I am interested in connecting the theory of combinatorial objects, such as matroids and stable trees, with their algebraic geometric counterparts, using bands and other generalization of rings.
A band scheme model for the moduli space of genus-0 stable curves
We unify various combinatorial objects associated with the moduli space of genus-0 stable curves with n marked points by introducing a band scheme model. We study its rational point sets over various bands. For instance, its tropical valued points coincide with extended tropical curves and its rational points over a field recover the usual moduli space of stable curves over the field. This project is advised by Professor Melody Chan.
Union of matroids over bands
We define the union of matroids over bands with certain additive structure, which unifies the existing definitions of ordinary, oriented, valuated matroids union and direct sum of linear spaces. Under mild assumptions, we show that the union is still a matroid over bands. As a corollary, we obtain a multiplicative structure for bimatroids over bands. Since valuated bimatroids can be viewed as a tropical analog of linear transformations, we hope to utilize bimatroid multiplication over bands to study tropical representation theory and its extension to bands. This project is in collaboration with Victoria Schleis at Durham University, UK.
My interest also includes quasi-symmetric functions and its application on scheduling problems, such as graph arboricity.
Directed arboricity
Directed arboricity describes signed edge-coloring in a directed graph (or an oriented matroid) that avoids monochromatic signed circuits. This modifies the existing notion of arboricity for an undirected graph. The number of valid signed edge-colorings forms a polynomial, despite not satisfying contraction-deletion recursion. This polynomial is a specialization of a type D quasisymmetric function, defined by viewing directed arboricity as a scheduling problem of type D. This project is advised by Professor Caroline Klivans.
Paper 'From Boxes to Polynomials: A Story of Generalization' (with G. Akhyar & L. Yuan), The Mathematical Intelligencer, Springer, Dec 2022 (link)
Some notes on hook-content formula, Integrality, and Elliptic dimension
Presentation and report on ‘KZ functor for rational Cherednik algebras’, AMSI Connect Conference, Australian Mathematical Sciences Institute, Feb 2022 (slides, poster and report)
Poster ‘Constructing minimum Euclidean skeletons for polygons with holes’, Vacation scholarship program, University of Melbourne, Feb 2021 (poster)