ARRA: Bruhut and balanced graphs, manifolds, partitions, and affine permutations

The PI is investigating and laying fundamental work in the areas of algebraic, geometric and topological combinatorics. The PI proposes five research projects in these areas. The first project is a continuation of the PI's successful research program of the cd-index of polytopes. Billera and Brenti showed the cd-index also exists for the strong Bruhat graph of Coxeter systems. The P1 has recently found a natural extension to balanced graphs and showed the Kazhdan-Lusztig polynomi- als, an important invariant in topology and representation theory, also extend to balanced graphs. The goal of the project is to settle conjectures about Kazhdan-Lusztig polynomials by working in the larger class of balanced graphs. The second project is to further develop the theory of flag f- vectors of regular subdivisions of manifolds. The overall objective is to consider fundamental results for polytopes and establish similar striking results for general manifolds. Project 3 is to extend enumerative results for the restricted partition lattice to topological and representation-theoretic settings. Project 4 concerns finding bijective proofs of determinants involving Dowling-Stirling numbers. These two last projects are work-in-progress with the P1's graduate student Ji Yoon Jung. Project 5, together with the PI's graduate student Eric Clark, is to extend permutations statistics to the group of affine permutations. Finally, the PI proposes to start the Combinatorial Coffee Club of Lexington which will meet three times a year. This will initiate an informal set- ting for graduate students and faculty to discuss research problems in geometric, topological and algebraic combinatorics. Algebraic combinatorics is the study of discrete structures occurring in the areas of algebra, geometry and topology. The PI's proposed research will simplify the currently difficult area of Kazhdan-Lusztig theory, will extend the breadth of combinatorial techniques for polytopes to more general manifolds, and will discover basic properties of permutations and posets. This research will build upon the P1's ability to develop sophisticated techniques and simplifications to tackle problems in algebraic combinatorics. Results and theoretical framework developed in this proposal and the broader field of algebraic combinatorics will potentially have many applications, including to phylogenetic trees in biology, coding theory in computer science, and techniques in optimization. The PI will continue to present his results at national and international conferences, to inspire graduate students (the PI was a keynote speaker at the first Graduate Student Combinatorics Conference), and to give talks at local colleges and high schools. The P1 directs undergraduate research projects, develops new undergraduate and graduate courses and most recently, organizes the Math Movie of the Month to educate the community about math and attract new majors. One of the PI's current doctoral students is female. This continues the University of Kentucky's tradition of graduating large numbers of female Ph.D.'s. which is only surpassed by Stanford.