How AI Is Shaping Everyday Research

By Laura P. Schaposnik and Maximilian J. Telford

In 1953, Roald Dahl wrote The Great Automatic Grammatizator, a tale in which a machine begins to churn out novels with such skill and speed that it threatens to overshadow human creativity. Created by a young engineer, the Grammatizator is a mechanical marvel that learns to compose stories following the strict rules of grammar and style, until it becomes indistinguishable from a real author’s work. What begins as a curiosity quickly evolves into a disruptor, raising unsettling questions about the essence of human ingenuity. About a decade later, in 1965, Stanisław Lem published The Cyberiad in which Trurl’s Electronic Bard is able to automate poetry from simple prompts: after some training, Trurl would ask for the following:

Let’s have a love poem, lyrical, pastoral, and expressed in the language of pure mathematics. Tensor algebra mainly, with a little topology and higher calculus, if need be. But with feeling, you understand, and in the cybernetic spirit.

He quickly received a mathematical poem:

Come, let us hasten to a higher plane, Where dyads tread the fairy fields of Venn, Their indices bedecked from one to $n$, Commingled in an endless Markov chain!

While Dahl’s and Lem’s stories were fantastical critiques of automation, they hold surprising relevance today as we grapple with the role of AI in traditionally human domains. Today, in a Lemian spirit, ChatGPT answers to the above prompt gently:

Oh, let our topologies intertwine, In compact spaces, love divine. A Möbius kiss, a Klein bottle’s hue, Boundaries vanish where love is true.

Soon after OpenAI released the public version of ChatGPT, Dahl’s Great Automatic Grammatizator and its modern (nonfictional) incarnations became a subject of fascination to the fellows of All Souls College, Oxford—would writers and researchers be obsolete before too long? Would the origin of life be understood before we even made any new discoveries? ChatGPT, in its most Dahlian guise as a writer of prose, soon made an entrance into our academic worlds. This came at first in the form of student essays—there were a few poorly written examples that were entirely reliant on the technology for content and structure, as well as much more encouraging examples of excellent written work where AI had been used to polish and to help an able student whose first language was not English.

While AI’s impact on education offers a glimpse into its potential, its applications in scientific research reveal further possibilities for collaboration between humans and machines. In 2024, the broader impact of AI on science was recognized by the award of two Nobel Prizes—in physics and in chemistry. The physics prize went to John Hopfield and Geoffrey Hinton “for foundational discoveries and inventions that enable machine learning with artificial neural networks.” And half of the award in chemistry went to Demis Hassabis and John Jumper “for protein structure prediction.” That brief phrase is rather an inadequate description of the pair’s work to develop an AI model called AlphaFold2.

AlphaFold2 has resulted in an extraordinary leap in the abilities of computers to predict the complex three-dimensional structures of hundreds of millions of proteins using the linear sequences of twenty different amino acids that they are composed of. As with biologists and chemists, physicists and mathematicians have not been put off by the fear of potential misuses of AI, and practitioners of many different branches of science are now focusing on the multitude of beneficial ways in which AI might help advance our fields in the immediate future.

Indeed, unlike Dahl’s automaton, which adhered strictly to the grammatical rules it was programmed with, today’s AI models go far beyond mere mimicry. They’re no longer restricted to predefined patterns: They are beginning to generate insights, suggest new research directions, and, in some cases, even outperform human intuition in fields like mathematics and physics. Now that we have passed the second anniversary of the public release of ChatGPT, we look into some highlights of how AI, in general, has become a valuable research partner, in some instances pushing past the limits of human capability. Whether it’s helping researchers navigate the high-dimensional topology of moduli spaces, optimizing the complex geometry of Calabi-Yau manifolds, or discovering new patterns in algebraic structures, AI is fundamentally reshaping the process of mathematical and physical exploration.

Modern AI tools, such as neural networks and support vector machines, excel at identifying patterns and making predictions across complex datasets. This article presents specific examples of how AI is integrated into the daily research activities of mathematicians and physicists, emphasizing practical, real-world applications rather than providing a broad overview of the field. We will first examine some examples of how AI has advanced geometry, algebra, and number theory and then explore the particular case of Calabi-Yau manifolds. Finally, we encourage readers to view ChatGPT as an inspiring muse, even if it is currently a fairly limited general problem-solving tool.

Laura P. Schaposnik is a professor of mathematics at the University of Illinois, Chicago. Her email address is schapos@uic.edu.

