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Math and Machine Learning Reunion Workshop

September 8, 2025 @ 9:00 am - September 10, 2025 @ 5:00 pm

Math and Machine Learning Reunion Workshop

Dates: September 8–10, 2025

Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA

Machine learning and AI are increasingly important tools in all fields of research. In the fall of 2024, the CMSA Mathematics and Machine Learning Program hosted 70 mathematicians and machine learning experts, ranging from beginners to established leaders in their field, to explore ML as a research tool for mathematicians, and mathematical approaches to understanding ML. More than 20 papers came out of projects started and developed during the program. The MML Reunion workshop will be an opportunity for the participants to share their results, review subsequent developments, and develop directions for future research.

Registration required

In-person registration

Zoom meeting registration

 

Invited Speakers

Organizer:Michael Douglas, CMSA

Schedule

Monday Sep. 8, 2025

9:00–9:30 am Morning refreshments
9:30–9:45 am Introductions
9:45–10:45 am Angelica Babei, Howard University
Title: Predicting Euler factors of elliptic curves
Abstract: Two non-isogenous elliptic curves will have distinct traces of Frobenius at a large enough prime, and a finite set of $a_p(E)$ values determines all others. However, even when enough $a_p(E)$ values are provided to uniquely identify the isogeny class, no efficient algorithm is known for determining the remaining $a_p(E)$ values from this finite set. Preliminary results show that ML models can learn to predict the next trace of Frobenius with a surprising degree of accuracy from relatively few nearby entries. We investigate some possible reasons for this performance. Based on joint work with François Charton, Edgar Costa, Xiaoyu Huang, Kyu-Hwan Lee, David Lowry-Duda, Ashvni Narayanan, and Alexey Pozdnyakov.
10:45–11:00 am Break
11:00 am–12:00 pm Kyu-Hwan Lee, University of Connecticut
Title: Machine learning mutation-acyclicity of quivers
12:00–1:30 pm Lunch
1:30–2:30 pm Gergely Bérczi, Aarhus University
Title: Diffusion Models for Sphere Packings
2:30–2:45 pm Break
2:45–3:45 pm Randy Davila, RelationalAI, Rice University
Title: Recent Developments in Automated Conjecturing
Abstract: The dream of a machine capable of generating deep mathematical insight has inspired decades of research—from Fajtlowicz’s Graffiti program in graph theory and chemistry to DeepMind’s neural breakthroughs in knot theory. In this talk, we briefly trace the evolution of automated conjecturing systems and present recent advances that deepen our understanding of what it means for machines to conjecture—a pursuit long embodied by our system, TxGraffiti. Building on this legacy, we introduce a new framework that integrates optimization, enumeration, and convex geometric methods with creative heuristics and symbolic translation. This extended system produces not only conjectured inequalities, but also necessary and sufficient condition statements, which can then be automatically ranked by IRIS (Inequality Ranking and Inference System) model and translated into Lean 4 for formal verification. The result is a flexible architecture capable of generating precise, human-readable, and logically rigorous conjectures with minimal manual intervention.
We showcase results across a range of mathematical areas, including graph theory, polyhedral theory, number theory, and—for the first time—conjectures in string theory, derived from the dataset of complete intersection Calabi–Yau (CICY) threefolds. Together, these developments suggest that with the right blend of structure, strategy, and aesthetic, machines can generate conjectures that not only withstand scrutiny but invite it—offering a glimpse into a future where AI contributes meaningfully to the creative process of mathematics.
3:45–4:00 pm Break
4:00–5:00 pm Eric Ramos, Stevens Institute of Technology
Title: An AI approach to a conjecture of Erdos
Abstract: Given a graph G, its independence sequence is the integral sequence a_1,a_2,…,a_n, where a_i is the number of independent sets of vertices of size i. In the 90’s Erdos and coauthors showed that this sequence need not be unimodal for general graphs, but conjectured that it is always unimodal whenever G is a tree. This conjecture was then naturally generalized to claim that the independence sequence of trees should be log concave, in the sense that a_i^2 is always above a_{i-1}a_{i+1}. This stronger version of the conjecture was shown to hold for all trees of at most 25 vertices. In 2023, however, using improved computational power and a considerably more efficient algorithm, Kadrawi, Levit, Yosef, and Mirzrachi proved that there were exactly two trees on 26 vertices whose independence sequence was not log concave. They also showed how these two examples could be generalized to create two families of trees whose members are all not log concave. Finally, in early 2025, Galvin provided a family of trees with the property that for any chosen positive integer k, there is a member T of the family where log concavity breaks at index alpha(T) – k, where alph(T) is the independence number of T. Outside of these three families, not much else was known about what causes log concavity to break.In this presentation, I will discuss joint work of myself and Shiqi Sun, where we used the PatternBoost architecture to train a machine to find counter-examples to the log concavity conjecture. We will discuss the successes of this approach – finding tens of thousands of new counter-examples with vertex set sizes varying from 27 to 101 – and some of its fascinating failures.

