I work on spectra of higher Laplacians and combinatorial structure of graphs and 2-complexes. This site is deliberately “notes-first”: definitions, statements, proof sketches, and computational artifacts.
Fix a base graph $G=(V,E)$ (edges as oriented $1$-simplices). Choose a set of triangles $T$ (oriented $2$-simplices), hence a coboundary $\delta_1(T)$. Study the operator on edge-cochains $$L_1(T)=\delta_0\delta_0^{\mathsf T}+\delta_1(T)^{\mathsf T}\delta_1(T),$$ and how the smallest positive eigenvalue $\lambda_{\min}^+(L_1(T))$ evolves as triangles are added.
Where to start
- Foundations fixes notation for cochains, coboundaries, and Laplacians.
- Notes collects theorem-heavy pages (each printable as PDF).
- Papers is an honest status board: published / preprint / in progress / notes.
One-line definitions
The down Laplacian on $1$-cochains is $L_1^{\downarrow}=\delta_0\delta_0^{\mathsf T}$ (graph part), the up Laplacian is $L_1^{\uparrow}(T)=\delta_1(T)^{\mathsf T}\delta_1(T)$ (triangle part), and $L_1=L_1^{\downarrow}+L_1^{\uparrow}$.
A recurring technique: exploit the fact that adding a single oriented triangle produces a rank-one PSD update on the upper Laplacian. This makes interlacing and Courant–Fischer arguments unusually effective.