Maximilian J. Telford is the Jodrell Professor of Zoology and Comparative Anatomy at University College London. His email address is m.telford@ucl.ac.uk.

AI-Driven Discovery

By Yang-Hui He

In the twentieth century, major results in pure mathematics—such as the proof of the 4-color theorem or that of the Kepler conjecture—were obtained by reducing the work to computer checks on myriads of cases. Now already in the first two decades of our century, there has been a paradigmatic shift toward human activity being dependent on AI. This has been due to two decisive factors:

(1)

Data: The internet has made almost all of human knowledge available to every person’s fingertips.

(2)

Compute: Vast improvements of the CPU and GPU have rendered every laptop what once would have been considered a supercomputer.

A new era for mathematics

In the empirical sciences, AI, especially machine-learning (ML) methodologies, has rapidly become an integral component of research—for instance, the Higgs particle at CERN could not possibly have been discovered without ML detection by sifting through data. While this is clear for experiments, the role of AI in pure mathematics and theoretical physics had been more difficult to imagine until 2017, when ML was used to find new patterns in algebraic geometry to explore the landscape of string theory He17. In the past seven years, a flurry of activity has resulted in hundreds of papers in this emerging field of “AI for mathematics,” including some breakthroughs that have led to new mathematics DVB$^{+}$21HLOP22 (see GHR24He24 for reviews on these topics).

Indeed, with the rapid advances in ML for mathematical reasoning and formal derivation, notably with the likes of DeepMind using large language models (LMMs) to solve Math Olympiad problems to silver medal level, it is no wonder mathematicians of the highest profile are delivering major addresses (such as to the ICM) on the “future of mathematics” being vitally dependent on proof assistants and AI. In He24, we cataloged three ways, to which we alluded above, whereby AI can help mathematics: (1) Bottom-Up, i.e., formalization and proof copilots; (2) Meta, i.e., LLM for mathematics; and (3) Top-Down, i.e., AI-guided conjectures and intuition. In what follows we shall describe some instances of (3) where AI has been fruitful illustrating the advances through specific research articles.

Learning algebraic structures: One may wonder how well ML learns an algebraic structure, and this is what we considered in HK19. In particular, we considered the Cayley (multiplication) table of the first 100 finite groups of size $n$, so that each is an $n \times n$ Latin square where each row and column is strictly a permutation of 1 to $n$. We then zero-padded to the right-bottom so that we have a list of $100 \times 100$ integer matrices valued in $[0, n]$. Next, the matrices are labeled as 1 or 0 according to whether the associated finite group is simple or not. Now, the nonsimples will dominate by far; therefore, we also include permutations of the Cayley tables, giving an equivalent representation of a group, weighted toward the simples. In all, we thus establish a balanced (roughly equal number of 0 and 1) set of, say, 50,000 labeled matrices. If one plots these in a $100^2$-dimensional Euclidean space by vectorizing each matrix via flattening,⁠¹ a curious thing occurs: A support vector machine (SVM) finds a separating hypersurface between the simples and nonsimples! This is further corroborated later where a generator representation was used and an NN was applied. This protoconjecture was found purely by AI (there is no way to visualize $100^2$ dimensions)—proto since one needs to formulate it more precisely—and could give a novel approach in representation theory.

¹

Vectorizing each matrix (converting it from a grid-like structure into a linear array) allows machine learning algorithms to process the data efficiently.

✖

A murmuration conjecture: It is difficult to imagine an AI algorithm directly finding any new patterns within prime numbers, but after some initial experiments, the following fruitful experiment was conducted in HLOP22. Given (the isogeny class of) an elliptic curve $E$, fix a number $n$ (say, 100). Consider the vector of Euler coefficients $a_p$ for primes $p = p_1, \ldots , p_n$ (recall that $a_p = p+1- \#E(\mathbb{F}_p)$, the deviation of the number $\#E(\mathbb{F}_p)$ of points of $E$ over the finite field $\mathbb{F}_p$). It was found that all the standard neural classifiers were able to learn to use the vector $a_p$ to predict the rank of $E$ to almost 100% accuracy. While one knows that the Birch-Swinnerton-Dyer conjecture is fundamentally at work, it is unknown how to explicitly obtain the rank from $a_p$. An interpretation of a PCA (principal component analysis) finally gave rise to a conjecture: The average of the $a_p$ coefficients HLOP22, over the set $C$ of all elliptic curves of a given rank $r$ within a conductor range $[2^N, 2^{N+1}]$, is a function of the $n$th prime $f_r(n) \coloneq \frac{1}{|C|} \sum _{E \in C} a_{p_n}(E)$; this $f(n)$ oscillates in a precise way and converges to a well-defined shape and is expected to hold for all $L$-functions. This “murmuration conjecture” reflects a fundamental bias in the distribution of primes.

