Audience: Ph. D. students and post doctoral Fellows

Synopsis : This course will focus on several interconnected themes such as invariant theory of finite and reductive groups, Anand-Dumir- Gupta conjecture about magic squares, Briancon-Skoda Theorem in complex analytic geometry and its generalisation to rational singularities and Hochster-Roberts Theorem about Cohen-Macaulay property of the rings of invariants of reductive groups in characteristic zero. These apparently disconnected topics are connected in positive characteristic if one uses tight closure of ideals and the notions of F-regular and F-rational rings.

Prerequisites: Basic Commutative algebra as covered in Atiyah-Macdonald’s Text Book: Introduction to Commutative Algebra.

Course 1: Review of commutative algebra by H. Ananthnarayan, IIT Bombay

• Lecture 1+2 : Dimension Theory
• Lecture 3: Cohen-Macaulay rings
• Lecture 4: Gorenstein rings
• Lecture 5: Completion of local rings

Course 2: Invariant Theory of finite groups by J. K. Verma, IIT Bombay

• Lecture 1+2 : Noetherian and CM property of R^G, Noether’s bound
• Lecture 3: Examples of R^G where G is a symmetry groups of Platonic solids
• Lecture 4: Molien Series and its use in computation of R^G

By Manoj Kummini, CMI, Chennai

• Lecture 1: Macaulay-Stanley Theorem about Gorentein graded algebras
• Lecture 2: Gorenstein property of R^G when G is a finite subgroup of SL(n)
• Lecture 3: Shephard-Todd Theorem
• Lecture 4: Some examples and results in positive characteristic

Course 3: Introduction to reductive groups and their invariant theory By R.V. Gurjar, IIT Bombay

• Lecture 1: Basic theory of linear algebraic groups. Lie Algebras. Examples.
• Lecture 2: Reductive Groups. Several definitions using complete reducibility of finite dimensional representation, Linear reductivity, purely group theoretic definition involving the radical of the group. Examples of reductive and non-reductive group.
• Lecture 3: Reynold  operator. Finite generation of the ring of invariants. Discussion of Hilbert's 14th Problem.
• Lecture 4: Properties of the quotient morphism. Luna's Etale Slice Theorem.
• Lecture 5: Rationality and other properties of the singularities of the quotient.

Course 4: Positive Characteristic Commutative Algebra By Anurag Singh, University of Utah, USA

• Lectures 1+2 : Magic squares: Proofs of the Anand-Dumir- Gupta conjectures using positive characteristic methods.This provides an application of tight closure theory to an easily stated classical problem, introduces basic concepts such as the flatness of the Frobenius endomorphism, and illustrates the role played by F-regular rings in tight closure theory. The ADG conjectures lead to a fascinating open problem in tight closure theory, that would also be discussed here.
• Lecture 3+4 : The Hochster-Roberts theorem: Extensions of the above techniques to rings of invariants of linearly reductive groups; generic freeness, and reduction modulo p methods. These lectures would include several examples of invariant rings of classical groups such as determinantal rings and Grassmannians, and include some subtleties that arise in reduction modulo p techniques since the classical groups are linearly reductive in characteristic zero, but typically not in positive characteristic.
• Lectures 5+6 F-rational and F-regular rings: The Hochster-Roberts theorem establishes the importance of F-regular rings and F-rational rings as an object of study in their own right. We will discuss some results and quetions about these rings, as well as connections with singularities in positive characteristic including work of Smith, Hara, and Mehta-Srinivas. In these lectures, we will also discuss some important open questions in modular invariant theory that can be restated in terms of F-regular rings and splinters.
• Lectures 7+8: The Briancon-Skoda theorem: Proofs in positive characteristic, and in characteristic zero. The theorem has a rich history, and is yet another striking application of tight closure theory. It is also an opportunity to discuss powerful techniques including the Artin-Rotthaus theorem, and how it fits into the framework of general Ne’ron desingularization.
• Lectures 9+10 : Uniform bounds for symbolic powers: Theorems of Ein-Lazarsfeld- Smith and Hochster-Huneke. Bounds for symbolic powers of ideals constitute some of the most active areas of current research in commutative algebra. We will prove uniform bounds first in the case of positive characteristic, and then apply the Artin-Rotthaus theorem to obtain similar results in characteristic zero. The results as well as techniques tie in very nicely with the Briancon-Skoda theorem.

Time-Table for Lectures and Tutorials

 
Course Associates
1. Sagar Kolte, IIT Bombay
2. Mitra Koley, CMI, Chennai
3. Saurav Bhowmik, IIT Bombay
4. Tony Puthenpurakal, IIT Bombay
5. H. Ananthnarayan, IIT Bombay