 

Tuesday, Sep. 9, 2025

9:00–9:30 am Morning refreshments
9:30–10:30 am Maria Prat Colomer, Brown University
Title: From PINNs to Computer-Assisted Proofs for Fluid Dynamics
Abstract: Physics-Informed Neural Networks (PINNs) have emerged as an alternative to traditional numerical methods for solving partial differential equations (PDEs). We apply PINNs to the study of low regularity problems in fluid dynamics, focusing on the incompressible 2D Euler equations. In particular, we study V-states, which are a class of weak, non-smooth solutions for which the vorticity is the characteristic function of a domain that rotates with constant angular velocity. We have obtained an approximate solution of a limiting V-state using a PINN and we are currently working on a rigourous proof of the existence of a nearby solution through a computer-assisted proof. Our PINN-based numerical approximation significantly improves on traditional methods, a key factor being the integration of prior mathematical knowledge of the problem to effectively explore the solution space.
10:30–11:00 am Break
11:00 am–12:00 pm Sebastian Racaniere, Google DeepMind
Title: Generative models and high dimensional symmetries: the case of Lattice QCD
Abstract: Applying normalizing flows, a machine learning technique for mapping distributions, to Lattice QCD offers a promising route to enhance simulations and overcome limitations of traditional methods like Hybrid Monte Carlo. LQCD aims to compute expectation values of observables from an intractable distribution defined over a lattice of fields. Normalizing flows can learn this complex distribution and generate new configurations, improving efficiency and addressing challenges such as critical slowing down and topological freezing. Topological freezing, in particular, traps simulations in local minima and prevents exploration of the full configuration space, affecting accuracy. This approach incorporates the symmetries of LQCD through gauge equivariant flows, leading to successful definitions and good effective sample sizes on smaller lattices. Beyond accelerating configuration generation, normalizing flows also find application in variance reduction for observable calculation and exploring phenomena at different scales within LQCD. While further research is needed to apply these methods at the scale of state-of-the-art LQCD calculations, these advancements hold significant potential to improve the accuracy, efficiency, and reach of future simulations.
12:00–1:30 pm Lunch break
1:30–2:30 pm Sergei Gukov, Caltech
Title: On sparse reward problems in mathematics
Abstract: An alternative title for this talk could be “Learning Hardness.” To see why, we will explore some long-standing open problems in mathematics and examine what makes them hard from a computational perspective. We will argue that, in many cases, the difficulty arises from a highly uneven distribution of hardness within families of related problems, where the truly hard cases lie far out in the tail. We will then discuss how recent advances in AI may provide new tools to tackle these challenges. Based in part on the recent work with A.Shehper, A.Medina-Mardones, L.Fagan, B.Lewandowski, A.Gruen, Y.Qiu, P.Kucharski, and Z.Wang.
2:30–2:45 pm Break
2:45–3:45 pm Alyson Deines, IDA-CCR La Jolla; Tamara Veenstra, IDA-CCR La Jolla; Joanna Bieri, University of Redlands
Title: Machine learning $L$-functions
Abstract: We study the vanishing order of rational $L$-functions and Maass form $L$-functions from a data scientific perspective. Each $L$-function is represented by finitely many Dirichlet coefficients, the normalization of which depends on the context. We observe murmurations by averaging over these datasets. For rational $L$-functions, we find that PCA clusters rational $L$-functions by their vanishing order and record that LDA and neural networks may accurately predict this quantity. For Maass form $L$-functions, while PCA does not cluster these $L$-functions, we still find that LDA and neural networks may accurately predict this quantity.
3:45–4:00 pm Break
4:00–5:00 pm Mark Hughes, Brigham Young University
Title: Modelling the concordance group via contrastive learning
Abstract: The concordance group of knots in 3-space is an abelian group formed by the equivalence classes of knots under the relation of concordance, where two knots are concordant if they are the boundary of a smooth annulus properly embedded in the 4-dimensional product space S^3 x I. Though studied since 1966, properties of the concordance groups (and even the recognition problem of deciding when a knot is null-concordant, or slice) are difficult to study. In this talk I will outline ongoing attempts to model the concordance group using contrastive learning. This is joint work with Onkar Singh Gujral.

