Current Research

This page outlines the computational and theoretical aspects of representation learning, non-Euclidean geometry, and discrete homology currently under study.

1. Geometric Representation Learning & Hyperbolic Embeddings

The mathematical focus of this direction is the representation of hierarchical, tree-like structures (such as lexical taxonomies, syntactic parse trees, or social networks) in non-Euclidean spaces to minimize metric distortion.

By utilizing hyperbolic geometry — specifically the Poincaré disk and Lorentz models — we explore how constant negative sectional curvature matches the exponential volume growth of tree structures, allowing low-distortion embeddings.

Current Focus:

• Evaluation of Poincaré embeddings for hierarchical natural language representations.
• Lorentz model optimizations for relational knowledge graphs.
• Representations in mixed-curvature product spaces for heterogeneous relational data.

2. Topological Sequence Analysis & $fr$-Codes

This direction investigates the qualitative properties of discrete sequences using algebraic invariants. We study the non-commutative Magnus expansion in free group rings and the application of $fr$-codes (derived functors of limits over categories of free presentations of groups) to analyze syntactic constraints and structural dependencies.

Current Focus:

• Computing exact $fr$-code invariants over finite fields $\mathbb{Z}_p$ for sequence classification and coherence analysis.
• Evaluating the homological scaling exponent and topological compression rates of diverse text corpora.
• Developing Homological Feature Selection (HFS) methods for automated basis vocabulary extraction.

3. Graph Neural Networks & Geometric Priors

The focus here is on incorporating non-Euclidean geometric priors and topological invariants into Graph Neural Network (GNN) architectures.

By extending message-passing operations to Riemannian manifolds and injecting homological descriptors into graph convolutional layers, we study the expressiveness and generalization of structurally and geometrically informed learning models.

Current Focus:

• Manifold-constrained and curvature-aware message passing in graph-structured data.
• Integrating persistent homology descriptors into GNN message aggregation.
• Geometric clustering and non-Euclidean community detection methods on graph manifolds.

Interconnection of Methods

These three research directions are complementary:

• Hyperbolic geometry defines the underlying non-Euclidean manifold for graph neural network representations.
• Computational homology extracts qualitative topological features that can be integrated into graph convolutional architectures.
• Combinatorial graph structures serve as the common mathematical substrate for both topological analysis and representation learning.

The unification of these approaches aims to construct learning models that are simultaneously geometrically, topologically, and algebraically consistent.