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Generalized splines are an algebraic combinatorial framework that generalizes and unifies various established concepts across different fields, most notably the classical notion of splines and the topological notion of GKM theory. The former consists of piecewise polynomials on a combinatorial geometric object like a polytope, whose polynomial pieces agree to a specified degree of differentiability. The latter is a graph-theoretic construction of torus-equivariant cohomology that Shareshian and Wachs used to reformulate the well-known Stanley-Stembridge conjecture, a reformulation that was recently proven to hold by Brosnan and Chow and independently Guay-Paquet. This paper focuses on the theory of generalized splines. A generalized spline on a graph $$G$$ with each edge labeled by an ideal in a ring $$R$$ consists of a vertex-labeling by elements of $$R$$ so that the labels on adjacent vertices $u, v$ differ by an element of the ideal associated to the edge $uv$. We study the $$R$$-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: $1)$ for all graphs when the set of possible edge-labelings consists of at most two finitely-generated ideals, and $2)$ for cycles when the set of possible edge-labelings consists of principal ideals generated by elements of the form $(ax+by)^2$ in the polynomial ring $$\mathbb{C}[x,y]$$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in the theory of classical (analytic) splines. 

Hessenberg varieties H(X,H) form a class of subvarieties of the flag variety G/B, parameterized by an operator X and certain subspaces H of the Lie algebra of G. We identify several families of Hessenberg varieties in type A_{n−1} that are T -stable subvarieties of G/B, as well as families that are invariant under a subtorus K of T. In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Among the T-stable Hessenberg varieties, we identify several that are GKM spaces, meaning T acts with isolated fixed points and a finite number of one-dimensional orbits, though we also show that not all Hessenberg varieties with torus actions and finitely many fixed points are GKM. We conclude with a series of open questions about Hessenberg varieties, both in type A_{n−1} and in general Lie type.