Schedule
| Time | Wednesday Oct. 28 | Thursday Oct. 29 | Friday Oct. 30 | Saturday Oct. 31 | Sunday Nov. 1 |
|---|---|---|---|---|---|
| 9:00–10:00 | Orientation (9:00–9:30) | Matt Ballard | Nic Brody | Hood Chatham | Group work |
| 10:00–11:00 | Leilani Gilpin (9:30–10:30) | Adam Topaz | Steve Trettel | Group work | Group work |
| 11:00–12:00 | CLI/Install Session | Skills session | Visualization session | Group work | Presentations |
| 12:00–1:30 | Lunch | Lunch | Lunch | Lunch | Lunch |
| 1:30–2:30 | Gunnar Carlsson | Patricio Gallardo | Ben Antieau | Group work | Conference end |
| 2:30–3:30 | Eoin Mackall | David Eisenbud | Jared Duker Lichtman | Mohammed Abouzaid | |
| 3:30–4:00 | Coffee break | Coffee break | Group work | Coffee break | |
| 4:00–5:00 | Brainstorm project topics | Brainstorm, then get into groups for the 2nd half | Group work | Panel on future of math and AI |
Wednesday, October 28
Title: What to Know About AI
Abstract: Artificial intelligence has long been concerned with a deceptively simple question: How can we make machines solve problems intelligently? In this talk, I’ll introduce some of the foundational ideas of AI, including search, knowledge representation, and reasoning, and highlight their connections to familiar mathematical ideas. We’ll explore how machines can navigate large spaces of possible solutions, represent what they know, and draw conclusions from that knowledge. Finally, I’ll share some reflections on teaching AI in the age of generative AI, including our ongoing work examining how generative AI is changing the way students learn, solve problems, and think critically abou:Wt their own solutions.
11:00–12:00 · CLI/Install Session [Slides]
Session: An hour-long session to help participants familiarize themselves with the terminal and to get programs installed that they’ll use for the remainder of the workshop.
Title: Generalized Signal Processing for AI
Abstract: Signal processing, i.e. data transformation, is a fundamental component of the mathematical study of data in many classical situations, for example radar, sonar, etc. There are specialized tool kits utilizing Fourier analysis, Wavelet analysis, etc. which are critical to applications. In this talk, we will consider the corresponding problem for AI tools such as Convolutional Neural Networks and Transformer models for language processing. We will show by example what can be achieved in improving performance as measured by generalizability, robustness to noise, and in the case of LLMs average perplexity and feature interpretability. I will argue that methods such as these are key to addressing many of the problems with the technology, including AI safety and computational efficiency.
2:30–3:30 · Eoin Mackall [Slides]
Title: On Proofs and Progress
Abstract: What does experimental mathematics look like? In this talk I give one possibility by way of example: a program for constructing either a proof of, or a counterexample to, a stated conjecture through LLM inference. This program allows for distributed use by multiple users aiming to prove or refute a given conjecture, and it can run parallel searches simultaneously in any one instance. I’ll outline the architecture behind this program, my anecdotal experience using the program in some instances, and showcase some future directions that I have planned.
Thursday, October 29
Title: Working productively with AI
Abstract: I will discuss how I use AI as part of my research process.
Title: TBA
Abstract: TBA
Session: An hour-long session to get participants familiarized with skills files.
Title: How AI is changing my research and my students’ training
Abstract: I will describe how AI has changed the topics I work on and the way I mentor my students. The main case study is ongoing work relating neural networks to toric geometry, in which human and AI-proposed constructions were compared through explicit calculations. I will also discuss skills for pre-refereeing drafts, when students should and shouldn’t use AI, and the open questions of verification and cost.
Title: Weierstrass semigroups
Abstract: If X is smooth projective curve (or a compact Riemann surface) then the only globally defined regular functions on X are the constants; but if one point is removed there are many regular functions on the rest, each of which has a pole (locally 1/z^n for some n). Since the pole order of a product is the sum of the pole orders, these orders form a numerical semigroup—that is, a subset of the positive integers, closed under addition. Weierstrass proved around 1860 that this semigroup contains all but genus(X) non-negative integers; the semigroups is now called the Weierstrass semigroup of p in X, in his honor.
