Research Interest

My research interests lie in Algebraic Topology and Applied Topology (Cellular Sheaves).

Ongoing  Research

Pattern Formation on Networks via Algebraic Topology

My research explores how spatial patterns emerge on networks when coupling between nodes carries rich geometric structure. I work with cellular sheaves — objects from algebraic topology that encode directional, heterogeneous constraints across a network — and study how feedback between local dynamics and sheaf-geometric signals drives self-organization through mechanisms fundamentally different from classical theory. This work connects algebraic topology, dynamical systems, and network science. See an LMSRD Simulation using known reaction kinetics (Allen-Cahn, Gray-Scott, Schnakenberg, and FitzHugh-Nagumo).

Spectral Methods for Data on Networks

My research develops tools for analyzing structured data distributed over networks, using cellular sheaves. I study the spectral theory of the operators arising from this structure, focusing on how singular value decompositions reveal multi-scale patterns of approximate consistency and on the obstructions to passing between global spectral information and local sheaf-theoretic structure. I am also interested in dynamics constrained by sheaf structure and in stability guarantees ensuring robustness to noise. This work sits at the intersection of spectral graph theory, applied algebraic topology, and topological data analysis.

Sheaves and Heterogeneous Data Fusion on Networks

Our research develops mathematical frameworks for fusing heterogeneous data across networks, using cellular sheaves and Riemannian geometry on manifolds of positive-definite matrices. We study how local covariance structures can be consistently stitched together over a network topology, and what sheaf cohomology reveals when they cannot. This work combines ideas from algebraic topology, spectral theory, and manifold optimization.