Realizes the KLR algebra as affine sections of a coherent sheaf on a product of projective spaces. This geometrically explains the Elias–Qi \(\mathfrak{sl}_2\)-action, identifies its maximal finite-dimensional submodule via global sections, extends the action to the Witt algebra, and geometrizes the graded affine cellular structure.
Proves the expected asymptotic in Waring's problem over function fields, with a power-saving error, when the number of variables exceeds twice the degree under suitable characteristic and finite field size conditions. The proof uses an aggregate minor arc estimate, and the variable range is sharp for the expected asymptotic uniformly in the target polynomial.
Proves uniform exponential bounds for compactly supported Betti numbers of spaces of maps from fixed-genus curves to projective varieties. As an application, establishes a higher genus function field version of Manin's conjecture for split quartic del Pezzo surfaces.
Improves the number of variables required in Birch's theorem for smooth hypersurfaces over rational function fields from exponential in the degree to quadratic. The proof introduces multiplication rank to stratify the relevant exponential sums and combines it with a weighted degeneration of the Jacobian equations.
Shows there are no accumulating maps in the sense of geometric Manin's conjecture (i.e. pathological components of the moduli space of curves on a Fano variety) to general low degree hypersurfaces by adapting Sawin's approach to Waring's problem over function fields.
Proves that moduli spaces of smooth higher genus curves on smooth low degree hypersurfaces are essentially always singular (unless the hypersurface is a linear subspace, in which case we show the moduli spaces are always smooth).
Proves that Sawin's general multiple Dirichlet series over function fields are analytic and establishes a few functional equations. The former uses the decomposition theorem and general bounds for the cohomology groups of lisse sheaves on compactifications of configuration spaces, and the latter extends one of Sawin's examples employing a density trick with simple perverse sheaves.
Shows the moduli space of (arbitrary genus) curves on a smooth low-degree hypersurface is irreducible of the expected dimension by geometrically re-interpreting the Browning-Vishe circle method strategy. Applies this to obtain a converse of geometric Manin's conjecture.
Proves a sum-product bound for dual numbers and double numbers by extending Elekes's original strategy for complex numbers in a bootstrapping manner.