Current projects

Circle squaring: In 1990, Laczkovich showed that a ball and a cube of the same volume are equidecomposable, answering an old question of Tarski's. This has since been improved to show that the pieces in the equidecomposition can be Lebesgue measurable, and even Borel. I am currently working on studying these equidecompositions through so-called toast structures. The goal is to optimize the Borel complexity and the Minkowski dimension of the boundary of the pieces.
Sparse polynomial factorization: If a polynomial in n variables is sparse (has few monomials), must its factors also be sparse? The best-known results for the sparsity bound involve a dependence on n, the number of variables, when the conjecture is that it should only depend on the sparsity of the original polynomial. I am working on improving this bound for cofactors of multilinear polynomials.

Papers

Cardinal, Knauer, Micek, Pálvölgyi, Ueckerdt, Varadarajan, Colouring bottomless rectangles and arborescences, Computational Geometry 115< (2023), https://doi.org/10.1016/j.comgeo.2023.102020.
Dillon, Varadarajan, Explicit bounds for the layer number of the grid, 2023, arXiv:2302.04244.
Pattern problems related to the Arithmetic Kakeya conjecture (preprint), 2020.

Orthogonal projections for quantum channels and operator systems Developments in Computer Science, Budapest, 2021
Colouring bottomless rectangles and arborescences, 36th European Workshop on Computational Geometry, 2020 (program page)