# 2011

## Talk Summaries

Monday---Harmonic bundles day

C.S.: Harmonic bundles: How they give flat bundles and Higgs bundles, de Rham cohomology, complex of flat bundles, Dolbeault cohomology complex of Higgs bundles, semistability, laplacians, Kahler identities. Deformation theory of everything.

(1) From flat bundles to harmonic bundles: the space $GL(n)/U(n)$, equivariant harmonic maps, pluriharmonic maps and the Bochner formulae. Start towards the noncompact case.

(2) From Higgs bundles to harmonic bundles: Yang-Mills-Higgs theory, from a gauge theory viewpoint and from a Hermite-Einstein viewpoint. Nonlinear heat flow, Donaldson's functional, $L^2_1$ subsheaves, convergence

(3) Principal objects: $G$-bundles, real groups, the Toledo invariant, Milnor-Woods inequalities

(4) Orbifolds, DM-stacks, parabolic bundles Orbifolds, Deligne-Mumford stacks, topological stacks, fundamental group, cohomology, Chern classes, relationship with parabolic bundles

Tuesday---Moduli space day

C.S.: Moduli of representations:

(5) Moduli spaces of Higgs bundles: Hitchin fibration, parabolic Higgs bundles; spectral covers, cameral covers; Hitchin's Teichmuller component

(6) Moduli spaces of representations and connections: Moduli of vector bundles with $\lambda$-connection, the Riemann-Hilbert correspondence, symplectic structure of the moduli space.

(7) Homeomorphisms between the moduli spaces: The hyperk\"ahler structure, the twistor space, Deligne's construction via $\lambda$-connections. Harmonic bundles give prefered sections of the twistor space

(8) Hodge theory, Variation of Hodge Structures, Mixed Hodge Structures: Hodge theory, Kahler manifolds, Kahler identities, families of varieties, variations of Hodge structure, mixed Hodge theory

Wednesday---Topics day

(9)Factorization theorems of representations and the Shafarevich conjecture, rigid representations as variations of Hodge structure

(10) Goldman-Millson deformation theory, mixed Hodge structure on the local deformation space

(11) Cohomology of moduli spaces: Hitchin's method, connected components of real representation spaces, the conjectures of Hausel

(12) Model categories, simplicial presheaves: Quillen model categories, simplicial sets, simplicial presheaves, application to stacks, homotopy types.

Thursday---Higher homotopy day

C.S.: Higher nonabelian Hodge theory

(13) Mixed Hodge structures: Fundamental groups, relative Malcev completions, higher homotopy types in the simply connected case, local rings of representation spaces

(14) Nonabelian cohomology: Higher geometric stacks, $X_DR$ and $X_Dol$, the Deligne-Hitchin glueing, Griffiths transversality, derived stacks

(15) Schematic homotopy types: $S^1$-action, Galois actions, mixed Hodge structures

Friday---Noncompact day

C.S.: Local systems on noncompact varieties

(16) The noncompact case: Starting with curves, statements in higher dimensions, twistor D-modules

(17) The higher dimensional noncompact case: Takuro Mochizuki's theory

(18) Local systems with singularities on the projective line: Katz's middle convolution, wild singularities

C.S.: Discussion