I came across an interesting article on this site.
Building math AI startup: How 24-year-old Stanford dropout Carina Hong is attracting Big Tech talent
By Phong Ngo
Sun 12/21/2025, 02:07 am (PT)
China-born Carina Hong, a 24-year-old Stanford dropout, has drawn an elite group of AI researchers and a world-renowned mathematician to Axiom Math, her startup building an AI mathematician.
Hong’s company has employed 17 people, many of whom previously worked at Meta’s Fundamental Artificial Intelligence Research (FAIR) lab, as well as Meta’s GenAI team and Google Brain, which merged into DeepMind in 2023.
Founded in March, Axiom drew widespread attention after announcing it had solved two long-standing Erdos math problems that had resisted solutions for decades. In September, the company revealed it had raised $64 million in seed funding.
Hong says Axiom’s ambition, tackling the hardest problems in mathematics, has been key to attracting top talent. "One thing I heard from some of the top researchers and mathematicians I've recruited to Axiom is that solving for mathematical superintelligence will be their legacy," Hong told Business Insider. "When the problem is hard enough, talent density gets very high, and that makes you a magnet for other great thinkers."
Among Axiom’s most notable recruits is Ken Ono, a 57-year-old professor at the University of Virginia and one of the world’s leading mathematicians, as well as Hong’s former mentor. "Ken Ono is the idol of many math students," said Hong.
Ono has followed an unconventional path from the outset. Crushed by parental pressure as a child, he left high school without graduating, yet went on to attend college, discover his passion for mathematics, and spend decades teaching at the University of Wisconsin and Emory before joining the University of Virginia in 2019.
Along the way, he led one of the country’s premier research programs for elite undergraduates and mentored 10 Morgan Prize winners, including his current boss. In mathematics, Ono is best known for his contributions to number theory, spanning topics from Ramanujan’s congruences to the umbral moonshine conjecture.
"He’s a larger-than-life figure in mathematics," said Ken Ribet, a former president of the American Mathematical Society.
Ono said that curiosity and intellectual discovery, not financial gain, drove his decision to join Axiom. "I’m not doing this for the money," Ono said, explaining that he turned down more lucrative offers from larger AI companies.
At the startup, his task is to stretch Axiom’s AI systems to their limits by designing problems that demand deep mathematical insight and by developing benchmarks to assess their performance.
Hong’s company is named after the mathematical term for a basic truth that can be the starting point of an entire theory. She aims to create an "AI mathematician" capable of reasoning through established problems, generating new ones, and rigorously validating its conclusions through formal proofs. If successful, Axiom could help solve mathematical challenges that have baffled human thinkers for generations.
"One thing I heard from some of the top researchers and mathematicians I've recruited to Axiom is that solving for mathematical superintelligence will be their legacy," Hong said. "When the problem is hard enough, talent density gets very high, and that makes you a magnet for other great thinkers."
She said that she focused some of her early recruiting efforts on FAIR because "they consistently deliver amazing research work."
Hong grew up in Guangzhou, China, where she developed a love for mathematics, according to Forbes. She taught herself English as a child to read advanced math textbooks and competed in math Olympiad programs under intense time pressure.
"I was always very interested in mathematical discoveries," Hong told The Wall Street Journal.
After earning bachelor’s degrees in mathematics and physics from MIT, Hong won the prestigious Frank and Brennie Morgan Prize, awarded to the top undergraduate math researcher in America.
As a Rhodes Scholar at Oxford University, she earned a master’s in computational neuroscience. She then enrolled at Stanford University to pursue a joint law degree and Ph.D. in mathematics.
While at Stanford, Hong met Shubho Sengupta, an AI scientist at Meta, by chance in a coffee shop. As they chatted, they realized they could combine their expertise and collaborate on groundbreaking ideas.
After closing Axiom’s seed funding round in September, Hong dropped out of Stanford and made Sengupta the company’s first hire. He now serves as Axiom’s chief technology officer.
I asked Gemin's Deep Research to tell me more about the "Umbral Moonshine Conjecture" and below is what it came up with. I don't pretend to understand the Mathematics involved but it's an interesting read and Ken Ono's ephiphany is especially interesting, reminding me of Saul's conversion on the Road to Damascus.
