Chern-Schwartz-MacPherson classes of graph hypersurfaces and Schubert varieties

This dissertation finds some partial results in support of two positivity conjectures regarding the Chern-Schwartz-MacPherson (CSM) classes of graph hypersurfaces (conjectured by Aluffi and Marcolli) and Schubert varieties (conjectured by Aluffi and Mihalcea). Direct calculations of some of these CSM classes are performed. Formulas for CSM classes of families of both graph hypersurfaces and coefficients of Schubert varieties are developed. Additionally, the positivity of the CSM class of certain families of these varieties is proven.;The first chapter starts with an overview and introduction to the material along with some of the background material needed to understand this dissertation.;In the second chapter, a series of equivalences of graph hypersurfaces that are useful for reducing the number of cases that must be calculated are developed. A table of CSM classes for all graph hypersurfaces corresponding to graphs with 6 or fewer edges are explicitly computed. This table also contains Fulton Chern classes and Milnor classes for the graph hypersurfaces. Using the equivalences and a series of formulas from [AM09a], a new series of formulas for the CSM classes of certain families of graph hypersurfaces are deduced. I prove positivity for all graph hypersurfaces corresponding to graphs with first Betti number of 3 or less. Formulas for graphs equivalent to graphs with 6 or fewer edges are developed (as well as cones over graphs with 6 or fewer edges).;In the third chapter, CSM classes of Schubert varieties are discussed. It is conjectured by Aluffi and Mihalcea [AM09b] that all Chern classes of Schubert varieties are represented by effective cycles. This is proven in special cases by B. Jones [Jon09]. I examine some positivity results by analyzing and applying combinatorial methods to a formula by Aluffi and Mihalcea. Positivity of what could be considered the "typical" case for low codimensional coefficients is found. Some other general results for positivity of certain coefficients of Schubert varieties are found. This technique establishes positivity for some known cases very quickly, such as the codimension 1 case as described by Jones, as well as establishing positivity for codimension 2 and families of cases that were previously unknown. An unexpected connection between one family of cases and a second order PDE is also found. Positivity is shown for all cases of codimensions 1-4 and some higher codimensions are discussed.;In both the graph hypersurfaces and Schubert varieties, all calculated Chern-Schwartz-MacPherson classes were found to be positive.