Algebraic Geometry and Number Theory

Algebraic geometry studies the shapes defined by polynomial equations, and when combined with number theory, it connects the geometry of these shapes to deep questions about integer and rational solutions. Moduli theory organizes families of geometric objects—curves, surfaces, maps—into spaces that can themselves be studied, while tools like Gromov-Witten invariants, intersection theory, and motivic cohomology provide ways to extract numerical and structural information from those spaces. Tropical geometry, which replaces smooth curves with piecewise-linear combinatorial shadows, has opened unexpected bridges between classical geometry and discrete mathematics. Active directions include understanding the birational classification of higher-dimensional varieties through minimal model theory, resolving singularities in a controlled way, and using geometric structures to constrain the rational points on varieties—a path toward some of the hardest unsolved problems in Diophantine equations.

Works

115,665

Total citations

950,106

Keywords

Moduli TheoryGromov-Witten TheoryLog Canonical SingularitiesMotivic CohomologyMinimal ModelsTropical Geometry