1. (history and making of) varieties, manifolds, schemes, and ringed spaces

2. local homeomorphisms and surjective submersion as a generalisation of open covers, sheaf theory

3. descent, stack, Brylisnki-gerbe and the third cohomology class

4. Murray's bundle gerbe

1. introduction

   1. tensors

   2. exterior derivative, de Rham cohomology

   3. Riemannian volume form and integration

   4. Stokes theorem  (Poincare duality)

2. complex manifolds

   1. complex manifolds: equivalent definitions, cotangent space m/m^2 definition

   2. almost complex structure and equivalent condition of integrability

   3. real vs complex vs holomorphic - functions, vector field and differential forms

   4. Dolbeault cohomology

3. vector bundles and sheaves

   1. real/complex/holomorphic vector bundles, (hermitian) metrics, connections and curvatures

   2. Cauchy-Rieman operators/pseudo-holomorphic structures

   3. Chern classes

   4. sheaves and Cech cohomology

   5. line bundles and divisors

4. Kahler and Hermitian manifolds 

   1. harmonic theory

   2. Kahler identities

   3. Hodge decomposition

   4. Lefschetz theorems

   5. Ricci form, dd^c-lemma

5. Kodaira embedding and vanishing theorems, projective manifolds

6. Kodaira-Serre duality, related results

7. Calabi-Yau and Aubin-Yau theorems, related results

• Brylinski - Loop spaces, characteristic classes, and geometric quantisation

• Bunk - Gerbes in geometry, field theory, and quantisation

• Murray - (An introduction to) Bundle gerbes

• ncatlab.org

• stacks.math.columbia.edu/browse

• principles of algebraic geometry - Griffits, Harris

• Einstein manifolds - Besee

• complex geometry - Huybrechts

• lectures on Kahler geometry - Moroianu

• lectures on Kahler manifolds - Ballmann

• a survey of the hodge conjecture - Lewis

1. categories, functors, natural transformations

2. adjoints

3. Yoneda lemma

4. limits and adjoints

5. more limits and Kan extensions

6. outlook on abelian categories, derived categories and derived functors

1. flat connections

2. foliations

3. fundamental groups

4. universal covering spaces

5. classification of flat connections

6. sheaves and sheaf cohomology

• Categories for the Working Mathematician - Saunders Mac Lane

• Basic Category Theory - Tom Leinster

• Notes on Category Theory - Paolo Perrone

• Algebra I & II - Alexey L. Gorodentsev

• Taubes - Differential geometry

1. Overview

2. vector, principal and fibre bundles 

3. connections and characteristic classes 

4. introductory notions in complex and Kähler geometry, Lefschetz theorem on (1,1)-classes

5. G-structures

Homework:

• categories and functors, tensor products, fibration examples (torus/group invariant), partition of unity, induced metric on tensor bundles, bundle reductions,

• Principles of algebraic geometry - Joe Harris and Phillip Griffiths 

• Differential forms in algebraic topology - Loring W. Tu and Raoul Bott

• Differential Geometry: Connections, Curvature, and Characteristic Classes - Loring W. Tu

• Fibre Bundles - D. Husemöller

• The topology of fibre bundles - Norman Steenrod

• Complex Geometry: An Introduction -  Daniel Huybrechts

• Modern geometry: methods and applications II, III - B.A. Dubrovin, A.T. Fomenko and S.P. Novikov

• Hodge theory and complex algebraic geometry I, II - Claire Voisin

• Basic bundle theory and K-cohomology invariants - Husemöller, Joachim, Jurčo, Schottenloher

• Compact manifolds with special holonomy - Joyce

1. review of Lie group theory and symplectic geometry

2. Hamiltonian actions

3. syplectic reduction

4. Morse theory

5. Delzant correspondence

• Lectures on Symplectic Geometry - Ana Cannas da Silva

• Torus Actions on Symplectic Manifolds - Michèle Audin

• Introduction to Toric Varieties  - William Fulton

• Moment Maps and Combinatorial Invariants of Hamiltonian $\mathbb{T}^n$-spaces - Victor Guillemin

• Introduction to Symplectic Topology - Dusa McDuff and Dietmar Salamon

• Introduction to Smooth Manifolds - John M. Lee

• Convexity and commuting Hamiltonians - M.F. Atiyah

• Convexity properties of the moment mapping - V. Guillemin and S. Sternberg

• Hamiltoniens periodiques et images convexes de l'application moment - T. Delzant

• Kähler structures on toric varieties - V. Guillemin

• Kähler metrics on toric orbifols - Miguel Abreu

• Hamiltonian torus actions on symplectic orbifolds and toric varieties - Eugene Lerman and Susan Tolman