# Workshop Programme

## for period 5 - 14 January 2011

### School on Moduli Spaces

5 - 14 January 2011

Timetable

Wednesday 5 January | ||||

09:00-09:55 | Registration | |||

09:55-10:00 | Welcome from Sir David Wallace (INI Director) | |||

10:00-11:00 | Hoffmann, N (Freie Universität Berlin) |
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Augmented bundles - I | Sem 1 | |||

The notion of augmented bundle comprises various extra structures on vector bundles. The lectures deal with examples like quasiparabolic bundles, coherent systems, framed bundles, Higgs bundles, and holomorphic triples, all on a compact Riemann surface. I'll explain their coarse moduli spaces and discuss the role of the stability parameters. |
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11:00-11:30 | Morning coffee | |||

11:30-12:30 | Gómez, T (Universidad Complutense de Madrid) |
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Principal bundles - I | Sem 1 | |||

This is an introduction to principal bundles in algebraic geometry. We will start with the basic definitions (associated fibrations, reduction and extension of structure group, stability, etc...) and then give a sketch of the construction of the moduli Notes: The following are the notes of some lectures I gave in Warsaw. In that school I gave 5 lectures, so this time I won't be able to cover all this material, but this could serve as a reference: T. Gomez, Lectures on principal bundles over projective varieties, Pragacz, P. (Ed.) Algebraic cycles, sheaves, shtukas, and moduli. Impanga lecture notes. Trends in Mathematics. Birkh\"auser Verlag, Basel, 2008. viii+236 pp., 45--68. Lectures on Principal Bundles over proyective varieties. arXiv: 1011.1801v1 |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Doran, B (ETH Zürich) |
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Geometric Invariant Theory - I | Sem 1 | |||

15:00-15:30 | Afternoon tea | |||

15:30-16:30 | Nitsure, N (Tata Institute of Fundamental Research) |
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Deformation Theory : an introduction to the algebraic approach - I | Sem 1 | |||

This series of three lectures aims to give an introduction to the algebraic approach to deformation theory, as developed by Grothendieck, Schlessinger, Illusie and Artin, illustrated with a few key examples. The topics will include (i) infinitesimal theory over Artin local rings including Schlessinger's theorem, tangent-obstruction theories and use of cotangent complex, (ii) algebraization via the Grothendieck existence theorem for formal schemes and the Artin approximation theorem, and (iii) construction of global moduli as quotient algebraic spaces. |
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16:45-17:45 | Tutorials | |||

17:45-18:45 | Welcome wine reception | |||

18:45-19:30 | Dinner at Wolfson Court (residents only) |

Thursday 6 January | ||||

10:00-11:00 | Hoffmann, N (Freie Universität Berlin) |
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Augmented bundles - II | Sem 1 | |||

The notion of augmented bundle comprises various extra structures on vector bundles. The lectures deal with examples like quasiparabolic bundles, coherent systems, framed bundles, Higgs bundles, and holomorphic triples, all on a compact Riemann surface. I'll explain their coarse moduli spaces and discuss the role of the stability parameters. |
||||

11:00-11:30 | Morning coffee | |||

11:30-12:30 | Gómez, T (Universidad Complutense de Madrid) |
|||

Principal bundles - II | Sem 1 | |||

This is an introduction to principal bundles in algebraic geometry. We will start with the basic definitions (associated fibrations, reduction and extension of structure group, stability, etc...) and then give a sketch of the construction of the moduli Notes: The following are the notes of some lectures I gave in Warsaw. In that school I gave 5 lectures, so this time I won't be able to cover all this material, but this could serve as a reference: T. Gomez, Lectures on principal bundles over projective varieties, Pragacz, P. (Ed.) Algebraic cycles, sheaves, shtukas, and moduli. Impanga lecture notes. Trends in Mathematics. Birkh\"auser Verlag, Basel, 2008. viii+236 pp., 45--68. Lectures on Principal Bundles over proyective varieties. arXiv: 1011.1801v1 |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Doran, B (ETH Zürich) |
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Geometric Invariant Theory - II | Sem 1 | |||

15:00-15:30 | Afternoon tea | |||

15:30-16:30 | Nitsure, N (Tata Institute of Fundamental Research) |
|||

Deformation Theory : an introduction to the algebraic approach - II | Sem 1 | |||

This series of three lectures aims to give an introduction to the algebraic approach to deformation theory, as developed by Grothendieck, Schlessinger, Illusie and Artin, illustrated with a few key examples. The topics will include (i) infinitesimal theory over Artin local rings including Schlessinger's theorem, tangent-obstruction theories and use of cotangent complex, (ii) algebraization via the Grothendieck existence theorem for formal schemes and the Artin approximation theorem, and (iii) construction of global moduli as quotient algebraic spaces. |
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16:45-17:45 | Tutorials | |||

18:45-19:30 | Dinner at Wolfson Court (residents only) |

Friday 7 January | ||||

10:00-11:00 | Gothen, P (Universidade do Porto) |
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Representations of surface groups and Higgs bundles - I | Sem 1 | |||

Classical Hodge theory uses harmonic forms as preferred representatives of cohomology classes. A representation of the fundamental group of a Riemann surface gives rise to a corresponding flat bundle. A Theorem of Donaldson and Corlette shows how to find a harmonic metric in this bundle. A flat bundle corresponds to class in first non-abelian cohomology and the Theorem can be viewed as an analogue of the classical representation of de Rham cohomology classes by harmonic forms. |
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11:00-11:30 | Morning coffee | |||

11:30-12:30 | Diaconescu, E (Rutgers) |
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Moduli spaces and physics | Sem 1 | |||

12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Doran, B (ETH Zürich) |
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Geometric Invariant Theory - III | Sem 1 | |||

15:00-15:30 | Afternoon tea | |||

15:30-16:30 | Nitsure, N (Tata Institute of Fundamental Research) |
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Deformation Theory : an introduction to the algebraic approach - III | Sem 1 | |||

This series of three lectures aims to give an introduction to the algebraic approach to deformation theory, as developed by Grothendieck, Schlessinger, Illusie and Artin, illustrated with a few key examples. The topics will include (i) infinitesimal theory over Artin local rings including Schlessinger's theorem, tangent-obstruction theories and use of cotangent complex, (ii) algebraization via the Grothendieck existence theorem for formal schemes and the Artin approximation theorem, and (iii) construction of global moduli as quotient algebraic spaces. |
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16:45-17:45 | Tutorials | |||

18:45-19:30 | Dinner at Wolfson Court (residents only) |

Monday 10 January | ||||

10:00-11:00 | Doran, B (ETH Zürich) |
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Geometric Invariant Theory - IV | CMS MR2 | |||

11:00-11:30 | Morning coffee - served at INI | |||

11:30-12:30 | Gothen, P (Universidade do Porto) |
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Representations of surface groups and Higgs bundles - II | CMS MR2 | |||

A Higgs bundle on a Riemann surface is a pair consisting of a holomorphic bundle and a holomorphic one-form, the Higgs field, with values in a certain associated vector bundle. A theorem of Hitchin and Simpson says that a stable Higgs bundle admits a metric satisfying Hitchin's equations. Together with the Theorem of Corlette and Donaldson, the Hitchin-Kobayashi correspondence generalizes the classical Hodge decomposition of the first cohomology of the Riemann surface, providing a correspondence between isomorphism classes of Higgs bundles and representations of the fundamental group of the surface. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Huybrechts, D (Bonn) |
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Introduction to derived categories and stability conditions - I | CMS MR2 | |||

The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of t-structures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and Kontsevich-Soibelman) and discuss examples. The relevant literature is accessible on the arxiv. |
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15:00-15:30 | Afternoon tea - served at INI | |||

15:30-16:30 | Alexeev, V (Georgia) |
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Introduction to moduli of varieties - I: Generalities on moduli spaces of varieties | CMS MR2 | |||

The yoga of moduli spaces. Main methods of construction. Compactifications and Minimal Model Program. Stable pairs (slc & ample K+B). Honest work vs "accidental" examples. |
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16:45-17:45 | Tutorials | |||

18:45-19:30 | Dinner at Wolfson Court (residents only) |

Tuesday 11 January | ||||

10:00-11:00 | Behrend, K (British Columbia) |
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Introduction to stacks - I | CMS MR2 | |||

Introduction to algebraic stacks as they are relevant to moduli theory. |
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11:00-11:30 | Morning coffee - served at INI | |||

11:30-12:30 | Gothen, P (Universidade do Porto) |
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Representations of surface groups and Higgs bundles - III | CMS MR2 | |||

In the final lecture, we give some applications of Higgs bundle theory to the study of the geometry and topology of character varieties for surface groups, via the identification with moduli of Higgs bundles. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Huybrechts, D (Bonn) |
|||

Introduction to derived categories and stability conditions - II | CMS MR2 | |||

The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of t-structures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and Kontsevich-Soibelman) and discuss examples. The relevant literature is accessible on the arxiv. |
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15:00-15:30 | Afternoon tea - served at INI | |||

15:30-16:30 | Alexeev, V (Georgia) |
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Introduction to moduli of varieties - II: Stable toric and semiabelic varieties | CMS MR2 | |||

Moduli of stable toric varieties vs toric Hilbert scheme. Convex tilings and secondary polytopes. Degenerations of abelian varieties. Semiabelian and semiabelic varieties. Compactifications of moduli spaces of abelian varieties. Refs: http://arxiv.org/abs/math/9905103, http://arxiv.org/abs/math/0207272, http://arxiv.org/abs/math/0207274. |
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16:45-17:45 | Tutorials | |||

18:45-19:30 | Dinner at Wolfson Court (residents only) |

Wednesday 12 January | ||||

10:00-11:00 | Behrend, K (British Columbia) |
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Introduction to stacks - II | CMS MR2 | |||

Introduction to algebraic stacks as they are relevant to moduli theory. |
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11:00-11:30 | Morning coffee - served at INI | |||

11:30-12:30 | Reineke, M (Bergische Universität Wuppertal) |
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Ringel-Hall algebras and applications to moduli - I | CMS MR2 | |||

Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry. The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representation-theoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields. We will develop several such identities and discuss more recent applications to wall-crossing formulae. Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wall-crossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147. |
||||

12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Huybrechts, D (Bonn) |
|||

Introduction to derived categories and stability conditions - III | CMS MR2 | |||

The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of t-structures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and Kontsevich-Soibelman) and discuss examples. The relevant literature is accessible on the arxiv. |
||||

15:00-15:30 | Afternoon tea - served at INI | |||

15:30-16:30 | Alexeev, V (Georgia) |
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Introduction to moduli of varieties - III: Weighted hyperplane arrangements | CMS MR2 | |||

$\bar M_{0,n}$. Hassett's weighted stable curves. Compact moduli of weighted hyperplane arrangements. Refs: http://arxiv.org/abs/math/0205009, http://arxiv.org/abs/math/0501227, http://arxiv.org/abs/0806.0881 |
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16:45-17:45 | Tutorials | |||

19:30-22:00 | Conference dinner at Emmanuel College |

Thursday 13 January | ||||

10:00-11:00 | Behrend, K (British Columbia) |
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Introduction to stacks - III | CMS MR2 | |||

Introduction to algebraic stacks as they are relevant to moduli theory. |
||||

11:00-11:30 | Morning coffee - served at INI | |||

11:30-12:30 | Reineke, M (Bergische Universität Wuppertal) |
|||

Ringel-Hall algebras and applications to moduli - II | CMS MR2 | |||

Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry. The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representation-theoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields. We will develop several such identities and discuss more recent applications to wall-crossing formulae. Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wall-crossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147. |
||||

12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Huybrechts, D (Bonn) |
|||

Introduction to derived categories and stability conditions - IV | CMS MR2 | |||

15:00-15:30 | Afternoon tea - served at INI | |||

15:30-16:30 | Alexeev, V (Georgia) |
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Introduction to moduli of varieties - IV: Surfaces of general type related to abelian varieties and hyperplane arrangements | CMS MR2 | |||

Compact moduli of surfaces of general type derived from (1) abelian varieties, (2) line arrangements. Campedelli, Burniat, Kulikov surfaces. Refs: http://arxiv.org/abs/math/9905103, http://arxiv.org/abs/0901.4431 |
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16:45-17:45 | Tutorials | |||

18:45-19:30 | Dinner at Wolfson Court (residents only) |

Friday 14 January | ||||

10:00-11:00 | Behrend, K (British Columbia) |
|||

Introduction to stacks - IV | CMS MR2 | |||

Introduction to algebraic stacks as they are relevant to moduli theory. |
||||

11:00-11:30 | Morning coffee - served at INI | |||

11:30-12:30 | Reineke, M (Bergische Universität Wuppertal) |
|||

Ringel-Hall algebras and applications to moduli - III | CMS MR2 | |||

Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry. The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representation-theoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields. We will develop several such identities and discuss more recent applications to wall-crossing formulae. Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wall-crossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147. |
||||

12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Huybrechts, D (Bonn) |
|||

Introduction to derived categories and stability conditions - V | CMS MR2 | |||

15:00-15:30 | Afternoon tea - served at INI | |||

15:30-16:30 | Alexeev, V (Georgia) |
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Introduction to moduli of varieties - V: Surfaces of general type: the general case | CMS MR2 | |||

Several versions of the moduli functor. Problems, and some solutions. Steps in the construction of the moduli space. Ref: many older papers, http://arxiv.org/abs/0805.0576, http://arxiv.org/abs/1008.0621 (by Kollár) |
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16:45-17:45 | Tutorials | |||

18:45-19:30 | Dinner at Wolfson Court (residents only) |