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, AnandDumir Gupta conjecture about magic squares, BrianconSkoda Theorem in complex analytic geometry and its generalisation to rational singularities and HochsterRoberts Theorem about CohenMacaulay 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 Fregular and Frational rings.
Prerequisites: Basic Commutative algebra as covered in AtiyahMacdonald’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: CohenMacaulay 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: MacaulayStanley Theorem about Gorentein graded algebras
 Lecture 2: Gorenstein property of R^G when G is a finite subgroup of SL(n)
 Lecture 3: ShephardTodd 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 nonreductive 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 AnandDumir 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 Fregular 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 HochsterRoberts 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 Frational and Fregular rings: The HochsterRoberts theorem establishes the importance of Fregular rings and Frational 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 MehtaSrinivas. In these lectures, we will also discuss some important open questions in modular invariant theory that can be restated in terms of Fregular rings and splinters.
 Lectures 7+8: The BrianconSkoda 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 ArtinRotthaus theorem, and how it fits into the framework of general Ne’ron desingularization.
 Lectures 9+10 : Uniform bounds for symbolic powers: Theorems of EinLazarsfeld Smith and HochsterHuneke. 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 ArtinRotthaus theorem to obtain similar results in characteristic zero. The results as well as techniques tie in very nicely with the BrianconSkoda theorem.
TimeTable for Lectures and Tutorials
Date/Time 
9.00 10.15 
10.15 11.30 
11.30 11.45 
11.45 1.00 
1.00 2.30 
2.30 3.30 
3.30 4.00 
4.00 5.00 
5.00 5.30 
Mon 19 
Ananth 
Gurjar 
Tea

Verma 
Lunch

T1

Tea

T2

Snacks

Tue 20 
Ananth 
Gurjar 
Verma 
T3

T4


Wed 21 
Ananth 
Gurjar 
Verma 
T5

T6


Thu 22 
Ananth 
Kummini 
Verma 
T7

T8


Fri 23 
Ananth 
Gurjar 
Kummini 
T9

T10


Sat 24 
Kummini 
Gurjar 
Kummini 
T11

T12


Sun 25 











9.30 11.00 
11.00 11.30 
11.30 1,00 
1.00 2.30 
2.30 3.30 
3.30 4.00 
4.00 5.00 
5.00 5.30 
Mon 26 

Singh 
Tea

Singh 
Lunch

T13

Tea

T14 
Snacks

Tue 27 

Singh 
Singh 
T15

T16


Wed 28 

Singh 
Singh 
T17

T18


The 29 

Singh 
Singh 
T19

T20


Fri 30 

Singh 
Singh 
T21

T24

Tutorial Numbers 
Tutorial Instructor 
Associates 
T1 T5 T9 
H. Ananthnarayan 
Mitra Koley, Manoj Kummini 
T3 T7 T11 
RV Gurjar 
Sagar Kolte, Saurav Bhowmik 
T2 T4 T6 
JK Verma 
Ananthnarayan, Mitra Koley 
T8 T10 T12 
Manoj Kummini 
Mitra Koley, Ananthnarayan 
T 13  T 24 
Anurag Singh 
Tony Puthenpurakal anoj Kummini 
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