alejandrozarzuelo.com

Alejandro Zarzuelo Urdiales

AI+math/physics researcher working on optimization, graph theory, scientific reasoning evals, and AI-assisted research.

I was the lead AI+math/physics researcher at Archivara, where my work helped create the technical story behind its YC S26 path and reported $25M valuation. My strongest recent work is self-directed: a 16 x 16 matrix multiplication result and a clique-cluster graph theory program for modeling line-structured systems. My formal background is a double degree in Physics and Mathematics.

Signature workCompany fit
Signature workCompany fitTrajectoryResearchBenchmarksContact
2208variable multiplications in the 16 x 16 matrix result
OEISclique-cluster sequence family and graph theory reference trail
$25Mreported Archivara valuation story supported by technical work
YC S26Archivara trajectory in which I served as lead AI+math/physics researcher

Signature work

Original research selected for leverage, not routine execution.

These are the two results I want technical teams to notice first: one in optimization with infrastructure relevance, one in graph theory with new modeling power for structured systems.

Optimization and AI infrastructure

16 x 16 matrix multiplication

I identified matrix multiplication as a high-leverage optimization target and developed a 16 x 16 construction using 2208 variable multiplications. The underlying approach is generalizable: it is not merely a one-off table of constants, but a method for searching and structuring algebraic improvements in one of the central operations behind modern compute.

  • Original research direction selected for leverage, not assigned as routine benchmark work.
  • Relevant to accelerator, compiler, model-efficiency, numerical-kernel, and AI infrastructure teams where small algebraic improvements can compound across large systems.
  • Together with the cyclic convolution work, this points toward faster arithmetic primitives for FFT-like computation and other high-throughput workloads.
  • A landmark-style optimization result: important as an object in itself, and important because the method can transfer to adjacent multiplication and convolution problems.
16 x 16 resultCyclic convolution

New graph-theoretic structure for line-like systems

Clique-cluster graph theory

I developed clique-cluster graph theory as a way to bring line structure, weighted relationships, and commensurable organization into graph-theoretic language. This matters because many real systems are not only connected; they have routes, chains, layers, flows, precedence, distance, and operational order. Ordinary graph edges often erase that structure.

  • A self-directed expansion of graph theory rather than a narrow application of existing techniques.
  • Turns line-like structure into a graph-theoretic object, addressing a modeling gap in discrete systems.
  • Relevant to logistics, information processing, routing, scheduling, and operations teams that need richer structure than node-edge connectivity alone.
  • Further developments could give companies better mathematical tools for representing operations, bottlenecks, and multi-stage flows.
  • Public OEIS sequence family gives the work a durable mathematical reference trail.
Clique-cluster paperOEIS A395684OEIS A395695

Where this matters

Built for teams where mathematical initiative changes the product.

I am looking for environments where original research taste, optimization ability, and benchmark fluency can turn into concrete company advantage.

AI infrastructure and optimization

I look for technical bottlenecks where mathematical structure can translate into real system value: matrix multiplication, cyclic convolution, algebraic shortcuts, and evaluation methods for scientific reasoning.

Hardware-aware research teams

The matrix multiplication and cyclic convolution work is naturally legible to teams operating at the level of accelerators, compilers, model efficiency, FFT-like pipelines, and numerical kernels: places where a better algorithmic object can matter at scale.

Logistics and information processing

Clique-cluster methods are relevant to logistics, planning, routing, information processing, and structured operations because they preserve line-like or weighted organization, not only unstructured connectivity.

Startup value creation

At Archivara I was not only executing prompts. I was selecting problems, converting model output into research artifacts, and helping create the technical narrative behind a YC S26 path and reported $25M valuation.

Trajectory

From AI-assisted research to company-relevant technical value.

The through-line is simple: select meaningful problems, use frontier models where they are strongest, pressure-test them where they fail, and convert raw output into artifacts that survive review.

2025-2026

Lead AI+math/physics researcher at Archivara

Served as the core scientist for Archivara's AI-for-science research program, turning frontier-model output into checkable mathematical and physical results. My work supplied a central part of the technical and research substance behind Archivara's YC S26 trajectory and reported $25M valuation.

