Discrete Ricci Flow in Higher Dimensions.

摘要

The objective of this research to develop an efficient and justifiable algorithm to geometrize a given closed 3-manifold, and to show how its topologic characterization can be applied to complex networks. Hamilton's Ricci flow (RF) was developed in order to geometrize such 3-manifolds. The geometrization theorem (GT) states that each prime 3-manifold is either geometric or its simple pieces are geometric. The continuum approach is not numerically practical. Accordingly, we developed a discrete piecewise linear (PL) version of Hamilton's RF. It is the first dimensionally agnostic generalization of RF for PL geometries. We refer to our approach as simplicial Ricci flow (SRF). For a broad class of examples, the SRF equations reproduced the continuum RF. SRF provides an efficient approach to the 3-manifold recognition problem. Student S. Ray applied SRF to implement the 1916 Weyl isometric embedding problem. His results are being used to develop a discrete quasi-local measure of congestion in networks -- a possible filtration parameter to guide network reconfiguration.