Joey Beauvais-Feisthauer

Computing the stable homotopy groups of spheres is arguably the most central problem in stable homotopy theory. The best tool for it, the Adams spectral sequence, starts from a table (its E₂ page) with a long tradition of being pushed further and further. This run pushes it through stem 400, far past the previous best.

In all: 37 days, up to 45 GPUs, ~18 quadrillion Milnor products and ~1.2 sextillion bit operations of row reduction.

• Distributed the core algorithm across up to 45 GPUs of four kinds (H200, L40S, V100, A30) on a shared HPC cluster: one coordinator, stateless workers scheduled through Slurm, row reduction as a service. On a benchmark workload: 205.5 h on one node → 28.3 h on 23 GPUs.
• Found structure in the mathematics that exposes far more parallel work. Inside each step, the dependency graph grows exponentially wide but only polynomially deep. Between steps, the algebra allows a relaxation that halves the critical path. Together they took the run from ~20 GPUs’ worth of parallelism to ~190.
• Row reduction over 𝔽₂, built on the GPU matrix multiply below: a 1,048,576 × 1,048,576 matrix (137 GB) in 12 minutes on one H200, against ~35 hours extrapolated for the best CPU code. Production results are verified by Freivalds checks.
• Successive rewrites of the hottest GPU kernel: ~37× on a fixed batch of work. The last one prunes its search space instead of brute-forcing it, with 2.4× less state per thread and device memory from 90 GB to under 2 GB, which let older 16 GB cards join the run.
• Traded storage for recompute: 12 PB projected on disk → ~100 GB, and faster, because the parallel filesystem had been the bottleneck. Peak memory went from over 700 GB (out-of-memory kills) to 45 GB.
• Fault-tolerant on a shared cluster: over the 37-day run, workers came and went as their allocations ended and restarted, with checkpoint/restart throughout, and the coordinator was upgraded live ~20 times without losing completed work.

one node vs 23 GPUs Oct 2 → Nov 5 row reduction vs CPU 37× in three kernels 12 PB we didn’t need

The production code for this run lives on its own branch; the pieces that stand on their own are being backported to sseq.

• An open-source Rust library for large-scale computation in algebraic topology. It produced our published stem-256 dataset (2023) and the stem-400 run above.
• Rewrote the hottest loop to search only the candidates that can contribute, instead of enumerating everything and throwing most of it away: 18× faster on CPU at the largest size tested (1.9× at the smallest), with the gap widening steadily with size. The same idea drives the GPU kernel in the run above.
• Replaced a hash-map lookup in the inner loop with closed-form indexing: 2.7× faster at scale, and the core tables from 12.4 GB of memory to 2.1 GB.
• Hand-wrote a Hopper tensor-core matrix multiply over 𝔽₂, built on the 1-bit wgmma … and.popc instruction: 10,031 trillion bit operations per second on one H200, 14.5× the best other GPU code and 331× the best CPU code at the same size. Built up in 12 phases: TMA loads with a 128-byte swizzle, warp-specialized producer/consumer pipelining, a persistent grid with grouped tile rasterization, TMA multicast across clusters, and register blocking.
• Hand-wrote an AVX-512 microkernel for matrix multiplication over 𝔽₂: over a trillion bit operations per second on a single CPU core.
• Generalized the main algorithm to an entire family of related algebras (every prime, plus the variants used in motivic homotopy theory), built on one generic implementation instead of special-cased branches.
• In review: the rest of the GPU backend and the distributed solver from the run above.
• Type-safe finite field arithmetic for primes up to 2³¹ − 1 via const generics and SIMD intrinsics; non-prime fields via Zech logarithms.
• Wait-free atomic data structures with trie-based backing and write-once semantics.
• 400+ property-based tests, plus loom/miri against UB. Found a priority inversion via tracing; fixing it saved days of computation.

1,579 → 10,031 trillion ops/s 18× at degree 600

Since 2025, my code has been written with AI coding agents (Claude Code), from my designs and optimization ideas. Everything merged upstream is reviewed by hand by Hood Chatham and me.

• Turns JSON into self-contained interactive HTML+SVG spectral sequence charts — pan/zoom, KaTeX, multi-page collections, custom theming.
• Pydantic-introspected JSON schema; Nix flakes, uv, ruff, pyright; published on PyPI.
• A WebGL renderer for huge charts: the full stem-400 chart below, 1.1 million nodes and edges, is a 1.4 MB file that decodes in ~130 ms.

↑ live: the full 2-primary Adams E₂ page through stem 400, with all 253,204 generators and 884,312 h₀, h₁, h₂ products, drawn by SeqSee’s WebGL renderer. Scroll to zoom in, drag to pan.

The minimal resolution of 𝔽₂ over the Steenrod algebra through stem 400 and filtration 202: 253,204 generators in 81,002 bidegrees, of which 173,007 lie beyond the previous best chart (Wu, t ≤ 350). Complete agreement with Wu’s chart on the 50,399 bidegrees both cover.

A 5,300+ CPU-day distributed computation on 19 64-core nodes, producing 22.9 TB of compressed data. At release, the state-of-the-art dataset in computational stable homotopy theory.

An extension of the classic Milnor–Moore equivalence to certain bialgebras without antipodes.

Two novel algorithms for automated differential deduction: a local one built on careful bookkeeping of affine spaces, and a global one that rephrases the whole problem as SAT.