As it traditionally happens in the University of Duisburg-Essen, the phd students organize a seminar called "Baby seminar" which we use as a moment to study altogether some topic of common interest.
Important! We meet on February in the room WSC-N-U-3.05 on Thursday 12th, at 15.00-15.15 (after the meeting for the research seminar (as suggested by Chiara) to dicuss the proposals).
The next baby seminar will take place in the summer semester 2014/2015, parallel to the usual university semester. It now time to think about some proposals for the possible topics. I have already started to discuss with some of you about some possibilities, and I also have the list we outlined during our meeting.
Here is a copy of the list.
One possibility is to study the basics of étale cohomology and then to aim, for example, to understand and proof all the properties of l-adic sheaves that we used in Giuseppe's lectures on the Weil conjectures. This could include studying the cohomology of curves, the smooth base change theorem, some classical theorems on singular cohomology (Poincaré duality, the Künneth formula, ...), the étale fundamental group, the proper definition of constructible and l-adic sheaves and the Grothendieck-Lefschetz trace formula. We could also treat Grothendieck's six functor formalism.
Alternatively, we could skip/shorten some of the above and try to learn something about pro-étale topology and cohomology that was introduced by Bhargav Bhatt and Peter Scholze very recently (see http://arxiv.org/abs/1309.1198). In Leiden there was a seminar about this last semester (see http://pub.math.leidenuniv.nl/~jinj/2014/ec/).
A generalisation of Serre duality for coherent sheaves, using derived categories. The standard reference is Hartshorne's Residues and Duality, but we can also follow a more recent approach given by Amnon Neeman in his paper "Grothendieck duality via homotopy theory" (http://arxiv.org/abs/alg-geom/9412022). This uses for example Brown's representability theorem. There was a seminar about this in Leiden: http://www.math.leidenuniv.nl/~wzomervr/coco-2014/.
An important direction is the study of variations of Hodge structure. In order to understand this, one need to understand monodromy action, Lefschetz pencils and so on, (which are basic ingredients for Weil conjecture as well). Roughly speaking, VHS is focusing on change for Hodge structures when we considering a family of smooth manifolds. Especially, when the special fiber is singular. In some case, studying VHS is simpler than HS. For example, the proof of variational Torelli theorem is easier than the classical Torelli theorem. Indeed rational Torelli only use part of the geometry of the canonical map, while Torelli theorem uses a much richer geometry.
K3 surfaces (Kummer, Kähler, Kodaira) form a vast class and play a very important role in the study of surfaces (and higher-dimensional varieties as well). They do not have very special structures like group structure (abelian varieties), but still easy to deal with, and many things can be computed on them (Hodge structures, moduli spaces, Picard lattices and so on). They provide a very nice testing ground for results in algebraic geometry, complex geometry,etc. For example, Weil conjectures were proved in case of K3 surfaces by Deligne before he obtained the general result. K3 surfaces are extremely interesting from both geometric and arithmetic perspectives. Many recent works are focusing on K3 surfaces, with broad diversity of aspects: zero cycles, rational points, Diophantine equations, Bloch-Beilinson conjecture, automorphism groups, Picard goups, etc.Understanding K3 surfaces should be nice. Working with something "general enough, and computable" is a good idea.
The rough goal: Understand K3 surfaces by some different equivalent definitions; some basic properties; Torelli theorem and Hodge decomposition for K3 surfaces; understand Kuga-Satake construction of abelian varieties attached to a K3 surfaces; moduli spaces of K3 surfaces, etc.
Goal: to see how the cotangent complex classifies infinitesimal extensions, and some applications of this to give obstruction theories for classical deformation problems:
Rough program: homotopy theory of dg-algebras, Quillen's construction of the cotangent complex, basic properties of the cotangent complex (localization, Mayer-Vietoris), derivations and square-zero extensions, deformation problems, applications and examples.
Goal: existence and properties of various moduli schemes (Hilbert schemes, Quot schemes, Picard schemes).
Rough program: faithfully flat descent of Grothendieck, Grothendieck's existence theorem, quotient schemes, Hilbert schemes, Picard schemes.
It would be nice if the guys who proposed some topic could send me a brief description of the topic they have suggested; then I would upload it to this web page, and in this way they could make some advertising by better explaining their idea.
If you wish to partecipate to our seminar, and you also have some proposal for the topic, please send me an email and I will upload your proposal adding it to the list.
For any further information about the seminar you can contact me at lorenzo.mantovani'youknowwhattoputhere(and remove the apostrophes)'uni-due.de