My broad area of interest is algebraic topology, and particularly applications of homotopy theory to geometry. My thesis work enumerates unstable vector bundles over projective spaces, using Weiss calculus -- a version of functor calculi which features the usage of cohomology theories, i.e., stable homotopy theory, to resolve unstable problems in geometry.
In recent joint work with P. Bhattacharya, we developed an equivariant Weiss calculus, which will facilitate tools from equivariant stable homotopy theory to study a broader range of problems in equivariant geometry, equivariant K-theory, and equivariant homological stability. My research interest also lies in chromatic homotopy theory, and in particular its applications in the study of vector bundles. In the future, I plan to develop a motivic version of Weiss calculus, which will impact the study of algebraic vector bundles in algebraic geometry.
Here is a Research Statement. (Last updated: Oct 25, 2024.)
Publications and preprints.
Metastable complex vector bundles over complex projective spaces.
Trans. Amer. Math. Soc. 376 (2023), 7783-7814. (Link.)
Abstract: We apply Weiss calculus to enumerate rank $r$ topological complex vector bundles over complex projective spaces $\mathbb{CP}^n$ with vanishing Chern classes when $r = n-1$ and $r = n-2$, which are the first two cases in the metastable range.
Enumerating stably trivial vector bundles with higher real K-theory. Joint with Hood Chatham and Morgan Opie.
Submitted. ArXiv preprint (2024), arXiv:2403.04733. (Link.)
Abstract: We combine Weiss calculus and EO-theory to obtain more enumeration results for unstable topological complex vector bundles over $\mathbb{CP}^n$ that are stably trivial. We prove an EO-detection result to obtain systematic lower bounds for the counts.
Equivariant Weiss calculus. Joint with Prasit Bhattacharya.
Submitted. ArXiv preprint (2024), arXiv:2410.12087. (Link.)
Abstract: We set up a framework of $G$-equivariant orthogonal and unitary calculus, where $G$ can be any finite group. This theoretical framework may have potential applications in equivariant geometry.
Work in progress.
Enumerating corank 2 complex vector bundles over complex projective spaces. Joint with Morgan Opie.
In preparation.
Abstract. We give a complete enumeration of rank $n$ topological complex vector bundles over $\mathbb{CP}^{n+2}$.
Orientation order and chromatic defect. Joint with Prasit Bhattacharya and Christian Carrick.
In preparation.
Abstract. Orientation order and chromatic defect are two competing ideas measuring how far a ring spectrum is from being complex-orientable. We explore how the two ideas are (un)related.