The Umbral Moonshine Conjecture and the Dawn of Artificial Mathematical Intelligence
Executive Summary
The trajectory of modern mathematics is currently defined by two monumental narratives: the unification of disparate theoretical frameworks—specifically finite group theory, number theory, and string theory—and the disruptive emergence of artificial intelligence as a collaborative engine for discovery. This report provides an exhaustive analysis of the Umbral Moonshine Conjecture, a profound hypothesis that generalizes the mysterious connections observed in "Monstrous Moonshine" to a broader class of groups and functions known as mock modular forms. We detail the conjecture's rigorous proof by John Duncan, Michael Griffin, and Ken Ono, elucidating the critical role of "shadows"—the modular forms that restore symmetry to Ramanujan’s mock theta functions.
Parallel to this theoretical revolution is a structural transformation in how mathematical research is conducted. We investigate the recent, high-profile departure of Ken Ono from his tenured professorship at the University of Virginia to join Axiom Math, a silicon valley startup founded by his former protégé, Carina Hong. This document dissects the technical and philosophical underpinnings of Axiom’s quest for "mathematical superintelligence," contrasts its methodology with competitors like Harmonic, and clarifies the complex media narrative surrounding the automated resolution of Erdős problems. The synthesis of these elements suggests a new epistemological era where the "shadows" of modular forms and the "reasoning" of neural networks converge to illuminate the deepest structures of the mathematical universe.
1. The Historical and Theoretical Genesis of Moonshine
To comprehend the significance of the Umbral Moonshine Conjecture, one must navigate the historical current that connects the seemingly chaotic world of sporadic groups with the pristine order of modular functions. This relationship, termed "Moonshine," stands as one of the most enigmatic discoveries of the 20th century.
1.1 The Monstrous Precursor
The story begins in the late 1970s, a period of intense classification in finite group theory. Mathematicians were hunting for the "Monster," the largest of the sporadic simple groups. This group, denoted as $\mathbb{M}$, is a colossal algebraic object containing approximately $8 \times 10^{53}$ elements, a number larger than the count of atoms in the solar system. It sits as a supreme object of symmetry, yet for years, it appeared isolated from other branches of mathematics.
Simultaneously, number theorists were studying the $j$-invariant, $j(\tau)$, a modular function crucial to the classification of complex elliptic curves. The Fourier expansion of this function begins:
where $q = e^{2\pi i \tau}$.
In 1978, mathematician John McKay made a serendipitous observation that sparked a revolution: the coefficient $196884$ is exactly the sum of the dimensions of the two smallest irreducible representations of the Monster group ($1 + 196883$). Further inspection revealed that the next coefficient, $21493760$, was a linear combination of the dimensions of the Monster's representations.
This observation led John Conway and Simon Norton to formulate the Monstrous Moonshine Conjecture, which posited that there exists an infinite-dimensional graded module, $V^\natural$, acted upon by the Monster group, such that the graded traces of elements of the group (McKay-Thompson series) are modular functions. This conjecture was famously proved by Richard Borcherds in 1992, utilizing Vertex Operator Algebras (VOAs) and string theory, fundamentally linking the discrete symmetry of groups to the continuous physics of strings.
1.2 The "Umbral" Evolution
While Monstrous Moonshine was a triumph, it relied on classical modular forms—functions with perfect symmetry under the modular group $SL_2(\mathbb{Z})$. However, the mathematical universe contains objects that exhibit "near" symmetry, objects that were historically elusive and poorly understood.
Around 2010, physicists Eguchi, Ooguri, and Tachikawa observed a new Moonshine phenomenon involving the Mathieu group $M_{24}$ and the elliptic genus of K3 surfaces. Unlike the Monster case, the functions appearing here were not classical modular forms. They were mock modular forms.
This discovery suggested that Monstrous Moonshine was not a singularity but part of a larger, hidden archipelago of relationships. This realization birthed the Umbral Moonshine Conjecture, formulated by Miranda Cheng, John Duncan, and Jeff Harvey. It proposed that the Mathieu Moonshine was merely one of 23 instances of a generalized phenomenon, each connected to a specific root system of the 23 Niemeier lattices.
2. The Machinery of Shadows: Mock Modular Forms
The Umbral Moonshine Conjecture cannot be understood without a deep dive into the analytic nature of mock modular forms and the concept of the shadow. This mathematical machinery is the bridge between the 1920s insights of Ramanujan and 21st-century string theory.
2.1 Ramanujan’s Last Letter
In 1920, mere months before his death, Srinivasa Ramanujan wrote his final letter to G.H. Hardy. In it, he listed 17 functions he called "mock theta functions." He claimed they possessed asymptotic properties similar to theta functions (a type of modular form) but did not satisfy the standard modular transformation laws.
For eight decades, these functions were a tantalizing puzzle. They were computable and clearly significant, yet they lacked a definition that fit into the standard edifice of modular forms. They existed in the "umbra"—the shadow—of classical theory.
2.2 The Zwegers Revolution: Harmonic Maass Forms
The mystery was resolved in 2002 by Sander Zwegers in his doctoral thesis. Zwegers realized that Ramanujan's mock theta functions were the holomorphic parts of a broader class of non-holomorphic functions called Harmonic Maass Forms.
A Harmonic Maass Form $F(\tau)$ is a function defined on the upper half-plane that transforms like a modular form of weight $k$, but instead of being holomorphic (vanishing derivative with respect to $\bar{\tau}$), it is an eigenfunction of the weight-$k$ hyperbolic Laplacian operator.
Zwegers demonstrated that such a form can be canonically decomposed:
$f(\tau)$ - The Mock Modular Form: This is the holomorphic component containing the "interesting" arithmetic information, such as the partition counts or the dimensions of the Umbral modules. However, on its own, it breaks modular symmetry.
$g^*(\tau)$ - The Non-Holomorphic Correction: This component restores the modular transformation properties. It is defined via a period integral of a classical modular form $g(\tau)$.
2.3 The "Shadow" Explained
The function $g(\tau)$ appearing in the integral of the non-holomorphic part is technically termed the shadow of the mock modular form $f(\tau)$.
2.3.1 Technical Definition
If $f(\tau)$ is a mock modular form of weight $k$, its shadow $g(\tau)$ is a modular form of weight $2-k$. The relationship is governed by the differential operator $\xi_k = 2iy^k \overline{\frac{\partial}{\partial \bar{\tau}}}$. Specifically, the shadow is proportional to the image of the Harmonic Maass form under this operator:
Because the operator involves complex conjugation, the shadow is intimately linked to the non-holomorphic "defect" of the mock modular form.
2.3.2 Conceptual Analogy
To explain this to a non-expert (or "layman"), one might visualize a mock modular form as a three-dimensional object casting a two-dimensional shadow. The object itself (the Harmonic Maass Form) is perfect and symmetric (modular). However, if we only look at one slice of it (the holomorphic part, or mock modular form), the symmetry appears broken or distorted. To understand the full geometry and restore the symmetry, one must account for the "shadow" it casts. In Umbral Moonshine, the shadow is not just an error term; it carries the genetic code of the underlying symmetry group (the Niemeier lattice root system).
2.4 Physical Interpretation
In the context of string theory, the mock modular form counts the quantum states of a black hole (degeneracy), while the shadow ensures the theory remains consistent with the modular invariance required by the topology of the string worldsheet. The "shadow" represents the continuum of scattering states that renders the counting of bound states (the mock part) non-invariant on its own.
3. The Umbral Moonshine Conjecture: A Universe of 23 Cases
The Umbral Moonshine Conjecture posits that the phenomenon observed for the Monster group is not unique, but rather the most famous instance of a general structural truth connecting finite groups to modular forms.
3.1 The Role of Niemeier Lattices
The classification of even unimodular lattices in 24 dimensions reveals exactly 24 such lattices.
One is the Leech Lattice, which has no roots (vectors of norm 2). This lattice is the geometric heart of Monstrous Moonshine.
The other 23 lattices are the Niemeier Lattices, and each is uniquely characterized by its non-empty root system (a combination of A, D, and E root systems that sum to rank 24, e.g., $24A_1$, $12A_2$, $8A_3$, etc.).
3.2 The Conjecture Statement
For each of the 23 Niemeier root systems $X$, there exists:
An Umbral Group ($G_X$): A finite group derived from the automorphism group of the lattice modulo its Weyl group.
