Topological, Enumerative, and Algebraic Combinatorics

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PI John Shareshian studies problems in combinatorics that arise in or have consequences for other fields of mathematics. Combinatorics is the study of discrete, typically finite, mathematical structures. Such structures arise often in mathematics and the natural and social sciences, leading to various applications. For example, networks of various types are modeled by combinatorialists as graphs, which are simply collections of points, some pairs of which are considered to be related. Despite the simplicity of such models, mathematicians have derived many deep and applicable theorems about them. Perhaps surprisingly, there are close connections between combinatorics and other fields of mathematics in which non-discrete objects are studied, including topology and geometry. The work of PI Shareshian involves the close study of such connections, with the aim of solving problems about both discrete and non-discrete structures.PI Shareshian studies connections between combinatorics and other fields of mathematics, including algebra, topology, and geometry. In joint work with Michelle Wachs, he aims to prove that the graded Frobenius characteristic of a refined version of Stanley's chromatic symmetric function for a unit interval graph is in fact the cohomology of an associated regular semisimple Hessenberg variety, and to use such a result to attack a longstanding conjecture of Stanley and Stembridge about such symmetric functions. Shareshian studies connections between the structure algebraic objects and the combinatorial structure of lattices naturally associated to such objects. For example, he aims to provide a result on Lie algebras roughly analogous to his earlier result that a finite group is solvable if and only if the order complex of its subgroup lattice is shellable.