify increasingly complex structures, from Jean-Pierre Sauvage's 1989 trefoil knot to record-breaking tight molecular knots.

This episode explores the hidden mathematical heartbeat of the ocean, where vibrant underwater cities are being defended by an unexpected alliance of physicists and mathematicians. While coral reefs support a quarter of all marine species and over a billion human livelihoods, they are currently under siege from marine heatwaves, acidification, and invasive predators. The journey takes listeners from the depths of recursive fractal growth and logistic curves to the cutting edge of conservation technology. You will discover how researchers use "Degree-Heating-Weeks" to forecast mass bleaching, apply the "traveling salesman" puzzle to outsmart crown-of-thorns starfish, and utilize graph theory to reconnect isolated reef "nodes" via larval dispersal. From 3D-printed terracotta tiles with 95% survivorship rates to the future of quantum-accelerated fluid models and AI-driven digital twins, this exploration reveals how the clarity of numbers is providing a vital lifeline for the ocean's beating heart.

This episode investigates the mind-bending Banach-Tarski Paradox, a mathematical theorem that suggests you can take a solid ball, cut it into a finite number of pieces, and reassemble them into two identical balls of the same size as the original. Often called the "Pea and the Sun Paradox," this 1924 discovery by Stefan Banach and Alfred Tarski defies our common-sense understanding of volume and matter. You will learn how the "Axiom of Choice" allows mathematicians to create bizarre, infinite scatterings of points that don't have a measurable volume in the traditional sense. The journey explains how infinite sets—like the collection of all whole numbers—behave differently than finite ones, allowing a part to be as "big" as the whole. From the uncountably infinite points of a sphere to the "non-amenable groups" that make such rearrangements possible, this exploration reveals the strange logic of set-theoretic geometry where one plus one doesn't always equal two

This episode explores the hidden mathematical heartbeat of the ocean, where vibrant underwater cities are being defended by an unexpected alliance of physicists and mathematicians. While coral reefs support a quarter of all marine species and over a billion human livelihoods, they are currently under siege from marine heatwaves, acidification, and invasive predators. The journey takes listeners from the depths of recursive fractal growth and logistic curves to the cutting edge of conservation technology. You will discover how researchers use "Degree-Heating-Weeks" to forecast mass bleaching, apply the "traveling salesman" puzzle to outsmart crown-of-thorns starfish, and utilize graph theory to reconnect isolated reef "nodes" via larval dispersal. From 3D-printed terracotta tiles with 95% survivorship rates to the future of quantum-accelerated fluid models and AI-driven digital twins, this exploration reveals how the clarity of numbers is providing a vital lifeline for the ocean's beating heart.

In this episode, we step down into the sub-microscopic world of chemistry to explore the groundbreaking construction of a "Periodic Table of Molecular Knots". While statistical mechanics dictates that any long, agitated string will eventually tangle with 100% probability, nature relies on knots at the tiniest scales, tying loops into roughly 1% of our proteins and packing knots into the tight coils of our DNA. We look inside the cell to meet topoisomerases. These specialized biological untanglers cut, pass, and reseal our molecular threads to keep the genetic code from breaking or mutating under stress.But the real magic begins where fingers and tweezers are entirely useless. We follow the historic journey of chemists learning to tie individual molecules on purpose. Moving past the early 1989 Nobel Prize-winning synthesis of a simple three-crossing trefoil knot, modern chemistry has harnessed a brilliant technique called "directed self-assembly", using transition metal ions as charged scaffolding to orchestrate complex molecular weaving. We map out the mathematics of topological crossing numbers, look at the specialized Python software tools used to verify these structures, and marvel at the 2024 gold-based world record holder for the tightest knot ever tied by human ingenuity.

In this episode, we venture into the deeply dramatic history of infinite mathematics to unlock the enigmas of how we count things that never end. We begin in October 2018 with mathematician David Asperó on a vacation in Italy, experiencing an epiphany that would lead to a landmark proof alongside collaborator Ralf Schindler.Published in the Annals of Mathematics, their work gracefully unites two historically rival axioms, dealing a heavy theoretical blow to one of the most famous mathematical guesses of all time: the 1878 Continuum Hypothesis. We trace this battle of ideas back to 1873, introducing the brilliant, tortured genius Georg Cantor, the first man to systematically explore the scales of infinity. We walk through his logical mind-benders, utilizing an infinite auditorium metaphor to show how Cantor shattered common sense by proving that "half" of an endless set is the same size as the "whole". Finally, we pull apart his legendary "diagonal argument" thought experiment, demonstrating the breathtaking mathematical magic trick he used to reveal that decimals form a smooth, continuous line that can never be listed, transforming infinity from a single abstract concept into an intellectually exciting playground of competing mathematical foundations.

We step into a mind-bending, ruler-banned universe where objects behave like endlessly flexible play dough. IIn the world of topology, you can stretch, twist, or compress a shape across galaxies or down to a speck, but you can never tear the dough or poke a new hole. We uncover the fascinating mathematical rules that famously prove a coffee mug and a doughnut are structurally identical, transforming complex geometry into a robust form of dynamic arithmetic. We walk through the creation of a mathematical "hole scorecard" that pinpoints the shape's permanent DNA. To do this, topologists have to bypass everyday definitions of space and use the strict "rubber band test" to separate smoothable dents from permanent tunnels. We explore the brilliant system of Betti numbers, formalized by Henri Poincaré, and trace how mathematicians map out hierarchies of emptiness, from disconnected islands to deep tunnels and trapped, hollow cavities. Finally, we dive into the elegant framework of homology, discovering how scientists look for "nothing" by tracking the physical boundaries that surround it.