An Umbral Module ($K_X$): An infinite-dimensional graded vector space $K_X = \bigoplus_{n \in \mathbb{Q}} K_{X,n}$.
Mock Modular Forms ($H_X$): The conjecture asserts that the graded character of an element $g \in G_X$ acting on the module $K_X$ is given by a specific mock modular form $H_X^{(g)}(\tau)$.
Crucially, the shadow of this mock modular form $H_X$ is completely determined by the root system $X$ of the corresponding Niemeier lattice. This ties the analytic behavior of the function (its shadow) directly to the algebraic structure of the lattice.
Why "Umbral"?
The term "Umbral" was chosen by the authors (Cheng, Duncan, Harvey) specifically because the theory is governed by the shadows of the mock modular forms. It serves as a double entendre: referring to the "shadowy" nature of the mock theta functions and the fact that these moonshines exist in the "shadow" of the larger Monstrous Moonshine.
4. The Proof: From Guess to Theorem
The transition of Umbral Moonshine from a daring conjecture to a proven theorem is a landmark achievement in 21st-century mathematics, primarily attributed to the collaborative work of John F. R. Duncan, Michael J. Griffin, and Ken Ono.
4.1 The Proving Team
Ken Ono: A world-renowned expert in number theory, specifically in the areas of partitions, modular forms, and the legacy of Ramanujan. His expertise in the analytic properties of mock theta functions was the "key" to unlocking the proof.
John Duncan: One of the originators of the conjecture, bringing deep insight into the group-theoretic and lattice-theoretic aspects.
Michael Griffin: A brilliant young mathematician (Ono’s doctoral student at the time) who contributed significantly to the computational and analytic framework.
4.2 The Proof Strategy
The proof, published in Research in the Mathematical Sciences (2015) under the title "Proof of the Umbral Moonshine Conjecture," settled the existence of the modules for all 23 cases.
The proof required demonstrating two incredibly difficult properties for the conjectured mock modular forms:
Integrality: The coefficients of the $q$-expansions of these forms must be integers. This is non-trivial because the formulas for these coefficients (derived from Rademacher sums) naturally produce transcendental numbers involving $\pi$ and $e$. Proving they collapse to integers requires delicate cancellation analysis.
Representation Theoretic Structure: The integer coefficients must be decomposable into the dimensions of irreducible representations of the Umbral groups. That is, for every coefficient $c(n)$, one must satisfy $c(n) = \sum a_i \dim(V_i)$ with non-negative integers $a_i$.
4.3 Key Technique: Rademacher Sums
The team employed a powerful generalization of the Rademacher sum. In classical theory, Hans Rademacher found an exact convergent series for the partition function $p(n)$. Duncan, Griffin, and Ono adapted this to mock modular forms. They constructed the candidate functions $H_X$ as specific Rademacher sums of the first kind.
By analyzing the analytic behavior of these infinite sums, they proved that the shadows of these constructed functions perfectly matched the root systems of the Niemeier lattices, as required by the conjecture. They then used bounding techniques to show that the multiplicities of the group representations were indeed non-negative integers.
4.4 Status and Significance
Ken Ono remarked on the speed of the resolution: "Things came together quickly... We have been able to prove it and it is no longer a guess".
This result unifies the "sporadic" chaos of finite groups with the "smooth" world of string theory vacua, suggesting that the universe's symmetries are dictated by the interplay of mock modular forms and their shadows.
5. Ken Ono: The Spirit of Ramanujan and the "NI" Epiphany
To understand the next phase of this narrative—the leap to Artificial Intelligence—one must examine the career and philosophy of Ken Ono, the mathematician at the center of the Umbral Moonshine proof.
5.1 Academic Pedigree and Ramanujan’s Legacy
Ken Ono is not merely a researcher; he is a historian and guardian of Srinivasa Ramanujan’s mathematical spirit.
Roles: He held the Marvin Rosenblum Professorship at the University of Virginia (UVA) and served as the Vice President of the American Mathematical Society (2018-2021).
Research: Beyond Moonshine, Ono has made breakthroughs in the theory of partitions, finding fractal structures in partition numbers that were previously thought to be random.
