Research

A generalized spectral correspondenceWe establish a strong categorical correspondence between isomorphism classes of sheaves of arbitrary rank on one algebraic curve and twisted pairs on another algebraic curve. In a particular application, we realize a generic elliptic curve as a spectral cover of the complex projective line $\mathbb{P}^1$ and then construct examples of semistable co-Higgs bundles over $\mathbb{P}^1$ as pushforwards of locally-free sheaves of certain small ranks over the elliptic curve. By appealing to a composite push-pull projection formula, we conjecture an iterated version of the spectral correspondence. We prove this conjecture for a particular class of spectral covers of $\mathbb {P}^1$. The proof relies upon a classification of Galois groups into primitive and imprimitive types. In this context, we revisit a century-old theorem of J.F. Ritt.

Very stable and wobbly loci for elliptic curvesWe explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus $1$. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We utilize semistability of trivially twisted very stable bundles to prove that the wobbly locus is always a divisor in the moduli space of semistable bundles on a genus $1$ curve. We prove, by extension, a conjecture regarding the closedness and dimension of the wobbly locus in this setting. This conjecture was originally formulated by Drinfeld in higher genus.

Research Interest: My current research interest lies in algebraic geometry of moduli spaces of sheaves and twisted Higgs sheaves. Right now I am exploring twisted very stable Higgs sheaves and holomorphic chains on curves under the guidance of Dr. Steven Rayan. You can find my recent research articles here.

A generalized spectral correspondence co-written with Steven Rayan (2023) is available online.
My PhD Thesis (2020 - 2024) supervised by Dr. Steven Rayan is available online.
Very stable and wobbly loci for elliptic curves co-written with Steven Rayan (2024) is available online.
Co-Higgs bundles of Schwarzenberger type and the determinant morphism (2025).

I presented talks and wrote introductory write ups during as a graduate student. Here are some.

Basic Algebraic Geometry (Riemann surfaces).pdf

Galois presentation.pdf

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