9504834 Yau This research seeks to combine expertise in differential geometry and spectral graph theory. The proposers will study the combinatorial Laplacian and its eigenvalues using ideas from differential geometry. Several major topics from differential geometry and mathematical physics are also to be investigated: holomorphic curves, complex and holomorphic vector bundles, and general relativity are some of the topics to be pursued. Spectral graph theory plays an important role in computational complexity theory and is heavily motivated by problems in communication theory. For example, optical fiber networks and Asynchronous Transfer Mode networks require robust and simultaneous connections, thus leading one to consider expander graphs. Graph theory envisioned here applies also to materials research: the molecular structure of macromolecules such as the new fullerene molecule define what are known as Cayley graphs.
