An algebraic variety is a geometric object defined as the zero set of a system of polynomial equations. A broad goal of birational algebraic geometry is to classify these objects via certain intrinsic invariants akin to topological genus (aka number of holes). This is done through a partly conjectural process called the log minimal model program, which contracts away anomalous curves to produce a more uniform variety, which can then be classified directly.
I am particularly interested in developing this program for varieties of interest to arithmetic geometry, namely those over finite fields and the integers. A major part of my work to date has been proving the existence of the log minimal model program for varieties of dimension three in these situations. The main difficulty in working with arithmetic varieties is that cohomology vanishing theorems fail, which are indispensable in characteristic zero. New techniques are needed to get around this, and the search for these often takes me into new areas of commutative algebra, arithmetic geometry and derived algebraic geometry.
My research is supported by NSF Grant #2401279 and Simons Foundation Gift #850684. .
Perfectoid pure singularities, with Bhatt, Ma, Patakfalvi, Schwede, Tucker and Witaszek.
Test ideals in mixed characteristic: a unified theory up to perturbation, with Bhatt, Ma, Patakfalvi, Schwede, Tucker and Witaszek.
Purely inseparable Galois theory I: The fundamental theorem with Lukas Brantner.
Connectedness principle in characteristic p>5 with Stefano Filipazzi. Michigan Math. J., 74 (2024) 675-701
Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic with Bhatt, Ma, Patakfalvi, Schwede, Tucker and Witaszek. Publ. Math. IHES, 138 (2023) 69-227
On the log minimal model program for 3-folds over imperfect fields of characteristic p>5 with Omprokash Das. J. Lond. Math. Soc. 106 (2022) 3895-3937
An analog of adjoint ideals and PLT singularities in mixed characteristic with Ma, Schwede, Tucker and Witaszek. J. Algebraic Geom., 31 (2022) 497-559
Singularities of general fibres and the LMMP with Zsolt Patakfalvi. Amer. J. Math. 144(2) (2022) 505-540
Structure of geometrically non-reduced varieties with Lena Ji. Trans. Amer. Math. Soc. 374 (2021) 8333-8363 (2021)
On the abundance problem for 3-folds in characteristic p>5 with Omprokash Das. Math. Z. 292(3-4) (2019) 937-946
The LMMP for log canonical 3-folds in char p in Nagoya Math. J. 230 (2018) 48-71
Finite generation of the log canonical ring for 3-folds in char p in Math. Res. Lett. 24(3) (2017) 933-946
Existence of Mori fibre spaces for 3-folds in char p with Caucher Birkar. Adv. Math. 313 (2017) 62-101