Geometric properties coded in the long-time asymptotics for the heat equation on 𝑍ⁿ

This paper investigates connections between the long-time asymptotics of heat distribution on a body Ω Ω in Z n Z^n , and various geometric properties of Ω Ω , starting from an initially constant heat distribution supported on Ω Ω . We use combinatorial and differential geometric methods. We begin the paper with a result in R n R^n .