# Talbot 2011: Non-Abelian Hodge Theory

## Mentored by Carlos Simpson

### Notes

Marcy Robertson's notes: File 1 | File 2 | File 3

Chris Elliott's notes: File 1 | File 2 | File 3 | File 4 | File 5

There are three sets of notes available for the 2011 Talbot. There are notes by Chris Elliott and by Marcy Robertson. Below you will also find individual files of notes---this is the third set, but it is incomplete. Some of these notes were taken by Hiro Tanaka, then edited by the speakers.

## Talk Schedule

 Talk Description Team Captain Teammate(s) Notes by Monday---Harmonic bundles day 1 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. Carlos Simpson 2 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. Sushmita V Aaron Smith 3 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 Aaron Smith Joshua Batson 4 Principal objects: $G$-bundles, real groups, the Toledo invariant, Milnor-Woods inequalities Jesse Wolfson Harold Williams 5 Orbifolds, DM-stacks, parabolic bundles Orbifolds, Deligne-Mumford stacks, topological stacks, fundamental group, cohomology, Chern classes, relationship with parabolic bundles Jie Xia Daniel Halpern-Leistner Xia Tuesday---Moduli space day 6 Moduli of representations: Carlos Simpson 7 Moduli spaces of Higgs bundles: Hitchin fibration, parabolic Higgs bundles; spectral covers, cameral covers; Hitchin's Teichmuller component Harold Williams Andrew Kontaxis 8 Guest Talk: Affine cubic surfaces and relative $SL(2)$ - character varieties Domingo Toledo 9 Moduli spaces of representations and connections: Moduli of vector bundles with $\lambda$-connection, the Riemann-Hilbert correspondence, symplectic structure of the moduli space. Pietro Tortella Hendrik Orem Hendrik , Hiro 10 Homeomorphisms between the moduli spaces: The hyperkähler structure, the twistor space, Deligne's construction via $\lambda$-connections. Harmonic bundles give prefered sections of the twistor space Pavel Safronov Jacob Lurie Pavel 11 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 Alex Waldron Dario Beraldo Alex , Dario Wednesday---Topics day 12 Factorization theorems of representations and the Shafarevich conjecture, rigid representations as variations of Hodge structure Mike Skirvin Sushmita V 13 Goldman-Millson deformation theory, mixed Hodge structure on the local deformation space Hiro Tanaka Peter Dalakov Peter 14 Cohomology of moduli spaces: Hitchin's method, connected components of real representation spaces, the conjectures of Hausel David Jordan Marcy Robertson 15 Model categories, simplicial presheaves: Quillen model categories, simplicial sets, simplicial presheaves, application to stacks, homotopy types. Sheel Ganatra Marcy Robertson Thursday---Higher homotopy day C.S. Higher nonabelian Hodge theory Carlos Simpson 16 Mixed Hodge structures: Fundamental groups, relative Malcev completions, higher homotopy types in the simply connected case, local rings of representation spaces Peter Dalakov Saul Glasman, Tobias Barthel Peter 17 Non-Abelian Cohomology, I Jacob Lurie 18 Nonabelian cohomology, II: Higher geometric stacks, $X_DR$ and $X_Dol$, the Deligne-Hitchin glueing, Griffiths transversality, derived stacks Jacob Lurie Sam Gunningham 19 Schematic homotopy types: $S^1$-action, Galois actions, mixed Hodge structures Anthony Blanc Ting Chen, Tobias Barthel Anthony , Tobi Friday---Noncompact day 20 Local systems on noncompact varieties Carlos Simpson 21 The noncompact case: Starting with curves, statements in higher dimensions, twistor D-modules Chris Elliot Botong Wang Chris 22 The higher dimensional noncompact case: Takuro Mochizuki's theory Botong Wang Aaron Smith, Joshua Batson 23 Local systems with singularities on the projective line: Katz's middle convolution, wild singularities Toly Preygel Sam Gunningham 24 Discussion Everyone

MONDAY (Harmonic Bundles Day)
9-10 AM Harmonic Bundles
10:30-11:30 AM From Flat to Harmonic Bundles
12 - 1 PM From Higgs to Harmonic Bundles
1 PM - 5:30 PM Lunch Break, Dinner Prep
5:30 - 6:30 PM Principal Objects (G-bundles, Toledo invariant, etc)
6:30 PM - 8 PM Dinner
8 PM - 9 PM Orbifolds, DM Stacks.
9 PM - xx PM, Free Time

TUESDAY (Moduli Space Day)
8:30 - 9:15 AM Moduli of Representations
9:30-10:30 AM Moduli of Higgs Bundles
11 AM-12 PM Moduli of Representations and Connections
12 PM - 1:30 PM Lunch Break
1:30 PM - 2:30 PM Guest Talk: Domingo Toledo
2:30 PM - 3 PM Break
3 PM - 4 PM Homeomorphisms between Moduli Spaces
4 PM - 5 PM Dinner prep, Break.
5 PM - 6 PM Hodge Theory.
6 PM - 7 PM Cocktail Hour, Dinner Prep
7 PM - xx PM Dinner, Free Time

WEDNESDAY (Topics Day)
9 - 10 AM Factorization Theorems, Shafarevich Conjecture
10:30 - 11:30 AM Goldman-Millson and MHS on Local def. space
12 PM - 1 PM Cohomology of Moduli Spaces
1 PM - 7 PM Free Time, Hike.
7 PM - 8 PM Dinner
8 PM - 9 PM Model Categories
9 PM - xx PM, Free Time.

Thursday (Higher Homotopy Day)
9 - 10 AM Higher non-Abelian Hodge Theory
10:30 AM - 11:30 AM Mixed Hodge Structures
12 PM - 1 PM Derived Alg. Geometry
1 PM - 5:30 PM Break, Dinner Prep
5:30 PM - 6:30 PM non-abelian Cohomology
6:30 PM - 8:00 PM Dinner
8 - 9 PM Schematic Homotopy Types
9 PM - xx PM Free Time.

Friday (Noncompact Day)
9 - 10 AM Local systems on noncompact varieties
10:30 AM - 11:30 AM The noncompact case
12 PM - 1 PM Mochizuki's Theory
1 PM - 4 PM Break
4 PM - 5 PM Local systems with singularities on the projective line
5 PM - 7 PM Dinner
7 PM - 8 PM Discussion
8 PM - xx PM Free Time