Themes
A detailed program of specific courses (for each academic year) that support the study of such subjects also depends on the interests of the students. Advanced knowledge of commutative algebra and of schemes is always a plus in this area but not really necessary, to a first approach, if not in the relatively elementary aspects. Here we set a list of broad themes that can be afforded gradually & quite independently:
Homological Algebra Resolutions, derived functors, and derived categories. Spectral sequences. Dualizing complexes, Cartier duality & Grothendieck-Verdier duality. Local & étale cohomology.
Homotopical Algebra Weak homotopy equivalences, simplicial structures and geometric realization. Model categories, homotopy categories and Quillen functors. Localization of model categories.
Algebraic K-Theory Projective modules and Picard group. The K_{0} of Grothendieck. Definition and various constructions of the higher K-theory. Fundamental theorems of K-theory. Riemann-Roch algebra & Chern classes.
Motivic Homotopy Correspondences and algebraic cycles. Homotopy invariant presheaves and motivic cohomology. Motivic complexes and A^{1}-localization.
Motives Abelian & triangulated categories of motives. Pure & mixed motives. Voevodsky motives, Chow motives, Grothendieck motives & standard conjectures. Nori motives & T-motives. The theory of 1-motives & Deligne conjectures.
Theses All students are welcome to just ask for a dissertation topic search. For the choice of dissertation topics all students are welcome to make a preliminary discussion on some subjects either during the courses of the master or after. Any new approach to the issues of mutual concern is included. Several topics can be obtained by browsing the above themes and looking at the references listed below. Master & Ph D Thesis of the past under my supervision are listed by clicking
here Click
here to see my Mathematics Genealogy.