Statement of David Galvin , July 31 , 2012

نویسنده

David Galvin

چکیده

The main thrust of my research is the use of techniques from information theory, probability and combinatorics to study structural, enumerative and algorithmic aspects of graph homomorphisms and related models, both in particular instances (independent sets and colorings, for example) and in general. Graph homomorphisms are important objects in graph theory, where they generalize a number of central and apparently unconnected notions, and allow for their common study. In statistical physics, they provide a natural language for the study of an important class of models, the hard-constraint spin systems. Graph homomorphisms also arise naturally in the study of communication networks. In what follows I will explain the notions just mentioned, and then describe my most important results. Broadly, the main questions I answer fall into three classes: • Structural: describing the typical appearance of a randomly chosen homomorphism, a question which relates directly to the important ones of Gibbs measures in statistical physics models and spatial unfairness in communication networks. • Algorithmic: studying how efficiently one may sample from the space of homomorphisms, a significant question from a theoretical computer science perspective. • Enumerative: counting or asymptotically estimating the number of elements in a space of homomorphisms, a source of many fascinating problems and a burgeoning research area in discrete mathematics. As will be seen in what follows, these questions are intimately interrelated.

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