Topological and Algebraic Combinatorics

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ABSTRACTPrincipal Investigator: Braun, Benjamin J.Proposal Number: DMS - 0758321Institution: University of Kentucky Research FoundationTitle: Topological and Algebraic CombinatoricsThe PI investigates problems in topological and algebraic combinatorics. The theory of Ehrhart polynomials is of broad interest to the mathematics community due to connections with commutative algebra, algebraic geometry, combinatorics, discrete and convex geometry, and number theory. The PI studies roots and coefficients of Ehrhart polynomials, with particular focus on reflexive polytopes. The PI also investigates problems regarding graph and poset homomorphism complexes, continuing the application of algebraic topology to combinatorics. The PI studies possible homotopy test graphs and investigates connections between poset homomorphism complexes and poset order dimension. Finally, in joint work with Richard Ehrenborg, the PI studies simplicial subcomplexes of the boundary complexes of associahedra arising from triangulations of non-convex polygons.Mathematics has historically been driven by the interplay between discrete and continuous structures. Contemporary problems arising from the interaction of combinatorics, algebra, and topology continue this tradition. The study of polytopes began in antiquity, with roots in the solid geometry of Euclid. Ehrhart theory is a contemporary approach to studying polytopes from a combinatorial and algbraic perspective, producing from a polytope with integer vertices a polynomial counting lattice points in integral dilates of that polytope. The roots and coefficients of these polynomials are known to carry some combinatorial and geometric data, but there are many open questions about exactly how far this line of investigation can be taken. The study of graphs began in the 1700's with investigations by Euler, and has found modern day applications in most areas of science and engineering. Studying chromatic numbers of graphs by moving to the continuous world leads to investigations of topological spaces with symmetries arising from symmetries of the graphs under consideration. Investigations in this direction have been successful so far, and many open questions remain.