The Birch Test: ChatGPT passed the Turing test in 2023. It is expedient to devise a more stringent test for AI in mathematics; this was called the Birch Test. An AI-assisted mathematical discovery must satisfy:

(A) Automaticity

It is completely made by AI from pattern spotting, without any human intervention.

(I) Interpretability

Any statements—conjectures or conclusions—must be precise to a human mathematician, who cannot distinguish it from one given by a human colleague.

(N) Nontriviality

It is nontrivial enough that the community of human experts will work on it.

Despite all the progress in AI-assisted mathematics over the last seven years, the Birch Test is strict enough that so far nothing has passed all three parts of the test. The geometry experiments of He17 suffer from the typical problem of deep neural networks: There is no interpretable formula one could extract. Thus, they fail Birch Test (I). Likewise HK19 is not quite precise enough and also fails Birch Test (I). The knot invariant relations found by saliency analyses DVB$^{+}$21, and the Reidemeister moves for complicated knots GHR24, though novel, interesting, and precise, were either readily proven or have not become sufficiently influential in the field; thus they fail Birch Test (N). The murmuration conjectures HLOP22 passed (I) and for the first time passed (N). However, the discovery fails (A) in that human mathematicians intervened in the process by choosing to study the weight matrices of the PCA.

Nevertheless, we are optimistic that the Birch Test will be completely passed in the near future. In any event, it is undeniable that AI is beginning to play and will continue to play a pivotal role in partnership with the human mathematician. Indubitably, for the next generation of mathematics students, machine learning and perhaps even Python programming, will together with statistics be part of the core undergraduate curriculum.

Yang-Hui He is a fellow and professor at the London Institute for Mathematical Sciences, Royal Institution, United Kingdom. His email address is yang-hui.he@merton.ox.ac.uk.

Machine Learning for Calabi-Yau Metrics

By Lara B. Anderson

Over the past four decades, the geometry of Calabi-Yau manifolds has played a key role in many aspects of progress in differential and algebraic geometry as well as the physics of string theory. Despite this rich history, significant challenges and open questions persist in both fields. In particular, while Yau’s theorem guarantees the existence of a Ricci-flat metric in such manifolds, explicit analytical expressions for these metrics remain elusive. Recently, machine learning (ML) has emerged as a promising tool for approximating Calabi-Yau metrics, sparking considerable excitement within the community, wondering about what questions (and answers) may now be in reach. I’ll try to highlight these ideas below (see AGG$^{+}$21 and references therein for further details and AGL2312 for a review).

Within string theory, the underlying physical theory is formulated in more than the $3+1$ dimensions of spacetime we observe in our universe. Instead, string theory can be formulated, for example, in $10$ spacetime dimensions of the form $\mathbb{R}^{1,3} \times X$, where $X$ is a $6$- (real)-dimensional compact manifold. If the volume/length scales in $X$ are sufficiently small, this could be consistent with physical observation. This is known as a “string compactification.” The result is that any 4-dimensional physics that arises is dependent on the properties of $X$, the compact extra dimensions, and in particular the notion of length in that space as encoded by its metric. This process impacts nearly every aspect of the $4$-dimensional theory including not just gravitational physics but the masses of particles and the strengths of their interactions. In order to decide whether such theories can agree with observations in particle physics and cosmology, the metric plays a crucial role.

Figure 1.

A 3-dimensional projection of a 6-dimensional Calabi-Yau manifold. This CY is the quintic hypersurface defined by $\sum _{i=0}^4 z_i^5=0$ in $\mathbb{C}\mathbb{P}^4$.

Calabi-Yau manifolds

One simple example of a compact space that satisfies the equations of motion of string theory is a so-called Calabi-Yau (CY) manifold Yau78Yau85. A Calabi-Yau manifold is a complex, compact manifold with vanishing first Chern class (i.e., $[\mathrm{tr}(R)]=0$ where $R$ is the Ricci curvature) and whose metric can be simply defined in terms of two derivatives of a function $K$ called the Kähler potential. Through Yau’s theorem, one knows that for an $n$-dimensional compact, complex, Kähler manifold $X$ with vanishing first Chern class, in any Kähler class $[J]$, the manifold $X$ admits a unique Ricci flat Kähler metric.

