Advanced Topics In Geometry

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Spec and Proj

• Introduction
• Spec
• Proj
• Solution to exercises

Blow ups

• Introduction
• First Example: Blowing up C^2 at the origin
• Blowing up an affine variety along an ideal
• Blowing up a complex manifold
• Key examples
• The general definition
• Solution to exercises

Complex manifolds and the Kähler condition

• Almost Complex structures
• Integrability
• The Kähler Condition
• Exercises
• Solutions to exercises

Chern Classes

• Introduction and Basic Definitions
• Bundles on spheres
• Chern classes as degeneracy loci of generic sections
• Suggested Reading
• Additional exercises
• Solutions to exercises

Classifying spaces, equivariant cohomology and localisation

• Classifying spaces
• Equivariant cohomology
• Localisation
• Solutions to exercises

Symplectic reduction

• Basic Symplectic Geometry
• Hamiltonian group actions and moment maps
• Symplectic reduction

Geometric Invariant Theory

• Introduction
• Reductive groups
• Introduction and first examples
• Quotients
• GIT in the projective setting
• Solutions to exercises

Symplectic toric varieties

• Introduction
• Symplectic toric manifolds from polytopes
• Another way to get toric symplectic manifolds
• Where is the fan?
• Toric blow-up
• Solution to exercises

Lefschetz pencils in algebraic and symplectic geometry

• Lefschetz hyperplane theorem
• Local study of ordinary double points
• Monodromy
• Hard Lefschetz Theorem

Moduli Spaces and Stable Bundles

• Introduction
• Brief aside on schemes
• Back to Moduli Spaces
• Moduli Space of Bundles

Mixed Hodge Structures

• Introduction
• Linear Algebra and Hodge Theory
• Mixed Hodge Structures
• The open smooth case
• Applications

The Weil conjectures

• Introduction
• The Weil conjectures
• Introduction
• Transfer
• Counting Points
• Zeta functions
• The Weil Conjectures

Atiyah-Singer Index theorem

• K-theory
• Fredholm Operators
• The Atiyah-Singer Index formula (INCOMPLETE)

Tropical Curves

• Introduction
• Amoebas of Curves
• The Graph of an Amoeba
• Corner Loci and Tropical Curves
• The Tropical Semiring
• Weights and the Balancing Condition
• Degree and the Number of Tentacles
• Newton Polygons
• Puiseux Series

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