RUI: Moduli Spaces and Combinatorial Algebraic Geometry

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The investigator will study various combinatorial and enumerative problems in algebraic geometry. Fundamentally linked to these problems are the study of moduli and parameter spaces and their cohomology theories, the study of objects combinatorially rich in structure, and degeneration techniques and their applications. In particular, the investigator proposes to investigate equivariant quantum cohomology theories, affine Grassmannians, Hessenberg varieties, flag varieties, moduli spaces of curves, quot schemes, GKM spaces, and stacks and their toric degenerations.The proposed research problems are concerned with fruitful interactions between modern methods in algebraic geometry and new developments in combinatorics, symplectic geometry, and physics. The proposed projects have significant potential impact on enhancing the understanding of fundamental objects in algebraic geometry and their connections to one another. Through research, education, advising, and organizing, the investigator will promote interactions and collaborations between students and faculty of colleges and universities.