Moreover, nontrivial⁠² CY manifolds have no continuous isometries, which means that there are no guiding symmetries to appeal to in writing down a metric. As a result, although Yau’s theorem guarantees its existence, no analytical expressions for the metric are known for $n>1$. However, there is another way to approach the problem (which also played a role in Yau’s proof of the theorem mentioned above) which is simpler than solving the curvature condition directly, and this is a Monge-Ampere-type condition arising on special differential forms on $X$. In addition to the Kähler form, one can also define the associated Kähler class $J_{CY}=J+\partial \bar{\partial }\phi$ for some smooth, zero-form $\phi$. Finally, a CY manifold gives rise to a unique holomorphic $(3,0)$-form, $\Omega$. The relevant Monge-Ampere equation arises from the observation that the volume (top) $(3,3)$-form on $X$ is essentially unique and as a result, the following expression must hold:

²

Excluding $n$-tori.

✖

$$\begin{equation} J_{CY} \wedge J_{CY} \wedge J_{CY}=\kappa \Omega \wedge \bar{\Omega }, {\tag{1}} \end{equation}$$

where $\kappa$ is a constant. This condition has been at the center of efforts to numerically approximate CY metrics for many years. The rough idea is that if one can generate points on the CY manifold and compute $\Omega$ directly, then one could attempt to solve the second-order Monge-Ampere equation for $\phi$.

The condition in 1 also provides an error measure on how far any metric (in the same Kahler class as the CY metric) is from the Ricci-flat one. There have been robust attempts in the literature over the years to approach this problem via an approximating class of algebraic Kähler potentials using a varied array of techniques, including a limiting scheme with proven convergence using so-called balanced metrics developed by Donaldson Don09 and approaches based on functional minimization HN13. Though results have been obtained using these approaches, they have proven computationally costly and not generalizable to generic points in moduli space.⁠³

³

CY manifolds have a moduli space of fluctuations $\delta g$ of the metric that preserve Ricci-flatness. These include volume-changing Kähler deformations and “shape-changing” complex structure deformations.

✖

Looking ahead

ML algorithms provide a novel approach to this problem, using either supervised or direct learning. In particular, it is possible to directly minimize a loss function built out of the Monge-Ampere constraint using the method of gradient descent. With any numerical approach we must ask, “Was our approximation ‘good enough’?” The answer to this of course depends on what one wants to do next. In the case of string theory applications for CY metrics, there are many exciting avenues to explore and some of them have already been realized. These include studying gauge theories over CY backgrounds including numeric/ML approaches to solutions of the Hermitian Yang-Mills equations or Hitchin’s equations for Higgs bundles over subvarieties (as introduced in the next section), as well as the solution of bundle-twisted Dirac equations. Additionally, it will be interesting to see whether these tools can aid in geometric explorations, including mirror symmetry and the study of special Lagrangian cycles.

One important step in physics is the computation of the masses and couplings of particles in the 4-dimensional gauge theories resulting from string compactification. Exciting progress in this latter direction has just been completed with the first direct computation of quark masses in a CY string compactification using these ML tools BMPT$^{+}$241CFTH$^{+}$242. While the physics of this background does not yet match the particle physics observed in nature, the ability to compute masses/couplings is a remarkable step forward.

Results like those above are only the first steps in utilizing new ML tools in this context. It will be interesting to see what further problems are now within reach.

Lara B. Anderson is an associate professor of physics in the Particle Theory Group and an affiliate professor in the mathematics department at Virginia Tech. Her email address is lara137@vt.edu.

A Short ChatGPT Adventure

By Steve Bradlow

In the end ChatGPT did not contribute a significant amount to our project, but it gave us the confidence to proceed, which made a world of difference.

The goal of the affected project was to identify and count all the pieces, i.e., the connected components, of objects known as moduli spaces of Higgs bundles. There is one such moduli space, denoted $\mathcal{M}(G,\Sigma )$, for each pair $(G,\Sigma )$, where $G$ is a Lie group and $\Sigma$ is a Riemann surface.

The spaces $\mathcal{M}(G,\Sigma )$ are high-dimensional complex analytic varieties which have many interesting features. In particular, these spaces may have multiple connected components, some of which can be attributed to the topology of the group $G$ or the surface $\Sigma$, but others have more mysterious origins and the exact number of components is still unknown in many cases, particularly for the exceptional simple Lie groups $F_4, E_6, E_7$, and $E_8$.

