Algebraic combinatorics: symmetric orbit closures and Schubert calculus

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The goal of this project is to construct combinatorial models. Combinatorial problems arise in many areas of mathematics and have applications that include optimization, computer science, and statistical physics. The central focus of this research is the study of polynomial equations that appear at the interface of the study of discrete mathematics, geometry, and symmetries. A key component of this project will be the training of students (high school, undergraduate, and graduate) and postdoctoral faculty. Thereby, we wish to help strengthen the STEM education and research infrastructure in the United States.Earlier work of the investigator has led to precise connections of equivariant cohomology with spectra of Hermitian matrices, combinatorial commutative algebra with Kazhdan-Lusztig polynomials, combinatorial K-theory with longest increasing subsequences in random words, and partition combinatorics with bibliometric indicators. Such examples motivate finding broader and more unified combinatorial laws, which is the goal of this project. This research will both deepen our understanding of such connections and help discover novel relationships.