Research
Broadly: spectral and homological invariants of graphs and $2$-complexes. I focus on quantities that are simultaneously (i) structurally meaningful, (ii) computable, and (iii) stable under natural operations.
Higher Laplacians from a fixed graph
With a fixed orientation of edges, $\delta_0$ is the signed vertex–edge incidence operator. A choice of triangles $T$ determines $\delta_1(T)$ and hence $$L_1(T)=\delta_0\delta_0^{\mathsf T}+\delta_1(T)^{\mathsf T}\delta_1(T).$$
Choose $T$ to make $\lambda_{\min}^+(L_1(T))$ large while keeping $|T|$ small. This raises monotonicity questions (“does adding triangles always help?”), stability bounds (how much can one triangle change $\lambda_{\min}^+$), and saturation/diminishing-returns phenomena.
Structured families
For highly symmetric families (strongly regular graphs, association-scheme settings, simplicial designs), one can sometimes compute or tightly bound spectra of $L_1^{\uparrow}$ by reducing to small matrix blocks.
What belongs on this site
- Foundations and “folklore” to keep notation honest and reusable.
- Notes pages that read like the front half of a paper: definitions, lemmas, proof skeletons.
- Computation pages that document experiments that motivate conjectures and counterexamples.