Delta Sets from Quivers 

A quiver is a multi-graph where additionally also loops are allowed. Quivers are important because they can be upgraded to finite categories (when adding an associative structure), because they appear historically: the first graph of Euler in the Koenigsberg Bridge problem was a quiver, they appear in physics as feynman graphs, they appear in chemistry if one wants to model multiple bonds , they appear in pop culture like the Good will hunting graph. The reason for me to use quivers is to get a simpler proof of the upper bound for ALL eigenvalues of a quiver. The reason is that the principal submatrix of the Kirchhoff matrix of a quiver is again the Kirchhoff matrix of a quiver, allowing induction. Also the lower bound for ALL eigenvalues given first by Brouwer and Haemers for graphs can be extended to quivers without multiple connections (also quivers without multiple connections have the property that princial submatrices of their Kirchhoff matrix is again a Kirchhoff matrix of a quiver without multiple connection), again allowing induction. But ahis is all old water under the bridge. This week I wondered whether quivers can produce higher dimensional delta sets similarly as graphs produce higher dimensional simplicial complexes. To get one dimensional delta sets is no problem because there is a gradient F and divergence F^T and the Kirkhoff matrix is of the form K = div grad = F^T F. This can be rephrased that there is a Dirac matrix defining a one dimensional delta set with vertices and edges as elements. Note that we treat loops as edges.
Pictures were taken on Friday morning (Saturday was again a bit rainy). In the intro shot, I was visited by the Harvard peregrine Falcon whom I have seen the third time already while flying around. It might be one who nests at Memorial Hall. See this Gazette article:
news.harvard.e...
(A home fit for a king). They are great to control the number of pigeons in urban areas.