The research program

Lixel’s research program is the heart of what the practice is for. It centers on hard, structural problems in mathematics and computation: questions where the path forward involves both deep combinatorial reasoning and serious computational validation.

Active threads

Graph coloring and the Four Color Theorem. Working on partition and interface methods that combine combinatorial structure with computational checks, with an eye toward proof architectures that are both rigorous and intelligible.

Hadwiger’s conjecture. Active work on one of the longest-standing open problems in graph theory. Details after publication.

Partition and interface methods more generally. A family of approaches in which a complex global property is verified by decomposing a structure into well-understood blocks and reasoning carefully about how those blocks interact across their interfaces. The same machinery underlies several of the threads above and connects naturally to the kind of large-scale, validated computation Lixel does on the consulting side.

Why this is the flagship

These problems sit at the intersection of pure mathematics and serious computation. They reward both careful proof work and well-engineered software for searching, verifying, and visualizing combinatorial structure. That is exactly the kind of work Lixel exists to do.

From research to product

Research is upstream of Studio. Each thread here is chosen partly because it has a plausible path from idea to algorithm to product:

• Graph coloring work (Four Color Theorem, partition methods) feeds toward fast, practical graph-coloring algorithms and the tooling around them.
• Latent procedural graphics, growing out of the same partition and interface family, becomes the procedural-generation engine behind The Dimming and other Studio work to come.

New ideas tend to start here as proofs and prototypes, then mature into something people can use.

Notes from the research

Occasional posts about the math, the methods, and the engineering behind it appear under Writing.