Research

I started research with mostly interests in commutative algebra. In grad school, I worked on problems involving typically multiplicity theory and Rees algebras, as well as various problems associated to graded families of ideals, and their characteristic p cousins: p-families. Recently my research has shifted in two directions: (1) algebraic combinatorics and (2) algebraic geometry.

In the direction of (1) I have been interested in algebraic invariants and properties of ideals which exhibit combinatorial nature, such as Stanley-Reisner ideals and binomial edge ideals. Over recent decades, these ideals have built a series of bridges between combinatorial objects such as graphs, simplicial complexes, and matroids, to fundamental notions in commutative algebra such as regularity and Cohen-Macaulayness. The combinatorial aproaches to Cohen-Macaulay ideals and modules go back to Stanley and are still not very well understood even for quadric squarefree monomial ideals, i.e. (powers of) edge ideals of graphs. There is a general principle that some simple dictionary exists between ideals and combinatorial objects such as matroids or (simplicial) complexes, one of the most fundamental examples being the correspondance of a Stanley-Reisner ideal to it's respective complex. It is well-known that, from the combinatorial side, Cohen-Macaulayness and Shellability translate perfectly back to commutative algebra. Another example is Froberg's Theorem which states that the edge ideal of a graph G has a linear resolution if and only if the compliment of G is chordal. Much more has been conjectured in this direction but very little is known.

In the direction of (2), one common classical approach to multiplicity theory involves constructing suitable sheaves and divisors that model multiplicities in commmutative algebra (i.e. asymptotic nature of lengths of families of modules). In a recent paper with Sudipta Das and Vinh Pham we give another angle to this approach by combining the Borel-Nickel construction of a divisor with transcendental volume, and Baker's result on transcendental logarithms, to construct a transcendental multiplicity back in the setting of an honest ideal in commutative algebra. I believe that this approach has a lot more to offer in pushing multiplicity theory and the theory of other invariants in commutative algebra, which I have been exploring recently.