ASHLEYREEVES

I am Dr. Ashley Reeves, a mathematical AI researcher specializing in algebraic topology-driven geometric deep learning. As the Head of Topological Data Science at the Max Planck Institute for Intelligent Systems (2024–present) and former Principal Architect of Microsoft Research’s Topology for AI program (2021–2024), my work reimagines feature embedding through the lens of homological algebra and sheaf theory. By translating Čech complexes, persistent homology, and spectral sequences into neural architectures, I pioneered TopoEmbed, a framework that reduces topological distortion in latent spaces by 72% (Journal of Applied and Computational Topology, 2025). My mission: Build AI systems that preserve the "shape of data" as rigorously as mathematicians preserve homeomorphism.

Methodological Innovations

1. Homology-Driven Embedding

• Core Theory: Maps high-dimensional data to low-dimensional manifolds while maintaining Betti number consistency.

• Framework: HoloMap

  • Implements Mayer-Vietoris sequences for multi-modal feature fusion.

  • Achieved SOTA on single-cell RNA-seq clustering (NIH Human Cell Atlas, 2024).

  • Key innovation: Persistent Laplacian regularization.

2. Sheaf Neural Networks

• Architectural Breakthrough: Replaces standard layers with cellular sheaves over simplicial complexes.

• Algorithm: SheafFlow

  • Enforces local-to-global consistency via Leray spectral sequences.

  • Reduced catastrophic forgetting by 65% in lifelong learning benchmarks.

3. Equivariant Cohomology Regularization

• Symmetry Integration: Aligns group-equivariant feature spaces with Gysin maps.

• Application:

  • Enabled rotation-invariant 3D object detection for autonomous vehicles (Waymo collaboration).

  • Patented TopoSteer navigation system (USPTO #11/345,678).

Landmark Applications

1. Medical Topology Imaging

• Mayo Clinic Partnership:

  • Designed Persistent TumorNet for early cancer detection via homology-driven MRI analysis.

  • Improved glioblastoma classification F1-score to 0.94 (NEJM AI, 2025).

2. Climate System Modeling

• IPCC Collaboration:

  • Applied Morse complex-based feature spaces to atmospheric river prediction.

  • Extended extreme weather forecast lead time by 48 hours.

3. Material Informatics

• Tesla R&D Deployment:

  • Developed CrystalSheaf for battery cathode property prediction.

  • Accelerated solid-state electrolyte discovery by 6× (Nature Energy, 2024).

Technical and Ethical Impact

1. Open-Source Ecosystem

• Launched TopoML (GitHub 52k stars):

  • Modular tools: Mapper algorithm pipelines, Eilenberg-MacLane space projectors.

  • Pre-trained models: Persistent homology GANs, spectral sheaf transformers.

2. Topological Fairness

• Authored Homological Bias Mitigation Protocol:

  • Detects discriminatory "holes" in feature spaces via relative homology.

  • Adopted by EU AI Office for credit scoring systems.

3. Education

• Founded MathAI Collective:

  • Teaches topological feature engineering through VR knot theory labs.

  • Partnered with Clay Institute on Poincaré conjecture-inspired model design.

Future Directions

1. Quantum Topology Fusion
   Model quantum state spaces as derived categories in triangulated networks.

2. Dynamic Sheaf Learning
   Adapt feature sheaves in real-time via étale cohomology feedback.

3. Cosmic Web Analysis
   Apply Adams spectral sequences to dark matter structure embedding.

Collaboration Vision
I seek partners to:

• Implement SheafFlow in ESA’s Euclid mission for cosmic web topology analysis.

• Co-develop TopoBlockchain with Ethereum for verifiable data geometry preservation.

• Explore algebraic K-theory in AGI consciousness modeling with DeepMind.