Mentorship: He is a legendary mentor, having guided numerous students to the Morgan Prize (the highest honor for undergraduate research), including the founder of Axiom Math, Carina Hong.
5.2 Beyond the Blackboard: Swimming and Data
Ono is unique among pure mathematicians for his engagement with the physical world. He served as a performance consultant for the UVA swimming team and elite Olympians (including Kate Douglass). By applying mathematical modeling to stroke mechanics and fluid dynamics, he optimized athletic performance. This experience with "messy" real-world data likely primed him for the statistical nature of modern AI.
5.3 The "NI" vs. "AI" Stance
For most of his career, Ono was a skeptic of Artificial Intelligence in the realm of creative mathematics. He famously opened his lectures with a joke:
"My name is Ken Ono, and I am NI—Naturally Intelligent."
He believed that while computers could compute, the "spark" of intuition—the ability to see a "shadow" where none was visible—was uniquely human.
5.4 The Epiphany
The turning point occurred in 2024. Ono was invited to a symposium to curate "unsolvable" research-level problems to test the limits of new AI models. He expected the machines to stumble. Instead, he was "profoundly shaken" by their performance. The AI did not just retrieve data; it displayed a rudimentary form of reasoning and pattern recognition that mirrored his own "epiphanies".
He realized that the "hallucinations" of AI—often cited as a flaw—were algorithmically similar to the creative "guesses" human mathematicians make before finding a rigorous proof. If these digital hallucinations could be tethered to a "shadow" of formal verification, AI could become a super-mathematician.
This realization led him to retire from his tenured position at UVA (effective 2025) and move to Silicon Valley, stating that the pursuit of "Mathematical Superintelligence" was now the most important frontier in science.
6. Axiom Math: The Quest for Mathematical Superintelligence
Axiom Math (legally Axiom Quant Inc.) is the startup that successfully recruited Ken Ono. It stands at the vanguard of the "Industrialization of Mathematics," a movement seeking to transition mathematical proof from an artisanal craft to a scalable, automated industrial process.
6.1 Carina Hong: The Prodigy Founder
The driving force behind Axiom is Carina Hong (Letong Hong), a figure of intimidating intellect and swift execution.
Trajectory: Born in Guangzhou, China. She graduated from MIT in three years with a near-perfect GPA (4.9/5.0), earned a Master’s from Oxford as a Rhodes Scholar, and entered a joint JD/PhD program at Stanford before dropping out to found Axiom.
Accolades: She is a winner of the Morgan Prize (2023) and the Alice T. Schafer Prize (2022).
Connection: Hong was Ken Ono’s student and mentee. Her ability to recruit her former mentor—effectively her "academic father"—validates the seriousness of her venture. Ono serves as the "Founding Mathematician," ensuring the rigour of the AI's output.
6.2 Corporate Structure and Funding
Headquarters: Palo Alto, California.
Funding: The company emerged from stealth with approximately $64 million in seed funding, valuing the pre-product company at $300 million.
Investors: The round was led by B Capital, with participation from Greycroft, Madrona, and Menlo Ventures.
6.3 The Mission: The "AI Mathematician"
Axiom’s goal is to build an "AI Mathematician" capable of:
Conjecturing: Identifying novel patterns in data (the "Moonshine" phase).
Proving: Generating rigorous, step-by-step proofs in formal languages like Lean or Coq (the "Zwegers" phase).
Self-Correction: Using formal verification to catch errors, eliminating the "hallucination" problem inherent in LLMs.
The "Axiom" in the name refers to the "fundamental truth that can be the starting point of an entire theory," signaling their ambition to rebuild math from the ground up.
7. The Landscape of AI Mathematics: Axiom vs. Harmonic
The emergence of Axiom has been accompanied by intense media scrutiny and significant confusion regarding its primary competitor, Harmonic. Understanding the distinction between these two entities is crucial for an accurate assessment of the field.
7.1 The "Erdős Problem" Controversy
In late 2025, reports circulated that Axiom Math had solved two famous open problems posed by Paul Erdős (Problem #124 and #481). This narrative was fueled by the simultaneous news of Ono’s hiring and Axiom’s funding. However, this attribution was incorrect.
The problems were actually solved (or partially solved) by Harmonic, a different AI startup co-founded by Vlad Tenev (CEO of Robinhood) and Tudor Achim. Harmonic’s AI model, Aristotle, achieved these feats.
