Nilpotent Hessenberg varieties are a family of subvarieties of the flag variety, which include the Springer varieties, the Peterson variety, and the whole flag variety. In this thesis I give a geometric proof that the cohomology of the flag variety surjects onto the cohomology of the Peterson variety; I provide a combinatorial criterion for determing the singular loci of a large family of regular nilpotent Hessenberg varieties; and I describe the equivariant cohomology of any regular nilpotent Hessenberg variety whose cohomology is generated by its degree two classes.

We study Hessenberg varieties in type A via their local defining equations, called patch ideals. We focus on two main classes of Hessenberg varieties: those associated to a regular nilpotent operator and to those associated to a semisimple operator. In the setting of regular semisimple Hessenberg varieties, which are known to be smooth and irreducible, we determine that their patch ideals are triangular complete intersections, as defined by Da Silva and Harada. For semisimple Hessenberg varieties, we give a partial positive answer to a conjecture of Insko and Precup that a given family of set-theoretic local defining ideals are radical. A regular nilpotent Hessenberg Schubert cell is the intersection of a Schubert cell with a regular nilpotent Hessenberg variety. Following the work of the author with Da Silva, Harada, and Rajchgot, we construct an embedding of the regular nilpotent Hessenberg Schubert cells into the coordinate chart of the regular nilpotent Hessenberg variety corresponding to the longest-word permutation in Bruhat order. This allows us to use work of Da Silva and Harada to conclude that regular nilpotent Hessenberg Schubert cells are also local triangular complete intersections. / Thesis / Master of Science (MSc) / Algebraic varieties provide a generalization of curves in the plane, such as parabolas and ellipses. One such family of these varieties are called Hessenberg varieties, and they are known to have connections to other areas of pure and applied mathematics, including to numerical linear algebra, combinatorics, and geometric representation theory. In this thesis, we view Hessenberg varieties as a collection of subvarieties, called coordinate charts, and study the computational geometry of each coordinate chart. Although this is a local approach, we recover global geometric data on Hessenberg varieties. We also provide a partial positive answer to an open question in the area.

The theory of Newton-Okounkov bodies can be viewed as a generalization of the theory of toric varieties; it associates a convex body to an arbitrary variety (equipped with auxiliary data). Although initial steps have been taken for formulating geometric situations under which the Newton-Okounkov body is a rational polytope, there is much that is still unknown. In particular, very few concrete and explicit examples have been computed thus far. In this thesis, we explicitly compute Newton-Okounkov bodies of some cases of Bott-Samelson and Peterson varieties (for certain classes of auxiliary data on these varieties). Both of these varieties arise, for instance, in the geometric study of representation theory. Background on the theory of Newton-Okounkov bodies and the geometry of flag and Grassmannian varieties is provided, and well as background on Bott-Samelson varieties, Hessenberg varieties, and Peterson varieties. In the last chapter we also discuss how certain techniques developed in this thesis can be generalized. In particular, a generalization of the flat family of Hessenberg varieties constructed in Chapter 6, which may allow us to compute Newton-Okounkov bodies of more general Peterson varieties, is an ongoing collaboration with H. Abe and M. Harada. / Thesis / Doctor of Philosophy (PhD)