Themes & Theses

General In arithmetic geometry - classically the study of solutions of polynomial equations and algebraic structures related to them - schemes, K-theory, étale cohomology and, finally, the motivic cohomology play a key role. These subjects have revealed deep connections between geometry, algebra and arithmetic by allowing the solution of important problems and conjectures. Such a new vision of mathematics traced by the monumental work of Alexander Grothendieck comes along with a key theory: motives. All this put its bases on commutative, homological & homotopical algebra.


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 K₀ 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 A¹-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.


References 
• Grothendieck work: EGA & SGA are key references. A recent good self-contained book on schemes is:
  • S. Bosch: Algebraic Geometry and Commutative Algebra Universitext, Springer, 2013, 504 p.


• Homological algebra: a good introduction with several applications is
  • C. Weibel: An introduction to Homological Algebra Cambridge Univ. Press, 1994. (as reported by J. Rotman)


• Grothendieck's local cohomology: there are R. Hartshorne notes of Grothendieck's seminar at Harvard published as a Springer LNM 41 (1961) & SGA 2 but also
  • M. P. Brodman & R. Y. Sharp: Local Cohomology Cambridge Univ. Press, 1998 (as reported by R. Hartshorne, including reference to Grothendieck's seminar at Harvard)


• K-theory: for the algebraic world & heritage good references are
  • C. Weibel: An introduction to Algebraic K-theory Graduate Studies in Math. vol. 145, AMS, 2013
  • E.M. Friedlander, D.R. Grayson (ed.) Handbook of K-theory Vol. 1, 2 Springer-Verlag
  • W. Fulton & S. Lang: Riemann-Roch Algebra Springer Grundlehren der mathematischen Wissenschaften Vol. 277, 1985


• Homotopical algebra: here we have
  • W.G. Dwyer & J. Spalinski: Homotopy theories and model categories in Handbook of Algebraic Topology I.M. James (ed.) North-Holland, 1995
  • M. Hovey: Model Categories Math Surveys & Monographs Vol. 63 AMS 1999.


• Motives: for the motivic world, motivic cohomology & homotopy we have
  • Y. André: Une introduction aux motifs (motifs purs, motifs mixtes, periodes) Panoramas et Syntheses Vol. 17 SMF, 2004 xii+261 pages
  • L. Barbieri Viale: A pamphlet on motivic cohomology Milan Journal of Math. Vol. 73 (2005) 53-73
  • C. Mazza, V. Voevodsky & C. Weibel: Lecture Notes on Motivic Cohomology Clay Mathematics Monographs, Vol. 2, 2006 (Notes of Voevodsky's seminar at Princeton)
  • B.I. Dundas, M. Levine, P.A. Ostvaer, V. Voevodsky: Motivic Homotopy Theory Lectures at a Summer School in Nordfjordeid (Norway, August 2002) Springer Universitext, 2007


• 1-motives: for the theory of 1-motives a good introduction is provided by the report 1-Motives of the Oberwolfach mini-workshop (December 18-22, 2010). We also have:
  • L. Barbieri Viale: On the theory of 1-motives in Algebraic Cycles and Motives London Math. Soc. Lecture Note Series, Vol. 343, Cambridge Univ. Press, 2007, 55-101
  • L. Barbieri Viale & B. Kahn: On the derived category of 1-motives Astérisque Vol. 381 (2016) xi + 254 pages


• Theoretical motives & Nori motives:
  • L. Barbieri-Viale: 𝕋-motives Journal of Pure and Applied Algebra Vol. 221, Issue 7 (2017) 1495-1898.
  • A. Huber & S. Müller-Stach: Periods & Nori motives Ergebnisse der Mathematik Vol. 65, Springer-Verlag, 2017.