The moduli spaces of Higgs bundles for these groups were the target when five of us⁠⁴ got together in Madrid in the spring of 2023. For those (and certain other) real Lie groups we had recently identified a class of special components in the spaces $\mathcal{M}(G,\Sigma )$ related to a feature of their Lie algebras that we called a magical $SL_2$-triple (but that’s another story). Our goal in Madrid was to prove that these Higgs bundle moduli spaces have no other “unexpected” connected components, i.e., to show that the known components provide a complete count.

⁴

Myself, Brian Collier, Oscar Garcia-Prada, Peter Gothen, and Andre Oliveira.

✖

The main weapon in our arsenal for attacking this objective was a real-valued Morse function introduced by Nigel Hitchin in his landmark 1987 paper where Higgs bundles were first introduced. The full conditions for classical Morse theory do not apply in this setting but the Hitchin function is at least a proper map and thus attains local minima on all connected components. We had previously identified local minima on the special connected components coming from the magical $SL_2$-triples but the possibility remained that there were other local minima and thus other connected components. Any further, as yet undetected, local minima could either be in the smooth locus of the moduli space or possibly at singular points. Our conjecture was that all connected components were accounted for, so the task was to rule out the existence of any such critical points.

The necessary conditions for the Hitchin function to be a Morse function do apply in the smooth loci of $\mathcal{M}(G,\Sigma )$, which means that local minima of the function can be detected by examining the index of the function’s Hessian at critical points. More precisely, the conditions amount to conditions on the dimensions of the pieces in a certain $\mathbb{Z}$-grading of the Lie algebra. This is good news!

A Lie algebra is a finite-dimensional vector space with the additional properties of an algebra, in which the algebra is fully and elegantly described using a set of vectors known as roots in the dual vector space. These root systems can be classified by Dynkin diagrams, and the key algebraic relations which determine the Lie algebra can be encapsulated in a poset diagram for a subset of the roots identified as the positive simple roots. With such data in hand it becomes possible in principle to check all possible $\mathbb{Z}$-gradings and thus search for those which might satisfy the conditions imposed at local minima of the Hitchin function.

In practice, given that the Lie algebras of $F_4$, $E_6$, $E_7$, and $E_8$ are complex vector spaces of dimensions 52, 78, 133, and 248, respectively, the required search cannot feasibly be done by hand. That is what computers are for! The problem for us was that our programming skills were either nonexistent or—at best—obsolete.

Before November 30, 2022, this obstacle would have most likely killed our project or at least stalled it until one of us dredged up enough courage to learn the basics of Python programming. But on November 30, 2022, ChatGPT 3 was launched, so when we got together in May 2023 we could say, “Let’s give the problem to ChatGPT.”

We had no idea how in fact that would work but we had read many articles about ChatGPT 3’s freakish capabilities. Our goal was now to get ChatGPT 3 to write a Python program implementing our search for $\mathbb{Z}$-gradings on the exceptional complex simple Lie algebras. Our first step was to compose a plain language description of an algorithm for doing the search, i.e., to produce the pseudocode for the program. Primed with its own version of suitably polished pseudocode, ChatGPT 3 almost instantly produced the Python program we needed. After just a few basic tips patiently delivered by a millennial coder with a tolerant disposition (aka, my son Henry) we were easily able to implement the program on our laptops. In a few days we had explored all the exceptional real forms and could confirm that the Hitchin function has no unexpected local minima in the smooth loci of the corresponding moduli spaces.

The final Python program was very simple, with no more than fifty lines of code, and the run times for the cases of interest were measured in minutes. In retrospect, it should not have been beyond our abilities to master the tools necessary for the automated search we needed to do. We should certainly have been able to manage it without help from a large language model like ChatGPT. The obstacles we faced were mostly psychological—trepidation caused by inexperience and, it must be admitted, a certain amount of intellectual inertia. Fortunately, the boost we got from ChatGPT was just enough to get us over these hurdles, and in this sense it was crucial.

Despite our ChatGPT-fueled success, our automated search did not fully prove our conjecture about the number of components—it left open the possibility of local minima outside the smooth loci of the moduli spaces—but it constituted a big step in that direction. The final step will require careful (somewhat tedious) analysis of the nonsmooth points in the moduli spaces. We live in hope that ChatGPT 4 will pave the way!

Steve Bradlow is a professor of mathematics at the University of Illinois at Urbana-Champaign. His email address is bradlow@illinois.edu.