Topological and Algebraic Combinatorics

The PI will investigate several problems in topological and algebraic comlnnatorics. He will continue his study of roots and coefficients of Ehrhart polynomials. I)[I.ilding on his previous results determining regions of the plane containing Ehrhart polyncJlto.ial roots and showing relationships among coefficients of Ehrhart polynomials of reflexive p(,lytopes. The theory of Ehrhart polynomials is of broad interest to the mathematics CI nmnunity due to connections with commutative algebra. algebraic geometry. combinatoric.; discrete and convex geometry. and number theory. The PI will also investigate problems r'~arding graph and poset homomorphism complexes. He will continue work on his conjel"t lfe on graph homomorphism complexes for stable Kneser graphs that would generalize result, of L. Lovasz and E. Babson-D. Kozlov. He will investigate connections between the topology of poset homomorphism complexes and poset order dimension. including his conjectu I!' generalizing a result of V. Reiner-V. \\'elker. The general study of hOlnomorphism compJe'Ces continues the application of algebraic topology to combinatorics. encouraging the CI )Uaboration of researchers in these disciplines. He will also undertake joint \vork with R. Ehrenborg on a conjecture regarding the homotopy type of "associahedron" -like complex1'~ arising from non-convex polygons. The PI will undertake this research as a faculty member in the discrete' mathematics research group at the University of Kentucky. The PI will seek out OppOi t mities to direct undergraduate research projects: Ehrhart theory in particular offers l!limy problems regarding quasipolynomials. sub-polytopes of the Birkhoff polytopes. roots of Ehrhart polynomials. and other topics that \vould be excellent for undergraduate research The PI will also seek out opportunities to mentor graduate students interested in discT,'te mathematics: he requests funds to provide summer support for such students. The P I will continue attending both research and teaching focused conferences such as the ann\l;.\ A:-'IS/:-'IAA joint meetings. A:-'IS and :-'IAA sectional meetings, and others.