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. I

In 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.

This episode explores the mathematical conflict between the Minimalist Conjecture and the chaotic data found in the study of numbers.

The story traces a 2,500-year quest to find rational solutions to equations, a pursuit that began with the Pythagorean obsession with fractions and the discovery of irrational numbers.

While mathematicians have mastered linear and quadratic equations, elliptic curves remain a stubborn mystery.

The narrative explains how these curves build rational points through a unique geometric trick: drawing a line through two known rational points to find a third, which is then reflected to create a new solution.

This ability to generate infinite solutions from a "starter kit" leads to the concept of rank, which measures the number of independent points needed to produce every other rational solution on the curve.

This episode explores the thirty-year quest to create a periodic table for the shape of space.

Mathematician William Thurston revolutionized geometry by proposing that every three-dimensional manifold is composed of pieces belonging to one of eight specific geometric environments.

While most categories are rare, the vast majority of spaces are hyperbolic—bizarre "dark matter" shapes that are larger on the inside than the outside and expand exponentially.

Thurston hypothesized that these chaotic hyperbolic worlds are secretly built upon a highly structured skeleton of "surface bundles," which only become visible when the space is "unrolled" through a mathematical tool called a covering space.

This obsession to find order within intense curvature remained a dream for decades because the wild nature of hyperbolic geometry tended to rip apart any surface researchers attempted to construct.

This episode explores the ambitious and arguably obsessive quest to prove the most self-evident fact in mathematics: $1 + 1 = 2$. At the turn of the 20th century, the mathematical world was thrown into turmoil by logical paradoxes, such as the famous Barber Paradox, which threatened the very foundations of certainty. In response, an unlikely duo of Cambridge mathematicians, Bertrand Russell and Alfred North Whitehead, spent a decade attempting to rebuild all of mathematics from scratch using pure logic. Their goal was to realize the centuries-old dream of a universal symbolic language where every truth could be mechanically calculated. This journey through "Logicism" required them to navigate the failures of predecessors and the complexities of "classes of classes," ultimately resulting in a monumental 360-page derivation just to reach the most basic arithmetic sum. It is a story of grand philosophical ambition, meticulous precision, and the staggering amount of work required to prove what we often take for granted.

Prime numbers are the building blocks of math—but their patterns remain one of the biggest unsolved mysteries. That’s where James Maynard comes in.From finding hidden structures in prime gaps to proving that some primes never contain a specific digit, Maynard has transformed our understanding of numbers. His work has massive implications, from strengthening cybersecurity to shaping the future of quantum computing.
This episode dives into the mind of a Fields Medalist who isn’t just solving old problems—he’s redefining the field. His discoveries are unlocking new ways to see randomness, patterns, and the very fabric of mathematics itself.
Numbers are everywhere, and thanks to Maynard, we’re finally beginning to understand them. Get ready for a deep dive into one of math’s greatest unsolved mysteries—the secret life of primes.

At first glance, it’s just a 2×2 grid. Four exponential numbers. No flashing lights, no smoke and mirrors. But hidden in that tiny setup is a mathematical riddle that’s resisted solution for nearly a century.
In this episode, we explore the Four Exponentials Conjecture, a quiet giant in the world of number theory. The idea is simple: if you pick two rationally independent numbers for your rows and two for your columns, and build exponentials from the combinations, at least one result must be transcendental—guaranteed.
That might sound like splitting hairs, but the implications are enormous. Proving this conjecture could unlock the deeper mysteries of exponential behavior, help us understand how "wild" numbers emerge, and even nudge open the gates to solving Schanuel’s Conjecture—one of math’s biggest unsolved problems.
We trace its origins from the 1940s to today’s cutting-edge attempts. You’ll hear how this compact problem bridges algebra, transcendence, and mathematical philosophy. Why can’t we trap all four numbers in the algebraic world? Why does this matter?
Because sometimes, proving one number is “weird enough” is all it takes to rewrite the rules.

What if geometry could guarantee a perfect shape—no matter how random your mess? Welcome to the world of the Happy Ending Problem, a mind-bending puzzle in combinatorial geometry that starts with just a handful of dots… and ends with a nearly century-old mystery still unsolved.
In this short documentary, we explore a charming-sounding problem with serious mathematical bite. Originally sparked by a group of Hungarian mathematicians in the 1930s—and rumored to have sparked a romance too—it asks: how many randomly placed points does it take to guarantee a convex polygon of a given size? We know the answer for small cases. But for larger shapes? It's still an open question.
We unravel why this simple-sounding puzzle hides deep complexity. From the ideas of Ramsey theory to breakthroughs in computational geometry, you'll hear how mathematicians—armed with clever algorithms and bold theory—keep pushing toward an answer.
At its heart, this is a story about inevitability: that in chaos, patterns will always emerge. Whether you're a math lover or just here for the beautiful strangeness of it all, you’ll find yourself hooked on the puzzle that promises a happy ending… but won’t tell us when.

Ever wondered what stacking oranges has to do with data encryption? Turns out, everything.For over 400 years, mathematicians struggled to prove the best way to pack spheres in space. Even Isaac Newton had theories but no proof. Then, Maryna Viazovska cracked the code—solving one of math’s oldest mysteries in just a few elegant pages.
Her breakthrough wasn’t just about stacking fruit—it has massive real-world impact. Her discoveries are revolutionizing data compression, improving security in cryptography, and even offering insights into quantum physics.
This episode explores the genius behind her work, the mathematical beauty of high-dimensional spaces, and why her Fields Medal win was a historic moment. Get ready for a mind-bending journey into a problem so simple it fooled mathematicians for centuries—until Viazovska solved it.

