Meeting times: MW, 2-3.30pm. The first meeting will be on Sept 6.

Room: 509 Lake.

If you are a Northeastern student planning to take this class, please consider doing this for credit and register.

Instructor: Prof. Ivan Losev

Office: 519 Lake

Office hours: by appointment.

0) A brief intro to Invariant theory: origins and goals.

I) Invariant theory of finite groups: finiteness properties, Noether theorem (a bound on degrees of generators), Chevalley-Shephard-Todd theorem (on invariants of complex reflection groups).

II) Birational invariants: separation of generic orbits by birational invariants.

III) Invariants under reductive group actions: a brief review of reductive groups, Hilbert's theorems on quotients and quotient morphisms. Algebro-geometric properties of quotients.

IV) Closed orbits in orbit closures: the Hilbert-Mumford theorem, Kempf's theorem of optimal destabilizing subgroups.

V) Connection to moment maps for compact group actions (the Kempf-Ness theorem) and applications.

VI) Computation of invariants: Classical invariant theory.

VII) Invariants under more general algebraic group actions, U-invariants.

VIII) GIT quotients: construction and properties. Linearization of line bundles. Applications to moduli spaces in Algebraic geometry.

IX) Local structure of the actions (Luna's etale slice theorem and applications).

X) Actions with good invariant theoretic properties and Vinberg's \theta-groups.

Homework 1 (corrected verison (typo in problem 3 fixed, thanks to Borya for noticing) due Oct 4, either in class or by e-mail by the end of the day; your official score for every homework is the minimum of 10 and the actual score). Hints. Solutions to problems 1 to 3 and a solution to the extra-credit problem.

Homework 2, due Oct 18. Hints. Solutions.

Homework 3, due Nov 1.

[B] N. Bourbaki, Lie groups and Lie algebras. Chapters 1-3, Chapters 4-6, Chapters 7-9, Springer.

[CM] D. Collingwood, W. McGovern. Nilpotent Orbits In Semisimple Lie Algebra: An Introduction, CRC Press, 1993.

[E] D. Eisenbud, Commutative algebra: with a view toward Algebraic geometry, GTM 150, Springer, 1995.

[K] H. Kraft, Geometrische Methoden in der Invariantentheorie, 1984. I'm afraid, there's no English translation, though there is a Russian translation.

[L] I. Losev, Kempf-Ness theorem and Invariant theory.

[MFK] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant theory, Springer 1994.

[OV] A. Onishchik, E. Vinberg, Lie groups and Algebraic groups.

[PV] V. Popov and E. Vinberg, Invariant theory in Algebraic geometry 4, Encyclopaedia of Mathematical Sciences, vol. 55, Springer Verlag.

[S] T. Springer, Invariant Theory, Springer 1977.

A very official syllabus can be found here. It contains an actual grading policy and grade cut-offs. In particular, there will be five homeworks.

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10) Oct 12, Thursday, 4.45-6.15, Dodge 070 (note unusual date, time and location, Dodge is building 43 on this map). We cover the Chevalley restriction theorem and discuss orbit closures and invariants for representations of tori. References include [PV, Sections 3.8, 5.3, 5.4]. Notes.

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