Enumeration in Graded Posets and Polytopes

Cornell University, Ithaca NY

Investigators

Abstract

Research on this project involves the use of algebraic, geometric and combinatorial methods to attack fundamental questions involving enumeration in convex polytopes and certain graded partially ordered sets. Questions considered involve enumerative properties of general polytopes and arrangements of hyperplanes, properties of related families of partially ordered sets, and connections of all of these structures to fundamental algebraic structures involving permutations. Recent connections between subalgebras of quasisymmetric functions and enumerative properties of families of partially ordered sets have made it likely that significant progress can be achieved on these questions. A major focus is to move closer to a characterization of the numbers of faces and chains of faces in polytopes and arrangements. In addition, the investigator and his colleagues will continue their work on the study of the geometry of the space of phylogenetic trees. An initial goal will be to determine an efficient scheme to compute distances between trees in this space. The type of enumerative information being sought here is of use in the design of geometric algorithms for problems in robotics and motion planning, and recently has come into play in the analysis of randomization schemes for the management of data. The associated permutation questions relate to questions arising when one studies efficient schemes for sorting large amounts of data. The questions involving phylogenetic trees bear on the development of statistical methods to deal with trees derived from DNA data. Such methods are of great interest to biologists studying the genetic relationship between different species or different diseases.