The contents of the course will depend on the audience' background and interests. I plan to start with exact sequences of abelian groups, complexes and their cohomology, the snake lemma, 5-lemma, and chain homotopy. Then we will discuss nonexactness of Hom in the category of modules over a (noncommutative) ring, projective and injective modules, resolutions, and the Ext functor. This should be enough for the first lecture, and then we will see. Other topics to be covered include additive and abelian categories, additive functors and their derived functors, the homotopy and derived categories of abelian categories, triangulated categories and the Verdier localization, and semiorthogonal decompositions arising in connection with the injective and projective resolutions. Spectral sequences may or may not be covered.
Bibliography: before starting on homological algebra, it may be instructive to learn a bit of basic algebraic topology. So the audience is encouraged to look into the first chapters of the book by Fuchs and Fomenko, where they discuss the basic properties of the homology and homotopy groups of topological spaces. Fuchs-Fomenko also have an excellent discussion of spectral sequences. Concerning homological algebra proper, there are introductory textbooks by Rotman and Weibel, and a much more advanced book by Gelfand-Manin. One can also read about triangulated categories in Verdier's article in SGA 4 1/2.
We will consider basics of birational geometry of surfaces and 3-folds with special regard to problems of rationality. The tentative plan of the lectures is the following:
Lecture 1. Blow ups, blow downs, canonical class, adjunction formula, rational surfaces, rationality criterion, quadric surfaces, cubic surfaces, two-dimensional Minimal Model Program, surfaces with a finite group action, surfaces over algebraically non-closed field of characteristic zero.
Lecture 2. Singularities of pairs, log canonical singularities, Kawamata log terminal singularities, mobile log pairs, canonical singularities, Nadel-Shokurov vanishing theorem, Inversion of Adjunction and its applications, Noether-Fano inequality, Pukhlikov inequality, Corti inequality.
Lecture 3. Non-rational surfaces over algebraically non-closed fields, non-rationality of smooth quartic 3-folds, finite subgroups in Cremona groups.
Everything we going to cover is contained in the following papers/books:
I leave the title and abstract as vague as possible, so that I can talk about whatever I feel like on the day. Many varieties of interest in the classification of varieties are obtained as Spec or Proj of a Gorenstein ring. In codimension <= 3, the well known structure theory provides explicit methods of calculating with Gorenstein rings. In contrast, there is no useable structure theory for rings of codimension >= 4. Nevertheless, in many cases, Gorenstein projection (and its inverse, Kustin-Miller unprojection) provide methods of attacking these rings. These methods apply to sporadic classes of canonical rings of regular algebraic surfaces, and to more systematic constructions of Q-Fano 3-folds, Sarkisov links between these, and the 3-folds flips of Type A of Mori theory.
For introductory tutorial material, see my website + surfaces + Graded rings and the associated homework.
For applications of Gorenstein unprojection, see "Graded rings and birational geometry" on my website + 3-folds, or the more recent paper.
Gavin Brown, Michael Kerber and Miles Reid, Fano 3-folds in codimension 4, Tom and Jerry (unprojection constructions of Q-Fano 3-folds), Composition to appear, arXiv:1009.4313
Reflection group is a discrete group of motions of a space of constant curvature (a sphere, Euclidean or hyperbolic space) which is generated by a set of reflections. Reflection groups appear remarkably often in various algebraic problems.
A very preliminary (and rather optimistic) plan of my lectures is as follows.
Lecture 1. Finite reflection groups. Examples: dihedral, permutation and hyperoctahedral groups. Regular tilings (kaleidoscopes) of a sphere, Euclidean space, Lobachevskii space. Coxeter cones and polytopes. Root systems, positive and simple roots.
Lecture 2. Classification of finite reflection groups by Coxeter graphs. An explicit construction of exceptional root systems. Crystallographic groups, Dynkin diagrams, relation to Lie algebras. Affine reflection groups. If time permits: hyperbolic reflection groups (an overview).
Lecture 3. A non-geometric example: representations of quivers, following Bernstein-Gelfand-Ponomarev. Indecomposable representations. Gabriel's theorem: the underlying graph of a finite type quiver is a simply-laced Dynkin diagram (ADE classification). Reflection functors. Bijection between the indecomposable representations and the positive roots. Independence on the orientation of a quiver. Kac's theorem.
2010-2011, Institute of Mathematics and Mechanics UB RAS, Ekaterinburg