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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