Bijective Combinatorics for Geometrical Structures

Brandeis University, Waltham MA

Investigators

Abstract

This project is in the field of combinatorics, which is a branch of mathematics concerned with the description and analysis of discrete data structures. Combinatorial analysis is a central tool to solve many problems in mathematics, and in related fields such as computer science and mathematical physics. This project includes several problems in which some seemingly complex structures seem to display some unexplained simple patterns. This suggest that these structures may have an alternative simpler description, a "bijection", explaining these patterns. Finding such a bijection can greatly simplify the mathematical analysis of the objects under consideration. Graduate students will be trained as part of this project. This projects aims at solving several important problems in mathematics using bijective techniques. The first problem is about unifying and extending a class of bijections between planar maps and lattice paths, in view of analyzing some statistical mechanics models on random lattices and relating them to Liouville quantum gravity and SLE curves. The second problem, motivated by graph theory, aims at finding the combinatorial structures underlying a mysterious enumerative identity for properly colored planar triangulations. The third project aims at using a generalization of Schnyder woods for planar graphs in order to design some graph-drawing algorithms. Other projects are about the combinatorics of hypergraphs, the encoding of faces in Coxeter hyperplane arrangements, and the enumeration of simplicial trees. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.