Combinatorics and commutative algebra of algebraic varieties with group actions

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Algebraic geometry is a central area of mathematics. It has importance not just in pure and applied mathematics, but also in the natural sciences, engineering, and beyond (eg. economics). At its core, algebraic geometry is the study of common zeros of collections of polynomials in multiple variables. A general theme in the field is to translate geometric questions about these zero-sets, or algebraic varieties, into equivalent algebraic questions. Examples of such geometric questions include "does this zero-set have multiple components?" and "are there any singularities?". ******The proposed research will address these types of geometric questions for important classes of algebraic varieties with many symmetries. For such algebraic varieties, it is often possible to further translate certain algebraic questions into combinatorics (eg. counting problems, or problems about discrete structures). Using and developing combinatorial tools to study algebro-geometric problems is the subject of combinatorial commutative algebra, the particular area of mathematics in which the proposal fits. ******Motivated by past successes of multiple mathematicians, I will use methods from combinatorial commutative algebra to study algebro-geometric properties of three classes of algebraic varieties carrying groups of symmetries: quiver loci of Dynkin quivers, Schubert varieties and related varieties, and certain Hilbert schemes. These varieties are important in pure mathematics, and some have found applications in other fields. For example, Schubert varieties are significant in both algebraic geometry and representation theory, and have applications in computer graphics and statistics; in recent joint work with Alex Fink and Seth Sullivant, we used properties of Schubert varieties to study conditional independence in algebraic statistics.******Results will be of interest to mathematicians, and will contribute to the literature on these important algebraic varieties. In certain instances, results will connect seemingly different mathematical objects, or communities of researchers studying different topics (eg. along the lines of my past joint works connecting type A quiver loci and Schubert varieties, and using Schubert varieties to study conditional independence). Results in particular directions will yield new insights into important open problems. Finally, the proposed research contains many projects suitable for students at all levels, and so the research program will have further impact through the training of highly qualified personnel.