Project Grant 2601624

The National Science Foundation Division of Mathematical Sciences awarded the University of Alabama $179,974 on August 1, 2026, under the Mathematical and Physical Sciences program (CFDA 47.049) to develop new algebraic and geometric tools for studying singular points in symmetric objects across three interconnected research components. The first component will develop algebraic complexes to compute the cohomology of toric vector bundles and prove sharp vanishing theorems, with applications to defining equations and syzygies of projective toric varieties. The second will apply Hodge-theoretic and representation-theoretic methods to compute local cohomology modules supported in Schubert varieties and characterize higher Du Bois and higher rational singularities. The third will develop a generalized homology theory for tautological systems of differential equations attached to reductive group actions. The theoretical advances will be implemented in Macaulay2, an open-source software system for algebra and geometry, to make the tools broadly accessible to researchers and students. The project includes training for undergraduate, graduate, and high school students in mathematical research. Work is performed in Tuscaloosa, Alabama, with a period of performance from August 1, 2026, through July 31, 2029. The award is classified as a Project Grant.