Last week, when looking up Henry McKean, I saw that he passed away on April 20, 2024. I don’t recall having met him in person but I have seen some of his work and met some of this students. Eugene Trubowitz was my Analysis 3 (complex analysis) and Analysis 4 …
A 2-manifold with boundary is a finite simple graph for which every unit sphere is a circular graph with 4 or more nodes or a path graph with 3 more nodes. The boundary is the set of vertices for which the unit sphere is a path graph, the interior is …
This is a presentation from Saturday, July 13, 2024. Curvatures are usually located on the zero dimensional part of space. I look here at curvature located on one or two dimensional parts of space. In the special case of a triangulation of a 2-dimensional surface, where the usual curvature is …
The smallest open sets in a finite topological space form the atom of space. It was almost 100 years ago, when one has turned away from non-Hausdorff topological spaces and decided they are less relevant (Hausdorff seems have convinced Alexandrov and Hopf to focus on Hausdorff property). This is unfortunate …
Here is some code illustrating the story. We take the 4 manifold (a favorite manifold of Heinz Hopf) and consider two random functions f,g. Now generate the two manifolds and . They are both 2 manifolds. It goes as follows: the sign data of {f,g} are in which are 4 …
In 1973 Leon Glass proved a discrete Poincare Hopf theorem for directed graphs embedded in a 2 dimensional manifold. Kate Perkins has related this to a discrete Poincare Hopf theorem of mine. This is a discussion of the connection.
We prove that the spectrum of the Hodge Laplacian dd* +d*d depends in a monotone way on the simplicial complex.
A finite abstraact simplicial complex or a finite simple graph comes with a natural finite topological space. Some quantities like the Euler characteristic or the higher Wu characteristics are all topological invariants. One can also reformulate the Lefschetz fixed point theorem for continuous maps on finite topological spaces.
