About
Research Focus
This space serves as a repository for independent research focused on the intersection of geometric deep learning, topological data analysis (TDA), and algebraic representation learning.
The core motivation of this work is to study how complex discrete and relational data — such as natural language, hierarchical networks, and taxonomies — can be modeled directly on appropriate mathematical manifolds without the metric distortion inherent in flat Euclidean representations. By applying methods from non-commutative algebra, group theory, and computational homology, we aim to investigate the qualitative shape and structural invariants of sequential and graph-structured data.
Key Research Directions
Geometric Representation Learning
- Hyperbolic Embeddings: Mapping hierarchical networks, taxonomies, and linguistic parse trees into non-Euclidean spaces (specifically, Poincaré and Lorentz models of hyperbolic space) to minimize metric distortion.
- Geometric Graph Neural Networks: Designing and evaluating neural architectures that incorporate Riemannian geometry priors for representation learning on manifolds.
Algebraic Topology of Discrete Sequences
- Computational Homology & fr-codes: Evaluating algebraic invariants (specifically $fr$-codes and standard complexes) as derived functors of limits over categories of free presentations of groups.
- Magnus Algebra Invariants: Applying non-commutative Magnus expansions in free group rings to compute exact topological ranks and identify structural syzygies (dependencies) in text.
Theoretical Foundations
Our computational models are grounded in the following mathematical disciplines:
- Riemannian Geometry: Manifolds, sectional curvature, geodesics, and hyperbolic spaces.
- Algebraic Topology: Homology, simplicial complexes, and persistent homology.
- Non-Commutative Algebra: Free group rings, filtration quotients, and Magnus expansions.
- Category & Group Theory: Functorial limits over free presentations, group homology.