Step into the strange and electrifying world where chaos meets quantum mechanics. The Quantum Maze unpacks the Berry–Tabor Conjecture—a decades-old mystery that suggests the universe’s most “predictable” systems might secretly follow rules of pure randomness. Sound impossible? That’s exactly why scientists are still obsessed with it.
Through immersive storytelling, this audio documentary explores the eerie parallels between classical order and quantum unpredictability. We trace how tidy, integrable systems—like a billiard ball rolling endlessly on a smooth table—might spawn quantum energy levels that behave like a cosmic game of chance. Along the way, you’ll uncover why degeneracies, symmetries, and tiny arithmetic quirks make or break the theory.
From the origins of chaos theory to real-world experiments with microwave billiards and quantum dots, we piece together the puzzle using expert insight, sharp metaphors, and big questions. What happens when random isn’t truly random? And what does it mean when order disguises chaos?
If you’ve ever wondered whether the universe is a well-oiled machine—or a glitchy simulation—this series is your backstage pass to the science of unpredictability.

What do melting ice, wildfires, and pandemics have in common? They all follow hidden mathematical rules—rules that one man uncovered.Meet Hugo Duminil-Copin, the mathematician who cracked the code of randomness and phase transitions. His discoveries explain how tiny shifts cause massive changes—like water turning to ice or a single spark igniting a forest fire.
His work isn’t just theoretical—it’s transforming how we predict disease outbreaks, design materials, and even understand financial crashes. From magnets to markets, his math reveals the tipping points that shape our world.
Join us as we dive into the work of Hugo Duminil-Copin and explore how his breakthroughs help us navigate an unpredictable world. If you thought math was just about equations, think again—this is the science of chaos itself

In the final episode, we explore the Poincaré Conjecture—the only Millennium Prize Problem that has been solved so far.

At its core, the conjecture asks a deceptively simple question: how can we tell if a shape in three-dimensional space is essentially a stretched-out version of a sphere? Though it sounds simple, this problem sits at the heart of topology, the study of shapes and spaces, and has profound implications for understanding the very structure of the universe.

After stumping mathematicians for over a century, it was finally cracked in 2003 by the enigmatic Grigori Perelman, who rejected both the million-dollar prize and global fame. Join us as we unravel the beauty of this groundbreaking solution and the fascinating story of the man who solved it.

In this episode, we dive into P=NP, the most important unsolved problem in computer science—a question so profound it could reshape technology as we know it. At its core, P=NP asks: can problems that are easy to check also be easy to solve? From cracking encryption to solving puzzles that would normally take centuries of computation, a solution to P=NP could unlock unimaginable computational power—or chaos.

The implications are staggering: a proof could revolutionize medicine, transportation, and artificial intelligence, or render our digital security obsolete overnight. It’s a problem that has baffled and captivated computer scientists for decades.

Join us as we break down the mystery, explore its real-world stakes, and imagine a world where P=NP is finally solved.

In this episode, we tackle the Hodge Conjecture—a grand mathematical puzzle that dares to explain the hidden structure of shapes, spaces, and higher dimensions. At its heart, the Hodge Conjecture is about understanding how complicated geometric shapes can be broken into simpler, more fundamental building blocks.

Solving this problem could illuminate the very fabric of geometry, with profound implications for fields like string theory, topology, and theoretical physics. It’s a mystery so deep that it has challenged some of the greatest mathematical minds.

Join us as we unravel this enigma and explore the beauty and complexity of dimensions far beyond our own.

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In this episode, we dive into the Navier-Stokes Equations, the mathematical key to understanding the chaos of air, water, and motion. These equations govern the flow of fluids—from ocean currents to turbulence in the air we breathe. But there’s a catch: proving that these equations always work without spiraling into chaos remains an unsolved mystery.

Even Hollywood couldn’t resist its allure. In the movie Gifted, the Navier-Stokes problem becomes the obsession of a mathematical prodigy, symbolizing the ultimate intellectual challenge. Solving it could transform everything from climate science to aerospace engineering.

Join us as we explore the math behind the chaos and why this problem has fascinated both geniuses and storytellers alike.

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Explore the Yang-Mills theory and the Mass Gap—a problem that has haunted mathematicians and physicists for decades. At the intersection of pure math and quantum physics, this enigma underpins our understanding of the universe's fundamental forces.

Yang-Mills theory describes the behavior of particles like quarks and gluons, the building blocks of matter, but here’s the twist: the theory predicts they should have mass, yet no one has been able to prove why—or how.

This "mass gap" is a mathematical black hole, critical to modern physics but maddeningly unsolvable. Solving it could revolutionize quantum field theory, bridging the gap between the theoretical and the physical. Join us as we delve into this tantalizing mystery, where science and math collide.

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In this episode, we uncover the Birch and Swinnerton-Dyer Conjecture—an enigma in mathematics so complex it’s often called “the most difficult problem you’ve never heard of.” At its heart lies a question about elliptic curves, abstract shapes with applications ranging from cryptography to string theory.  

Solving it could redefine how we approach the boundaries of number theory and computation.

Join us as we break down the mystery, meet the minds tackling it, and explore why it holds the key to understanding the infinity.

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This episode dives into one of the most elusive mysteries in mathematics: the Riemann Hypothesis. First proposed by Bernhard Riemann in 1859, it’s a conjecture that connects the primes—those fundamental building blocks of numbers—to a hidden rhythm in the infinite expanse of mathematics.

Cracking it is more than implications for number theory; it’s about unlocking the structure of the universe itself. From cryptography to quantum physics, the implications ripple through every corner of science.

In this journey, we’ll simplify the Hypothesis, peeling back the layers of abstraction to reveal the mesmerizing beauty and monumental stakes of solving the Riemann Hypothesis.

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