502 Johamhaean-ro, Jocheon-eup, Jeju-si, Jeju-do
Francois Charton (Axiom Math)
Xiaoyu Huang (Temple Univ.)
Byung-Hak Hwang (KIAS)
Helen Jenne (Pacific Northwest National Laboratory)
Donghun Lee (Korea Univ.)
Kyu-Hwan Lee (Univ. of Connecticut / KIAS)
Eric Ramos (Stevens Institute of Technology)
Christoph Spiegel (Zuse Institute Berlin)
Hongseok Yang (KIAS)
Byung-Hak Hwang (KIAS)
Taeyoung Kim (KAIST)
Bernard Lidický (Iowa State Univ.)
Deadline: March 31st
Registration link: here!
Accommodation and meals will be provided for up to 55 participants. The exact number of supported participants will depend on the final budget and room availability. All registrants will be notified by April 15 regarding support.
🍽️ Meals: all meals from Monday lunch through Friday lunch will be provided for supported participants. Breakfast will be provided only for supported participants staying at the workshop hotel.
🏨 Accommodation: supported participants will share a room with another supported participant. Those who prefer a single room may request one and will be required to cover half of the room cost.
Title: Developing an automata-theoretic approach to combinatorial interpretation with LLMs
Abstract: In the spring of this year, I had several discussion sessions with math colleagues on AI and
mathematics. We were all eager both to use and to advance AI for mathematical research, but the
impressive success stories coming out of big tech in this direction frankly scared us, and we went
looking for a research direction that would be safe from scooping and, we suspected, a better fit
for human–AI collaboration. Our tentative answer was automatic or semi-automatic theory building:
while big tech is strongly motivated to build models that crack famous open problems with
immediately legible payoffs, it is far less motivated to build models for theory building, whose
success is genuinely hard to evaluate. To test this conjecture, we used LLMs to develop a small
theory that brings algebraic combinatorics and computer science together.
Algebraic combinatorists study statistics on combinatorial objects such as Dyck paths, the balanced
sequences of up- and down-steps. A recurring theme is to find two different statistics that
nevertheless induce the same distribution of values, and then to explain the coincidence by a
combinatorial interpretation: a simple bijection on the objects that carries one statistic to the
other. The classic example is the area and dinv statistics on Dyck paths, equidistributed by way of
Haglund's zeta map.
Our goal was to measure the complexity of such bijections by the computational resources needed to
implement them, drawing on the rich hierarchy of machine models from automata theory. The theory we
arrived at places the zeta map just outside the classical hierarchy of string transducers and pins
down the single extra resource it requires: a "rank sweep," the level-by-level sorting along an
unbounded integer height that zeta performs. Within this framework one can prove genuine limits —
for instance, that no rank sweep can swap the area and dinv statistics, so any bijective proof of
their symmetry must reach beyond zeta-style sorting. In this talk I will describe both halves of the
story: the mathematics we built, and what working closely with LLMs throughout taught us about theory
building itself. My verdict is more optimistic than I expected at the outset: although today's LLMs
were not trained specifically for theory building, they proved far better than we anticipated at
proposing research questions, finding good definitions, and even inventing tools for proving
theorems, already very useful collaborators for a theory builder.
This is joint work with Jineon Baek, Byung-Hak Hwang, and Joonhyun La.
Title: Mathematical discovery by watching a model learn
Abstract: TBA
Title: Coloring the plane with neural networks
Abstract: We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance din the sixth color. Such colorings have previously been known to exist for 0.41 < √2−1 ≤d ≤1/√5 < 0.45. Our results significantly expand that range to 0.354 ≤d ≤0.657, the first improvement in 30 years. The constructions underlying this notably were derived by formalizing colorings suggested by a custom machine learning approach. This is joint work with Sebastian Pokutta, Konrad Mundiger, and Max Zimmer.
Title: Finding the right needle in a haystack
Abstract: Mathematical research often begins with a collection of examples and the search for a hidden pattern among them. In settings where only partial or aggregate information is available, this task becomes difficult, especially in combinatorics. In this talk, I will discuss such a situation and present a machine-learning-based approach for recovering the relevant combinatorial object from weak supervision. The emphasis will be on the mathematical application rather than the learning method itself. The main example will illustrate how AI can be used as a tool for structured discovery in combinatorial problems.
