# Talbot 2011: Non-Abelian Hodge Theory

## Mentored by Carlos Simpson

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May 1-7, 2011

Salt Lake City, Utah

### Notes

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 | |

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 |

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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 |

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 | |

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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 |

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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