The 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.
Mathematics has historically been driven by the interplay between discrete and continuous structures. Contemporary research arising from the interaction of combinatorics, algebra, topology, and geometry, such as the research supported by this award, continue this tradition. Projects involving the interaction of these areas are of interest to a variety of mathematicians and scientists. Topological and geometric interpretations of enumerative and algebraic problems are of great interest, in part due to the insights into combinatorial and algebraic structures that topological and geometric perspectives can offer. An example of this is a project supported by this award examining the geometric structure of MacMahon’s Partition Analysis, a one-hundred-year old computational tool developed by Percy MacMahon for studying power series identities that has seen a resurgence of use in the past fifteen years. Another example of this is a project supported by this award that studies the use of a type of space, called homomorphism complexes, to understand the algebraic structure of certain families of polynomials, called non-nesting ideals. A final example is a project supported through this award that uses another type of space, called a neighborhood complex, to study the structure of stable Kneser graphs, a special class of networks. A key component of mathematical activity is building mathematical community. This award has supported graduate and undergraduate student research, including student participation in conferences emphasizing research, teaching, and outreach. Through the support of graduate students, this award contributed to the development of mathematicians with an understanding of the complexities and depth of mathematical research. Through the support of undergraduate students, this award brought young mathematicians into contact with the boundaries of mathematical research, giving them a depth of mathematical experience transcending coursework; all the undergraduate researchers supported by this proposal plan to pursue teaching at the high-school or college level. This proposal supported the broader mathematical community, through travel support allowing the Principal Investigator to participate in conference organization.