2026

Independent research after Archivara

Continued producing original work outside the company structure, including the 16 x 16 matrix multiplication result and the clique-cluster graph theory program. This is where my initiative is clearest: I choose under-addressed problems and push them toward durable mathematical artifacts.

2026

Research-level math benchmarks with Ulam

Collaborated with Ulam around mathematical benchmark/data work, including ErdosBench-style evaluation: proof reliability, partial progress, obstruction finding, and the distinction between useful research behavior and unsafe solved claims.

2025

Early AI applications to open-problem mathematics

Produced the first partially original AI-assisted result on an Erdos problem, showing my tendency to look for new applications of frontier models early rather than waiting for a field to become conventional.

2025-2026

Spatial and multimodal evaluation

Contributed critically to SpatialBench-style multimodal/spatial reasoning evaluation work, where the important failure modes are grounded reasoning, symbolic abstraction, and planning over visual structure.

Now

San Francisco Bay Area

Meeting AI labs, eval teams, AI-for-science groups, optimization researchers, and infrastructure teams to find where I can be the most useful version of myself.

What I do

Useful at the boundary between mathematical taste and AI systems.

Original problem selection in AI+math and AI+physics
Algebraic optimization: matrix multiplication and cyclic convolution
Graph-theoretic modeling for line-structured systems
Scientific reasoning evaluations for frontier models
Proof auditing: gaps, overclaims, calibration, and repair
Research-level benchmark construction and data curation
Performance analysis in contest and research-math environments

Selected research

Public work across optimization, graph theory, evals, and physics.

A curated entry point into the work most relevant to AI labs and research teams: algebraic optimization, new graph-theoretic structure, reasoning evals, proof reliability, and scientific workflows.

16 x 16 matrix multiplication

A 2208-multiplication construction and related algebraic optimization work. This is one of my clearest examples of finding a hard technical problem, attacking it directly, and producing a result with infrastructure-level relevance. Because matrix multiplication sits under ML systems, simulation, graphics, and scientific computing, improvements here can matter far beyond the paper itself.

16 x 16 resultCyclic convolution

Clique-cluster graph theory

A new graph-theoretic program for representing clique-cluster, Rado-graph, line-like, and weighted structure. The goal is to make graph theory expressive enough for systems where ordinary edges lose too much information: logistics networks, staged operations, information flows, and systems where order and line structure carry real content.

Clique-cluster paperOEIS A395684OEIS A395691OEIS A395692OEIS A395693OEIS A395694OEIS A395695

AI mathematical reasoning

Evaluation of frontier models on research-level mathematics, including proof hygiene, partial progress, known-theorem use, obstruction finding, and model calibration. I was also responsible for testing the 86th Putnam exam and am familiar with environments where model performance needs careful analysis beyond simple leaderboard ranking.

Ulam ErdosBenchUlam data/evals

AI for physical science

Automated asteroid shape reconstruction, photometric inversion, symbolic physics, and equation discovery pipelines using AI-assisted scientific workflows.

Asteroid modelsPHYSMDT

Open-problem research

AI-assisted work around Erdos Problem 897 and other open-problem settings, with emphasis on turning model suggestions into auditable mathematical artifacts.

Erdos 897 counterexampleVisual proof support

Benchmarks and collaborations

Evaluation work beyond saturated public tests.

My benchmark interests are centered on research behavior: whether models can find obstructions, avoid false solved claims, reason over spatial structure, and generate outputs that humans can verify.

Ulam / ErdosBench

Research-mathematics evaluation work centered on proof reliability, private benchmark construction, hidden rubrics, partial-progress scoring, and model failure localization. This builds on my experience analyzing frontier-model performance on difficult math settings, including the 86th Putnam exam.

SpatialBench

Multimodal and spatial reasoning evaluation work focused on whether models can reason over visual/geometric structure rather than overfit shallow surface cues.

Archivara research corpus

A public record of AI-assisted math and physics outputs: optimization, graph theory, combinatorial geometry, formalization, and scientific discovery workflows.

Contact

For research roles, collaborations, evals, and AI-for-science work.

I am currently in the San Francisco Bay Area and am especially interested in teams working on AI infrastructure, mathematical reasoning, scientific AI, multimodal evals, logistics, and model reliability.

alejandrozarzuelo@gmail.com+1 (510) 570 9957LinkedInX / TwitterArchivara work