The plan of this project is to study toy models of quantum mechanics associated to finite graphs. These models allow for an understanding of part of the mathematical structure of quantum mechanics, without the analytical complications that arise in the study of infinite dimensional Hilbert spaces. The project will focus on computations and examples, and on the path integral approach to the discrete version of Selberg’s trace formula [1]. A good introduction to thes models is [3]. For much more on spectral graph theory see [2].
[1] P. Mnev, Discrete path integral approach to the trace formula for regular graphs, arXiv,org:0609028v1.
[2] F. Chung, Spectral graph theory,CBMS Regional Conference Series in Math-
ematics, No. 92, AMS.
[3] http://qchu.wordpress.com/2011/01/02/the-schrodinger-equation-on-a-finite-graph/
The study of homotopy invariant alebraic structures was introduced by Stasheff [4] in the 1960’ motivated by the study of the algebraic structure present on the based loop space. This subject developed into a rich field which is closely related to topology, combinatorics, mathematical physics and representation theory. The purpose of this project is to provide an introduction to some aspects of the theory of homotopy associative algebras and Stasheff polytopes. The original papers of Stasheff are [4]. An excellent introduction to $A_\infty$-algebras is [5]. For the Stasheff polytope, [6] is recommended.
[4] J. Stasheff, Homotopy associativity of H spaces I,II, Trans. Amer. Math. Soc. 108 (1963).
[5] B. Keller Introduction to A∞-algebras and modules, arXiv:math/9910179.
[6] J.L. Loday Realization of the Stasheff polytope, arXiv:math/0212126.