In 1892, Hurwitz asked: What semigroups can actually occur? The full answer is still not known/ I’ll explain the history of this problem, and some new results, based on extensive computations, which led to a new criterion using syzygies, applicable to a certain class of semigroups, from my recent work with Frank-Olaf Schreyer
Friday, October 30
9:00–10:00 · Nic Brody [Slides]
Title: AlphaCentaur: A workspace for human-AI collaboration
Abstract: I will introduce AlphaCentaur, a new tool which integrates existing mathematical research workflows into a workspace that can mediate between the human researcher and an AI agent. My research workflow historically has utilized many different technologies, including pdf readers, HTML/js webapps, TeX editors, Powerpoint, Notability, Lean, Jupyter notebooks, Obsidian, and more. AlphaCentaur creates a single ecosystem unifying these various tools into an AI-friendly format. We will demonstrate this app with a worked example, detailing progress on a research program that I previously believed would occupy the remainder of my research career. Quite possibly, this project will be finished in a few months. Given a finite set S of invertible 2-by-2 matrices with algebraic entries, can we find a presentation for the group generated by S? We will take a visual tour through algebraic, geometric, and arithmetic methods for answering this question.
Title: Computing the Light Field
Abstract: AI coding tools are making projects that once required months of programming practical to undertake. In this talk, we’ll explore one such project from mathematical computer graphics, where the goal is to produce photorealistic images by simulating light. There are two mathematical ingredients: how light travels and scatters through a scene, and how a camera collects it to form an image. Standard path tracing combines these into an unbiased estimator, spending substantial computation to produce an image for a fixed camera.
We’ll try something more ambitious: compute and store the entire stationary light field, then sample it with a simulated camera, producing photorealistic images in real time as we move through the scene or change focus and aperture. Under standard physical assumptions, this field satisfies an infinite-dimensional fixed-point equation. Computing and storing its solution requires enough GPU and memory-management machinery that I had long wanted to try this project but couldn’t justify the time commitment. Working with coding agents made it possible to hand off much of that implementation and focus on the mathematics itself.
11:00–12:00 · Visualization Session
Session: A session on making mathematical visualizations. After an hour we’ll share our progress with other participants.
1:30–2:30 · Benjamin Antieau [Slides]
Title: The forthcoming golden age of mathematics software
Abstract: I will discuss the process I’ve used for building robust software using agents and talk about a few specific projects, focusing on a new CAS called Noether. This is based on joint work with Deven Manam.
2:30–3:30 · Jared Duker Lichtman [Slides]
Title: Formalization and the future of mathematics
Abstract: We are poised to formalize all of known mathematics in the next year. This is the mathematical equivalent of the Human Genome Project. Autoformalization has scaled 5 orders of magnitude growth over the past year: from hundreds of lines of code to 13 million in the proof of Fermat’s Last Theorem. Rough estimates put tens of billions of lines, just 3 more orders of magnitude, to cover all mathematics textbooks and literature. Beyond verification for itself, this effort serves as the basis for future discovery.
Saturday, October 31
Title: Webassembly as a platform for mathematical communication
Abstract: Software written by mathematicians is often hard to use. Distributing software has historically been a specialized skill that few mathematicians have time to develop. WebAssembly and the Web platform is an extremely reproducible environment and is supported by practically every device. If it works on my machine it probably works everywhere. For this reason, it is very promising for scientific and mathematical communication and education. I will talk about my work getting software to run on WebAssembly. I will also speculate wildly about the potential impact of AI on this.
Title: Measuring AI capabilities in Research Mathematics
Abstract: In the past year AI systems have produced a handful of genuine research results in mathematics: a counterexample to the Jacobian conjecture, progress on two of Erdős’s problems. What is much less clear is how to measure what these systems can actually do. I will describe First Proof, which tests AI systems against unpublished mathematics (research mathematicians contribute problems with known but unpublished proofs, and the systems’ solutions are refereed and graded), and report what successive batches have taught us.
Title: The future of math and AI
Description: A panel on the future of math and AI with Mohammed Abouzaid, David Eisenbud, Patricio Gallardo, and Leilani Gilpin.
Sunday, November 1
No talks. Group presentations at 11:00.