Notes

Outline

"The Hodge Conjecture" The Hodge Conjecture Daniel S. Freed University of Texas at Austin

Introduction

Themes

The Greek Geometers

The Greek Geometers

Warmup

Warmup

Real Projective Line

Introducing Coordinates

Real Projective Line RP1 (Summary)

Real Projective Plane RP2

Coordinate Description of RP2

More Shapes

Intersection of Curves

Some Algebra

Complex Numbers in Geometry

Complex Projective Plane (CP2)

Algebraic Cycles

Enumerative Geometry in CP2

Higher Dimensions and Smoothness

Arena for the Hodge Conjecture

Reprise

Enter Topology

Topological Cycles

Topological Cycles

Topological Cycles

Intersection of Topological Cycles

The Problem

Scaling and Dimension

Integration

Integration and Topological Cycles

Integration in Complex Geometry

Hodge’s Contribution

The Hodge Conjecture (1950)

Progress and Related Conjectures

Is the Hodge Conjecture True?

The Last Word