In this talk, Jennifer Chayes, Distinguished Scientist and Managing Director of Microsoft Research New England and New York City, looks quite generally at some of the models we are using to describe networks, processes we are studying on the networks, algorithms we have devised for the networks, and finally, methods we are developing to indirectly infer network structure from measured data. In particular, she discusses models and techniques that cut across many disciplinary boundaries.
This is intended to be a survey talk, accessible to a general mathematical audience. The cdh topology was created by Voevodsky to extend motivic cohomology from smooth varieties to singular varieties, assuming resolution of singularities (for example complex varieties). The slogan is that every variety is locally smooth in this topology. Adapting this to algebraic K-theory has led to several breakthroughs in computation, and clarified connections between the "singular" part of K-theory and Hodge theory.
In this talk we will present an overview of the hypermatrix generalization of matrix algebra proposed by Mesner and Bhattacharya in 1990. We will discuss a spectral theorem for hypermatrices deduced from this algebra as well as connections with other tensor spectral decompositions. Finally if time permits we will discuss some applications and related open problems.