Dear Participants,
It will be better if you browse through these files and get some familiarity with them before the workshop. This will help you in following the workshop.
Rabeya Basu, Raja Sridharan, Manoj Keshari,Ravi Rao
This AIS is planned to introduce students to Euler class group theory over a Noetherian ring, and Quillen-Suslin theory for symplectic and orthogonal groups. There will be six streams of lectures (two per week).
Speakers:
- Rabeya Basu, IISER Pune
- Manoj Keshari, IIT Bombay
- Dilip Patil, IISc Bangalore
- Mrinal Das, ISI Kolkata
- Raja Sridharan, TIFR, Mumbai
- Ravi Rao, TIFR, Mumbai
- Selby Jose, Institute of Science, Mumbai
- Vinay Wagh, IIT Guwahati
Syllabus: We plan to cover the following topics in this workshop:
| Week 1 | |
| Speakers | Rabeya Basu, Dilip Patil and Manoj Keshari |
| Topics | Commutative Algebra : Noetherian rings, Artinian rings, Krull’s dimension theorem, Hilbert’s Nullstellensatz, projective and stably free modules, Suslin’s proof of Serre’s conjecture, Mohan Kumar’s theorem on complete intersections. Quillen-Suslin theory: Horrock’s and Quillen-Suslin theorem for unimodular rows, Normality of En (R) in SLn (R), En (R) acts transitively on Umn (R) for semilocal rings and for rings of dimension d with n ≥ d + 2. Transitivity theorems for R[X], n ≥ d + 2, d = dim(R). |
| Week 2 | |
| Speakers | Mrinal Das and Raja Sridharan |
| Topics | A simple introduction to Euler classes and complete intersections. Relation between projective modules and topology. Suslin’s n! theorem, Vaserstein symbol. Applications to Bass-Suslin conjecture for dimension 2. |
| Week 3 | |
| Speakers | Ravi Rao, Selby Jose and Vinay Wagh |
| Topics | Quillen-Suslin theory for Suslin matrices and some applications, Computational Quillen-Suslin. |
Time-Table:
| Date | 9:30-11:00 Talk |
11:30-13:00 Talk |
14:00-15:00 Tutorial |
15:30-16:30 Tutorial |
| May 21 | R Basu | D Patil | D Patil | R Basu |
| May 22 | R Basu | D Patil | D Patil | R Basu |
| May 23 | R Basu | M Keshari | M Keshari | R Basu |
| May 24 | M Keshari | D Patil | D Patil | M Keshari |
| May 25 | M Keshari | D Patil | D Patil | M Keshari |
| May 26 | M Keshari | D Patil | D Patil | M Keshari |
| May 28 | M K Das | R Sridharan | M K Das | R Sridharan |
| May 29 | M K Das | R Sridharan | M K Das | R Sridharan |
| May 30 | M K Das | R Sridharan | R Sridharan | M K Das |
| May 31 | M K Das | R Sridharan | R Sridharan | M K Das |
| June 01 | M K Das | R Sridharan | R Sridharan | M K Das |
| June 02 | M K Das | R Sridharan | R Sridharan | M K Das |
| Date | 9:30-10:30 Talk |
10:50:-11:50 Talk |
12:00-13:00 Talk |
14:00-16:00 Tutorial |
| June 04 | S. Jose | R. Rao | V. Wagh | V. Wagh + Ratnik (CL) |
| June 05 | S. Jose | V. Wagh | R. Rao | V. Wagh + Ratnik (CL) |
| June 06 | S. Jose | V. Wagh | R. Rao | R. Rao + Ratnik (CL) |
| June 08 | S. Jose | V. Wagh | R. Rao | S. Jose + Vijay (CL) |
| Date | 9:30-11:00 Talk |
11:30-13:00 Talk |
14:00-16:00 Tutorial |
| June 07 | S. Jose | Vineeth | S. Jose + Vijay |
| June 09 | S. Jose | R. Rao | V Wagh + Ratnik (CL) |
CL = Computer Lab
References:
- MPhil Thesis, Rabeya Basu
available at: https://rabeyabasublog.wordpress.com/notes/ - MPhil Thesis, Manoj Keshari
available at: https://www.math.iitb.ac.in/ keshari/Mphil_manoj.pdf - F. Ischebeck, R. Rao, Ideals and Reality - Projective Modules and number of generators of ideals, Springer Monograph in Mathematics.
Prerequisites:
Participants must be familiar with a graduate course in commutative algebra.