Michael R. Douglas

Michael R. Douglas · harvard

Artificial intelligence is making spectacular progress, and one of the best examples is the development of large language models (LLMs) such as OpenAI's GPT series. In these lectures, written for readers with a background in mathematics or physics, we give a brief history and survey of the state of the art, and describe the underlying transformer architecture in detail. We then explore some current ideas on how LLMs work and how models trained to predict the next word in a text are able to perform other tasks displaying intelligence.

Michael R. Douglas, Sarah Hoback, Anna Mei et al.

A foundational result in constructive quantum field theory is the construction of the free bosonic quantum field theory in four-dimensional Euclidean spacetime and the proof that it satisfies the Glimm-Jaffe axioms, a variant of the Osterwalder-Schrader axioms. We present a formalization of this result in the Lean 4 interactive theorem prover. The project is intended as a proof of concept that extended arguments in mathematical physics can be translated into machine-checked proofs using existing AI tools. We begin by introducing interactive theorem proving and constructive quantum field theory, then describe our formalization and the design decisions that shaped it. We also explain the methods we used, including coding assistants, and conclude by considering how AI assisted formalization may influence the future of theoretical physics. Our original release assumed three results, Minlos' theorem, the nuclear property of Schwartz space, and Goursat's theorem. In subsequent releases from our group and from contributors from the Lean community, these assumptions have been proven (or avoided), so that the OS/GJ axioms are now proven using only Lean and its library Mathlib.

Maissam Barkeshli, Michael R. Douglas, Michael H. Freedman

Recent progress in artificial intelligence (AI) is unlocking transformative capabilities for mathematics. There is great hope that AI will help solve major open problems and autonomously discover new mathematical concepts. In this essay, we further consider how AI may open a grand perspective on mathematics by forging a new route, complementary to mathematical\textbf{ logic,} to understanding the global structure of formal \textbf{proof}\textbf{s}. We begin by providing a sketch of the formal structure of mathematics in terms of universal proof and structural hypergraphs and discuss questions this raises about the foundational structure of mathematics. We then outline the main ingredients and provide a set of criteria to be satisfied for AI models capable of automated mathematical discovery. As we send AI agents to traverse Platonic mathematical worlds, we expect they will teach us about the nature of mathematics: both as a whole, and the small ribbons conducive to human understanding. Perhaps they will shed light on the old question: "Is mathematics discovered or invented?" Can we grok the terrain of these \textbf{Platonic worlds}?

Michael R. Douglas, Sergiy Verstyuk

We study long-run progress in artificial intelligence in a quantitative way. Many measures, including traditional ones such as patents and publications, machine learning benchmarks, and a new Aggregate State of the Art in ML (or ASOTA) Index we have constructed from these, show exponential growth at roughly constant rates over long periods. Production of patents and publications doubles every ten years, by contrast with the growth of computing resources driven by Moore's Law, roughly a doubling every two years. We argue that the input of AI researchers is also crucial and its contribution can be objectively estimated. Consequently, we give a simple argument that explains the 5:1 relation between these two rates. We then discuss the application of this argument to different output measures and compare our analyses with predictions based on machine learning scaling laws proposed in existing literature. Our quantitative framework facilitates understanding, predicting, and modulating the development of these important technologies.

Michael R. Douglas, Kit Fraser-Taliente

We review the problem of finding paths in Cayley graphs of groups and group actions, using the Rubik's cube as an example, and we list several more examples of significant mathematical interest. We then show how to formulate these problems in the framework of diffusion models. The exploration of the graph is carried out by the forward process, while finding the target nodes is done by the inverse backward process. This systematizes the discussion and suggests many generalizations. To improve exploration, we propose a ``reversed score'' ansatz which substantially improves over previous comparable algorithms.

Michael R. Douglas, Kyu-Hwan Lee

Can machine learning help discover new mathematical structures? In this article we discuss an approach to doing this which one can call "mathematical data science". In this paradigm, one studies mathematical objects collectively rather than individually, by creating datasets and doing machine learning experiments and interpretations. After an overview, we present two case studies: murmurations in number theory and loadings of partitions related to Kronecker coefficients in representation theory and combinatorics.

Michael R. Douglas, Michael Simkin, Omri Ben-Eliezer et al.

A knowledge graph (KG) is a data structure which represents entities and relations as the vertices and edges of a directed graph with edge types. KGs are an important primitive in modern machine learning and artificial intelligence. Embedding-based models, such as the seminal TransE [Bordes et al., 2013] and the recent PairRE [Chao et al., 2020] are among the most popular and successful approaches for representing KGs and inferring missing edges (link completion). Their relative success is often credited in the literature to their ability to learn logical rules between the relations. In this work, we investigate whether learning rules between relations is indeed what drives the performance of embedding-based methods. We define motif learning and two alternative mechanisms, network learning (based only on the connectivity of the KG, ignoring the relation types), and unstructured statistical learning (ignoring the connectivity of the graph). Using experiments on synthetic KGs, we show that KG models can learn motifs and how this ability is degraded by non-motif (noise) edges. We propose tests to distinguish the contributions of the three mechanisms to performance, and apply them to popular KG benchmarks. We also discuss an issue with the standard performance testing protocol and suggest an improvement. To appear in the proceedings of Complex Networks 2021.