Date	Topics	Comments
1/3	Motivation/Introduction	Abelian Gauss-Manin connection, Gauss's example, sketch nonabelian case, Betti and DeRham viewpoints
3/3	Review of differential geometry	References: Frank Warner, Foundations of Differentiable Manifolds and Lie Groups Libermann and Marle, Symplectic geometry and analytical mechanics
8/3	Lie group basics Some complex symplectic geometry	Left-invariant vector fields, exponential map, one-parameter subgroups, group actions, fundamental vector fields, conjugation, Adjoint action, adjoint action Darboux charts, Hamiltonian vector fields, induced Poisson structure
10/3	More symplectic geometry	cotangent bundles, coadjoint orbits and KKS theorem
15/3	More symplectic geometry	Moment maps, symplectic quotients
20/3	Bundles	Fiber bundles, vector bundles, clutching maps, constructions, vector bundles on the Riemann sphere, jumping lines
22/3	Connections on vector bundles	Definitions, covariant derivatives, local expressions, gauge transformations, curvature, flatness as condition for commuting covariant derivatives
24/3	More connections	"Nonabelian Poincare lemma", constructions, local system of solutions, monodromy
29/3	Basic Riemann-Hilbert correspondence	Flat holomorphic connections, Flat smooth connections, local systems, bundles with constant clutching maps, conjugacy classes of fundamental group representations. Connections on arbitrary fibre bundles (notions of flatness and completeness plus relation with constant clutching maps)
31/3	Logarithmic connections	Definitions, residues, restriction to divisor, Fuchsian systems, local non-resonant classification on curves.
5/4	Simple moduli spaces	stability, sufficient conditions for smoothness
7/4	Isomonodromic deformations	Local system of monodromy manifolds, De Rham interpretation, good trivialisations. Derivation of Schlesinger's equations
28/4	Schlesinger's equations	Time-dependent Hamiltonians, Malgrange's universality theorem
1/5	Sixth Painlevé equation	Some history, Painleve/Kowalevskaya property, First order equations with Painleve property, standard relation between PVI and Schlesinger equations
5/5	More PVI	Nonlinear monodromy, Fricke relation, cubic surfaces, mapping class and braid group actions Okamoto affine D4 and F4 Weyl group symmetries
30/5	Final Exam