This will be an introduction to how the graph (of a real symmetric matrix, or a general matrix) constrains the multiplicities of its eigenvalues. The case of trees is most interesting and this will be described in some detail, including the maximum multiplicity, the minimum number of distinct eigenvalues, the possible lists of multiplicities and how they come about. This talk will be an overview of the subject and a second talk in the GAG seminar will continue the description. The subject has just been covered in a new book from Cambridge University Press (same title), and REU students have helped to make some very important contributions over the years. There is still plenty of work to be done in the area, which has been of interest to algebraic graph theorists and numerical analysts, as well as matrix theorists.
We consider bootstrap percolation in tilings of the plane by regular polygons. First, we determine the percolation threshold for each of the infinite Archimedean lattices.
More generally, let T denote the set of plane tilings t by regular polygons such that if t contains one instance of a vertex type, then t contains infinitely many instances of that type. We show that no tiling in T has threshold 4 or more.
This material is self-contained, and requires no particular background. We'll share many open problems, as well as the intuition behind these results.
In the Hilbert space formulation, quantum states are density matrices, i.e., positive
semidefinite matrices with trace one, and quantum channels are trace preserving completely positive linear maps on matrices. In this talk, we will present some results on the existence of quantum channels that send certain quantum states to other quantum states. Additional conditioons on the quantum channels may be imposed to satisfy certain