3-Manifolds After Perelman: Topology, Geometry, and Effective Rigidity

Jeff Brock · 59:52

Building on the Ricci flow, this lecture delves into the profound implications of Perelman's work on 3-manifolds, connecting geometric analysis to the classification and rigidity of low-dimensional topological spaces.

The Great Wild Manifold Rodeo: Dennis Sullivan in Algebraic Topology

Michael J. Hopkins · 48:13

This lecture transitions to algebraic topology, surveying Dennis Sullivan's contributions to understanding manifolds through homotopy theory, thereby expanding the toolkit for studying topological invariants.

The Hodge Conjecture: Geometry and Analysis

Burt Totaro · 51:25

This lecture introduces the Hodge Conjecture, a central problem at the intersection of algebraic geometry and complex analysis, building on the geometric and topological foundations to explore deeper algebraic structures.

Hodge Theory of p-adic Varieties: Faltings' Legacy

Wiesława Nizioł · 48:14

This lecture advances into p-adic Hodge theory, extending the concepts of Hodge theory to arithmetic geometry and exploring the relationship between étale cohomology and differential forms over p-adic fields.

Matroids and the Integral Hodge Conjecture for Abelian Varieties

Philip Engel · 1:06:12

This final lecture explores the integral Hodge conjecture for abelian varieties, demonstrating its falsity in certain cases and showcasing cutting-edge research at the frontier of algebraic geometry and number theory.