Analysis of the "Solved" Problems:
Erdős Problem #124: Aristotle solved a formal statement of this problem. However, the formal statement contained a transcription error (a typo in the inequality condition) that rendered the problem significantly easier than Erdős intended. While the AI’s autonomous solution was valid for the stated problem, the original deep number-theoretic conjecture remains open.
Erdős Problem #481: Aristotle solved this problem autonomously. It was later discovered by the community that this problem had been solved by a human (David Klarner) in 1982, but the solution had been obscure and missed by database curators. Thus, Aristotle rediscovered the proof without human aid—a significant, albeit different, achievement.
7.2 Comparative Analysis: Axiom vs. Harmonic
The following table differentiates the two primary players in this space:
| Feature | Axiom Math (Axiom Quant Inc.) | Harmonic |
| Founder | Carina Hong (Mathematician, Rhodes Scholar) | Vlad Tenev (Robinhood CEO), Tudor Achim |
| Key Recruit | Ken Ono (Founding Mathematician) | -- |
| Flagship Model | "AI Mathematician" (Reasoning Engine) | "Aristotle" (Chatbot/Prover) |
| Primary Focus | Deep Theory, New Conjectures, Superintelligence | IMO Benchmarks, Reliability, Hallucination-free |
| Verification Tech | Formal Proofs (Lean/Coq) | Formal Proofs (Lean 4) |
| Funding | ~$64M Seed ($300M Valuation) | ~$295M (Series C, $1.45B Valuation) |
| Achievements | Solving "Hard" problems (Stealth/R&D) | IMO Gold Medal Level; Erdős #124 variant |
| Philosophy | "Talent Density" (Hiring from FAIR/Google) | "Scaling" (Synthetic Data, Self-Play) |
Source Integration:
7.3 Axiom’s Strategic Position
While Harmonic focuses on benchmarks like the International Mathematical Olympiad (IMO), Axiom appears to be targeting "theory building." Carina Hong has stated that their goal is not just to solve existing problem lists but to generate new mathematics—to find the next Umbral Moonshine. The recruitment of Ken Ono is strategic: his role is to guide the AI toward "tasty" and "deep" mathematics, preventing it from generating infinite trivialities.
8. Synthesis: The Shadows of Intelligence
The narrative arc from Umbral Moonshine to Axiom Math is not merely a sequence of events; it is a coherent evolution of how humanity interacts with mathematical truth.
8.1 The "Shadow" as a Metaphor for AI Verification
There is a profound symmetry between the mathematics of Moonshine and the architecture of AI reasoning.
In Moonshine: We have a "Mock" form—beautiful, information-rich, but structurally flawed (non-modular). To fix it, we need a "Shadow"—a rigid, modular form that restores consistency via an integral transform.
In AI: We have an "LLM"—creative, intuitive, capable of generating proofs, but prone to structural flaws (hallucinations/logical errors). To fix it, we need a "Verifier" (like Lean)—a rigid, formal system that restores consistency via logical checking.
Ken Ono’s move to Axiom suggests he recognizes this parallel. He has spent his life finding the shadows that make mock theta functions "true." Now, he is building the shadows that will make AI reasoning "true."
8.2 The Future of Discovery
The industrialization of this process—funded by $64 million at Axiom and nearly $300 million at Harmonic—suggests that the era of the lone genius (the Ramanujan figure) is ending. It is being replaced by the era of the Cyber-Human Symbiote.
Scientific Acceleration: These systems are not limited to pure math. They are already being applied to software verification, hardware design, and quantitative finance (finding arbitrage "shadows" in market data).
The Final Frontier: If Axiom succeeds, the next great conjecture—the successor to Umbral Moonshine—may not be found by a human noticing a pattern in a book of tables, but by a neural network analyzing millions of elliptic curves in high-dimensional space, guided by the intuition of Ken Ono and the code of Carina Hong.
In conclusion, the Umbral Moonshine Conjecture stands as a testament to the depth of the mathematical universe, while Axiom Math represents the tool we are forging to explore it. The "shadows" are no longer just obscure terms in a differential equation; they are the blueprint for the next intelligence.
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