Students are invited to prepare an optional expository paper for extra credit. The paper does not necessarily have to fully cover one of the items in the list below. You are also welcome to consult other sources besides those referenced here (see list of references below).
Please discuss your choice of material with the instructor before starting to work on the paper.
Possible Topics
- More on representable and adjoint functors: Criteria for representability of functors [22, §V.6-8]. Monads and comonads, Barr-Beck theorem [22, §VI.7]. For those familiar with triangulated categories: criteria for representability of homological functors [7].
- Koszul rings and Koszul duality: Koszulity and distributivity, Koszul rings and modules, PBW theorem, Veronese powers. [23]
- Operads, Koszul operads, Koszul duality for operads [20], [13].
- Derived categories and derived functors. [27, ch. 10], [11].
- Hochschild (co)homology: Hochschild-Kostant-Rosenberg theorem, Gerstenhaber operations, Batalin-Vilkovisky algebras. Deformation theory. Cyclic (co)homology. [27, ch. 9], [19].
- Some approaches to noncommutative geometry; see [12] for a possible general reference.
(a) Graded algebras as a generalization of a projective variety, [2], [3].
(b) Formal expansion at the commutative locus [15].
(c) Representation schemes [16], [6].
(d) Example of passing from noncommutative to commutative setting via characteristic p, [5]. - Brauer groups. Brauer groups in number theory: Brauer groups of global fields [9], Brauer-Manin obstruction [24, §8.2]. Generalizations of Brauer group, Brauer-Wall group [21], [18].
- Elements of invariant theory, variety of representations, Procesi theorem, [1, V.13-V.15].
- Growth of algebras and growth of groups, [17].
- Primitive ideals in enveloping algebras, Goldie rank polynomials. [8], [14]
- Golod-Shafarevich algebras and groups and their applications [10].
- Algebraic relation between groups and Lie algebras: Maltsev type completions in zero and positive characteristic [25, Appendix A3], [26] and applications; Jennings-Quillen theorem [4, §3.14].
References
- M. Artin, Noncommutative algebra, notes available in the Study Materials section
- M. Artin, W. Schleter, “Graded algebras of global dimension 3.”
- M. Artin, J. J. Zhang, “Noncommutative projective schemes.”
- D. Benson, Representations and cohomology, I
- A. Belov-Kanel, M. Kontsevich, “The Jacobian conjecture is stably equivalent to the Dixmier conjecture; Automorphisms of the Weyl algebra.”
- Y. Berest, G. Felder, A. Ramadoss, “Derived representation schemes and noncommutative geometry, Expository lectures on representation theory.”
- A. Bondal, M. van den Bergh, “Generators and representability of functors in commutative and noncommutative geometry.”
- J. Dixmier, Algebres enveloppantes
- J. W. S. Cassels, A. Fröhlich, ”Algebraic number theory."
- M. Ershov, ”Golod-Shafarevich groups: a survey.”
- S. Gelfand, Y. Manin, ”Methods of homological algebra.”
- V. Ginzburg, “Lectures on noncommutative geometry.”
- V. Ginzburg, M. Kapranov, “Koszul duality for operads.”
- J.-C. Jantzen, ”Einhullende Algebren halbeinfacher Lie-Algebren."
- M. Kapranov, “Noncommutative geometry based on commutator expansions.”
- M. Kontsevich, A. Rosenberg “Noncommutative smooth spaces.”
- R. Krause, H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension
- T. Y. Lam, Introduction to quadratic forms over fields
- J.-L. Loday, Cyclic homology
- J.-L. Loday, B. Vallette, Algebraic operads
- J. Lurie, lectures ”Brauer Groups in Chromatic Homotopy Theory,” lecture 1, available on YouTube.
- S. MacLane, Categories for the working mathematician
- A. Polishchuk, L. Positselski, Quadratic algebras
- B. Poonen, Rational Points on Varieties.
- D. Quillen, “Rational homotopy theory.”
- D. Quillen, “On the associated graded of a group ring.”
- C. Weibel, An Introduction to Homological Algebra.