Title: Implicit Neural Representation of Graphons for Extremal Graph Theory
Abstract:
Extremal graph theory problems can be studied asymptotically via graphons, effectively relaxing complex discrete challenges into continuous optimisation problems. This approach, however, necessitates solving an infinite-dimensional optimisation problem with non-convex objectives. To overcome this, we propose solving these graphon optimisation problems using neural implicit representations, a methodology adapted from recent advances in neural graphics. As a primary application, we focus on the problem of minimising target subgraph density subject to a fixed edge density constraint (e.g., 0.75). By parameterising the graphon with a neural network and applying architectural transformations to strictly enforce the density constraint, our framework reliably uncovers structured graphons and close-to-optimal solutions.
This tutorial begins with a comprehensive overview of our framework and core results. Following this, we introduce the practical prerequisites for implementation, including a primer on Python and PyTorch. The tutorial concludes with a hands-on session where we apply our framework to solve Razborov's classical triangle density minimisation problem.
This work was done in collaboration with Jineon Baek in KIAS, Joonkyung Lee in Yonsei University, and Hongseok Yang in KIAS.
Title: TBA
Abstract: TBA
Title: Datasets for AI-Driven Discovery in Algebraic Combinatorics
Abstract: The last five years have seen AI-driven breakthroughs in research-level mathematics, prompting excitement and discussion about the role of AI in mathematical discovery. This talk focuses on AI-based methods in algebraic combinatorics, a field particularly well-suited to computational methods because its core objects (e.g. permutations, graphs) are naturally represented by matrices. We give an overview of our work at Pacific Northwest National Laboratory, discussing two projects: the Algebraic Combinatorics Dataset Repository, a suite of datasets representing open and classical problems in the field, and OpenConjecture, a living dataset of conjectures extracted from recent arXiv papers. Finally, we discuss lessons learned from applying both specialized narrow models and LLM-based approaches to problems from these datasets. This talk represents joint work with Henry Kvinge, Davis Brown, Jesse He, Herman Chau, and Sara Billey.
Title: Exploring Evolve-Style Coding Agents
Abstract: This tutorial introduces a practical approach to mathematical research using LLM-based coding agents, with a focus on Evolve-style systems such as FunSearch and AlphaEvolve. We will discuss the basic workflow behind these agents, review representative results they have achieved, and examine the types of mathematical problems where they are most effective, as well as the settings where they still face limitations. The tutorial will then move from methods and examples to practice, with participants exploring how to apply these tools to a problem of their own.
Title: When Reinforcement Learning Needs Mathematicians to Find Rare Algebraic Graphs
Abstract: Many problems in mathematical discovery can be viewed as sparse searches through large combinatorial spaces, where successful examples are rare and off-the-shelf AI methods provide little guidance. In this talk, I will discuss a case study from Kalai’s algebraic Hirsch conjecture, where the goal is to find algebraic objects whose associated graphs violate a diameter bound. I will describe how hierarchical reinforcement learning (HRL), guided by mathematical structure, can identify useful intermediate bottlenecks and turn them into mechanisms for more effective exploration. This gives a first-of-its-kind application of HRL to a problem in commutative algebra.
Title: Minimal Graph Constructions for Sporadic Simple Groups Using AI
Abstract: In this talk, I will discuss how AI can be used to construct minimal graphs whose automorphism groups are isomorphic to finite sporadic simple groups, demonstrating a new AI-driven workflow.
Title: Pattern Boost and Axplorer
Abstract: Pattern Boost is a method developed for finding combinatorial constructions using machine learning. It combines local search and machine learning. It alternates between training a neural network to make examples and optimizing these examples using local search. While local search can get stuck in local optima easily, the neural network may extract some patters in the good constructions that help overcome the local optima problem. The original idea of Pattern Boost was recently reimplemented as Axplorer. In this tutorial we look at the tool from user perspective and experiment with possible applications.
Title: An AI enhanced approach to the tree Unimodality Conjecture
Abstract: Given a graph $G$, its independence sequence is the integral sequence $a_1,a_2,\ldots,a_n$, where $a_i$ is the number of independent sets of vertices of size $i$. In the late 80's Alavi, Erd\"os, Malde, Schwenk conjectured that this sequence is always unimodal whenever $G$ is a tree. This conjecture was then naturally strengthened 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 conjecture stood for many years, until in 2023, Kadrawi, Levit, Yosef, and Mizrachi proved that there were exactly two trees on 26 vertices whose independence sequence was not log-concave. In this talk, we will discuss an application of the AI architecture PatternBoost, developed by Charton, Ellenberg, Wagner, and Williamson to train a machine to find tens of thousands of new counter-examples to the log-concavity conjecture.
Ilkyoo Choi (Hankuk Univ. of Foreign Studies / DIMAG, IBS / KIAS)
Jeong Han Kim (KIAS)
Seog-Jin Kim (Konkuk Univ.)