Hodge theory is a rich subject at the intersection of algebraic, complex and differential geometry. Although it has numerous applications in fields as diverse as combinatorics, number theory, representation theory and mathematical physics, its biggest impact is undoubtedly felt in algebraic geometry, where it has been one of the standard tools for a very long time. The purpose of these lectures is to give a bird's eye view of Hodge theory, explain what it is all about, and demonstrate some of its more serious algebro-geometric consequences.
Tuesdays and Thursdays 11am - 12pm
Room 162, Peter Hall Building
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By request
Tuesday 30 July: Introduction
Thursday 1 August: Sheaves (Recording)
Tuesday 6 August: Cohomology and GAGA (Recording)
Thursday 8 August: Harmonic forms and the Hodge Theorem (Recording)
Tuesday 13 August: Kähler geometry and the Hodge decomposition (Recording)
Thursday 15 August: The Hodge-de Rham spectral sequence (Recording)
Tuesday 20 August: Deformations of Calabi-Yau varieties (Recording)
Thursday 22 August: The Kodaira Vanishing Theorem (Recording)
Tuesday 27 August: Positivity and the Hard Lefschetz Theorem (Recording)
Thursday 29 August: Mixed Hodge structures and the yoga of weights (Recording)
Tuesday 3 September: Variations of Hodge structure (Recording)
Thursday 5 September: Nearby and vanishing cycles (Recording)
Tuesday 10 September: More on nearby and vanishing cycles (Recording)
Thursday 12 September: Mixed Hodge modules (Recording)
The fundamentals of algebraic geometry, from an analytic perspective:
Phillip Griffiths and Joe Harris, Principles of Algebraic Geometry
The fundamentals of algebraic geometry, from an algebraic perspective:
Robin Hartshorne, Algebraic Geometry
David Eisenbud and Joe Harris, The Geometry of Schemes
(for the very brave) Alexandre Grothendieck and Jean Dieudonné, Éléments de Géometrie Algébrique
The "motivic" perspective on Hodge theory, Hodge theory beyond the smooth projective setting: