Anton Geraschenko's notes are posted in this directory.
Valery Alexeev (Georgia): Compact moduli of higher-dimensional varieties
The most famous geometrically meaningful compactifications of moduli spaces of varieties are Deligne-Mumford's moduli spaces of stable curves, and their generalizations to the relative case by Kontsevich and to the weighted case by Hassett. The Minimal Model Program offers a blueprint for extending the above to the case of higher dimensions. The lectures will concentrate on several recent examples where the program can be brought to fruition:
(1) varieties with group action (toric, abelian, spherical)
(2) hyperplane arrangements
(3) surfaces
Connections with the tropical geometry will be explored as well. The ideal prerequisites would be the familiarity with
(a) the one-dimensional case, e.g. Harris-Morrison's Moduli of Curves, Hassett's paper, another paper.
(b) toric varieties, most importantly -- the interplay between projective toric varieties and polytopes, see e.g. Fulton's Introduction to Toric Varieties or Oda's Convex Bodies and Algebraic Geometry.
(c) the singularities of pairs (X,B) appearing in the Minimal Model Program, such as log canonical and dlt, e.g. from Birational Geometry of Algebraic Varieties by Kollár and Mori.
Anton Geraschenko's notes are posted here
Kai Behrend (UBC): Foundations of Donaldson-Thomas theory
This will be an introduction to the basics of Donaldson-Thomas theory. Of particular importance in this theory is a certain constructible function on the moduli space. We will explain where this function comes from, and how it is used for calculations of invariants.
Topics covered will include: Symmetric obstruction theories, the virtual fundamental class, weighted Euler characteristics, the constructible function underlying Donaldson-Thomas theory, the equivariant case. As examples of applications, we will cover the Hilbert scheme of points on a Calabi-Yau threefold and the case of low degree curves on the quintic.
The material of these lectures will be taken from the three papers:
http://xxx.lanl.gov/abs/math/0507523
http://xxx.lanl.gov/abs/math/0512556
http://xxx.lanl.gov/abs/math/0601203
Anton Geraschenko's notes are posted here
Tom Bridgeland (Sheffield): Stability in triangulated categories
These five lectures will introduce the space of stability conditions on a triangulated category, give some geometric examples and touch on some of the recent work on wall-crossing behaviour in this context.
A rough plan (which may well change) is
(A) Brief recall of derived and triangulated categories. T-structures and tilting. Examples (e.g. P1).
(B) Stability in abelian and triangulated categories. Spaces of stability conditions. Examples (e.g. elliptic curve, K3 surface).
(C) Hall algebras. Reieneke's formula. Stokes factors and wall-crossing.
The main prerequisites apart from standard algebraic geometry will be a certain familiarity with derived categories, and the basic definitions concerning representations of quivers. The following two papers are useful introductions to derived categories:
Bernhard Keller `Derived categories and tilting' (available from his webpage)
Richard Thomas `Derived categories for the working mathematician' (on the arxiv)
Alternatively, consult the first chapter of
Robin Hartshorne `Residues and duality'
or the textbook
Gelfand and Manin `Methods of homological algebra'
The first section (pages 3-8) of William Crawley-Boevey's 'Lectures on representations of quivers' will give the necessary background on those. Also to be found on page 99 of Benson's book Representations and Cohomology I.
Anton Geraschenko's notes are posted here
Alessio Corti (Imperial): New methods in orbifold Gromov-Witten theory
I give an introduction to Gromov-Witten theory of Deligne-Mumford stacks and methods of calculation for toric stacks. The course will cover some of my recent work with Coates, Iritani, Li and Tseng. The focus will be on calculations with examples rather than complete proofs. My relationship with stacks is romantic rather than strictly professional. The course will have a list of problems for the student which is now at the bottom of my teaching page at:
http://www.ma.ic.ac.uk/~acorti/teaching.html
Desirable prerequisites:
(1) Basic knowledge of toric varieties, e.g. Chapter I of Fulton's "Introduction to toric varieties";
(2) Some familiarity with Gromov-Witten theory of manifolds, e.g. from "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande.
The course will cover some of the following topics:
1. Toric stacks, enhanced Picard group, Mori cone, the secondary fan.
2. GW invariants of stacks, simple examples.
3 GW theory of weak Fano toric stacks, the I-function and the mirror transformation, small quantum cohomology, simplest examples.
4. The B-model: Hori-Vafa, oscillating integrals, semi-infinite variation of Hodge structures, quantum cohomology and birational geometry, Givental's Lagrangian cone.
5. Representable toric morphisms from toric orbi-curves to toric stacks and recursion relations.
Anton Geraschenko's notes are posted here
Martin Olsson (Berkeley): Logarithmic structures with a view towards moduli.
In this mini-course, I have two main aims:
(1) to discuss the basic theory of logarithmic geometry in the sense of Fontaine, Illusie, and Kato, and
(2) illustrate through a number of examples the utility of log structures in the study of moduli spaces. The tentative plan for the lectures is the following.
-- Log structures and log schemes, charts, differentials.
-- Log smooth and log étale morphisms. Kato's structure theorem. Log de Rham cohomology.
-- Log deformation theory. Log cotangent complex.
-- Examples: the Deligne-Mumford compactification of Mg as a moduli space for log curves, local moduli for degenerating K3 surfaces, toric Hilbert scheme, broken toric varieties.
-- Connection with stacks. Twisted curves and log geometry.
The main prerequisite for the course, in addition to standard scheme theory and coherent cohomology, is some familiarity with the étale topology of schemes as can for example be found in Milne's book `Étale cohomology'. Students wanting a head start might also consult the article `Logarithmic structures of Fontaine-Illusie', by K. Kato Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 191--224, Johns Hopkins Univ. Press, Baltimore, MD, 1989.
For references click here.
Exercises for lecture 1
Exercises for lecture 2
Anton Geraschenko's notes are posted here.