 

 

Wednesday Sep. 10, 2025

9:00–9:30 am Morning refreshments
9:30–10:30 am Yang-Hui He, University of Oxford (Via Zoom)
Title: AI for Mathematics: Bottom-up, Top-Down, Meta-
Abstract: We argue how AI can assist mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and string theory in 2017, we summarize how this emerging field has grown over the past years, and show how various machine-learning algorithms can help with pattern detection across disciplines ranging from algebraic geometry to representation theory, to combinatorics, and to number theory. At the heart of the programme is the question how does AI help with theoretical discovery, and the implications for the future of mathematics.
10:30–11:00 am Break
11:00 am–12:00 pm Giorgi Butbaia, University of New Hampshire
Title: Computational String Theory using Machine Learning
Abstract: Calabi-Yau compactifications of the $E_8\times E_8$ heterotic string provide a promising route to recovering the four-dimensional particle physics described by the Standard Model. While the topology of the Calabi-Yau space determines the overall matter content in the low-energy effective field theory, further details of the compactification geometry are needed to calculate the normalized physical couplings and masses of elementary particles. In this talk, we present novel numerical techniques for computing physically normalized Yukawa couplings in a number of heterotic models in the standard embedding using geometric machine learning and equivariant neural networks. We observe that the results produced using these techniques are in excellent agreement with the expected values for certain special cases, where the answers are known. In the case of the Tian-Yau manifold, which defines a model with three generations and has $h^{2,1}>1$, we provide a first-of-its-kind calculation of the normalized Yukawa couplings. As part of this work, we have developed a Python library called cymyc, which streamlines calculation of the Calabi-Yau metric and the Yukawa couplings on arbitrary Calabi-Yau manifolds that are realized as complete intersections and provides a framework for studying the differential geometric properties, such as the curvature.
12:00–1:30 pm Lunch break
1:30–2:30 pm Eric Mjolsness, UC Irvine
Title: Graph operators for science-applied AI/ML
Abstract: Scalable, structured graphs play a central role in mathematical problem definition for scientific applications of artificial intelligence and machine learning. Qualitatively diverse kinds of operators are necessary to bring these graphs to life. Continuous-time processes govern the evolution of spatial graph embeddings and other graph-local differential equation systems, as well as the flow of probability between locally similar graph structures in a probabilistic Fock space, according to rules in a dynamical graph grammar (DGG). Both kinds of dynamics have biophysical application eg. to dynamic cytoskeleton, and both obey graph-centric time-evolution operators in an operator algebra that can be differentiated for learning. On the other hand coarse-scale discrete jumps in graph structure such as global mesh refinement can be modeled with a “graph lineage”: a sequence of sparsely interrelated graphs whose size grows roughly exponentially with level number. Graph lineages permit the definition of substantially more cost-efficient skeletal graph products, as versions of classic binary graph operators such as the Cartesian product and direct product of graphs, with analogous but not identical properties. Application to deep neural networks and to multigrid numerical methods are shown.
These two graph operator frameworks are interrelated. Further graph lineage operators allow the definition of graph frontier spaces, accommodating graph grammars and supporting the definition of skeletal graph-graph function spaces. In return, “confluent” graph grammars e.g. for adaptive mesh generation permit the definition of graph lineages through iteration. I will also sketch the design of compatible AI for Science systems that may exploit DGGs.
Joint work with Cory Scott and Matthew Hur.
2:30–3:00 pm Break
3:00–5:00 pm Panel and Discussion Group: Jordan Ellenberg, Tamara Veenstra, Sébastien Racaniere, Kyu-Hwan Lee, Sergei Gukov

 

 

 

 

Details

  • Start: September 8, 2025 @ 9:00 am
  • End: September 10, 2025 @ 5:00 pm
  • Event Categories: ,

Organizer

Michael R. Douglas (Harvard CMSA)

Venue