Introduction to Schemes: A Modern Primer
Schemes generalize algebraic varieties by attaching structure sheaves to topological spaces. This guide covers affine schemes, localization, and the spectrum of a ring.
Elliptic Curves and the Modularity Theorem
Exploring the deep connection between elliptic curves over Q and modular forms, culminating in Wiles' proof of Fermat's Last Theorem.
Sheaf Cohomology Explained
From Čech cohomology to derived functors: how sheaf cohomology bridges local data with global geometric properties in algebraic geometry.
Intersection Theory and Chow Rings
How to rigorously count intersections of algebraic cycles. Covers flat degenerations, excess intersection, and the structure of Chow groups.
Moduli Spaces and Stability Conditions
Constructing parameter spaces for algebraic objects. Explores GIT quotients, slope stability, and moduli of vector bundles.
Modern Developments in Derived Algebraic Geometry
Introducing dg-schemes and spectral algebraic geometry. How derived structures resolve intersection multiplicities and deformation problems.
Birational Geometry and the Minimal Model Program
Classifying algebraic varieties up to birational equivalence. Covers Mori theory, flips, flops, and the classification of Fano varieties.
Hodge Decomposition and Algebraic Cycles
How Hodge theory provides powerful constraints on the cohomology of projective varieties. The Hodge conjecture and its implications.