The three dimensional space is important because we live in it. With the scalar Laplacian in dimension q=3, the Hydrogen operator (leaving out constants) essentially explains the periodic system of elements and so the starting point of chemistry. The eigenvalue difference explain spectral lines (like Lyman (UV light) , Balmer …
Before we start, here is the code which generated the two matrices on the board of the talk. 5 lines for the Hodge Laplacian, 1 line for the Connection Laplacian. This works for any simplicial complex. Not only in physics, also in mathematics, there is a fundamental distinction between Fermionic …
Theorem: if a graph G=(V,E) with n=|V| vertices and m=|E| edges and maximal vertex degree d satisfies , then G satisfies the Brouwer conjecture . This adds confidence that the conjecture is true. As mentioned in the video (and now having worked on it for 3 weeks and wrapping up), …
A discrete geometry does not have a lot of symmetry as the automorphism group is in general empty. The isospectral set of the Laplacian or Dirac matrix is large enough however. Note that when dealing with a specific class of operators like Dirac matrices, then not all isospectral matrices qualify. …
While linear cohomology deals with functions on simplices, quadratic cohomology deals with functions on pairs of simplices that intersect. Linear cohomology is to Euler characteristic what quadratic cohomology is to Wu characteristic $w(G) = \sum_{x,y, x \cap y \in G} w(x) w(y)$. If the simplicial complex is split into a …
Quivers are graph in which multiple connections and loops are allowed. Since there is a Dirac operator d+d* with exterior derivative for them, they define a one-dimensional delta set (G,D,r), where G is the union of vertices and edges (loops count as edges) and r is the dimension function which …
We explain why the arboricity of 3 spheres can take values between 4 and 7 and mention that for 3 manifolds the upper bound is 9 (but believed to be 7).
We prove that the spectrum of the Hodge Laplacian dd* +d*d depends in a monotone way on the simplicial complex.
