Research
Research
My research interests chiefly comprise matroids, a vast combinatorial abstraction of hyperplane arrangements. Matroids unify the seemingly disparate ideas of linear independence of vectors, algebraic independence of field elements, and acyclicity of subgraphs. As time progressed, deeper connections with algebraic, tropical, and toric geometry were found.
One of my recent interests is to explore the valuativity of matroid invariants. Roughly speaking, a matroid can be identified with a certain convex polytope, and valuative invariants are those functions that are well-behaved with respect to the subdivisions of this polytope. In their 2010 paper Valuative Invariants of Polymatroids, Derksen and Fink studied such invariants. Amongst several significant contributions, they established the class of Schubert matroids as the main object of interest in this context. My candidacy exam report briefly surveys these developments. Recently, Eur, Huh and Larson identified the group of valuative invariants with the cohomology of the stellahedral variety, adding a geometric interpretation for these functions!
I am recently considering invariants that admit values beyond abelian groups and different ways to reformulate the notion of valuativity in this context.
Recently, I have become interested in topological combinatorics. I am exploring (jointly with K.S. Shukla and P. Deshpande) various topological aspects of the higher independence complexes of grid graphs.