Grant: Discovery Grants Program - Individual

My Discovery Grant Research Program studies problems in the fields of algebraic and enumerative combinatorics. The main objects of study or themes are two generalizations of the concepts of coloring and flows on graphs/networks. This proposal does not involves Indigenous communities, societies or individuals. Graph coloring is the problem of coloring the vertices of a graph with n colors so that adjacent vertices have distinct colors which models job scheduling. Stanley introduced a symmetric function, a multivariate polynomial that stays invariant under permuting variables, generalization of the chromatic polynomial that encodes graph coloring. We aim to study 30 year old conjectures of the positivity of these chromatic symmetric functions by Stanley, Stanley-Stembridge, and Shareshian-Wachs when expanded in certain basis of symmetric functions and new conjectures related to the "log-concavity" of the coefficients. The importance of advancing on these conjectures is that they indicate or use hidden structures from representation theory and geometry. A flow on a directed graph is the assignment of nonnegative integer values on the edges so that each vertex the incoming and outgoing flow is conserved which models transportation problems on a network and a special case includes the 5-flow conjecture of Tutte from 1954. If one allows edges to carry nonnegative real numbers then one obtains a convex polytope called a flow polytope of a graph whose lattice points correspond to the integer flows. These polytopes have rich connections in combinatorics and other areas like optimization and representation theory. Many questions remain about their volume, lattice points, and of an important way to subdivide them called triangulations. We aim to continue developing the theory of these flow polytopes and focus on answering open questions about their triangulations, the asymptotics of their number of lattice points and volume. The importance of these directions is that the "dual graph" of the triangulations can be used to realize geometrically important posets in the literature and bounds on their number of lattice points can help bound important quantities in representation theory. The anticipated outcomes for the firs theme are to bring recent advances on chromatic symmetric functions to the study of the more general Kazhdan--Lusztig elements of permutations, and showing that chromatic symmetric functions have the Lorentzian property of Brändén-Huh; Anari-Liu-Oveis Gharan-Vinzant. The outcomes of the second theme are to prove a conjectured lattice structure of the Danilov--Karzanov--Koshevoy triangulations of flow polytopes, find a rule to compute a metric invariant of such polytopes called the h* polynomial that refines the volume, and use Lorentzian polynomials to obtain sharp bounds on the number of lattice points. This program can easily integrate postdocs, graduate and undergraduate students at various stages and involve computer explorations.