Convener(s)
| Name: | Prof. Jugal K Verma | Prof. Krishna Hanumanthu | Dr. Parangama Sarkar |
| Mailing Address: | Visiting Professor, Department of Mathematics, Indian Institute of Technology Gandhinagar Palaj, Gujarat-382055 |
Professor, Department of Mathematics, Chennai Mathematical Institute H1, SIPCOT IT Park, Siruseri, Kelambakkam Chennai-603103 |
Assistant Professor, Department of Mathematics, Indian Institute of Technology Palakkad Kanjikode, Palakkad Kerala - 678623 |
| Email: | jugal.verma at iitgn.ac.in | krishna at cmi.ac.in | parangama at iitpkd.ac.in |
The aim of this AIS (Advanced Instructional School) is to introduce participants to powerful tools at the interface of Commutative Algebra and Algebraic Geometry. This school will emphasize how geometric ideas, such as schemes, coherent sheaves, and cohomology, together with homological methods in Commutative Algebra, lead to fundamental results in Commutative Algebra. To ensure that the school is accessible to PhD students whose research interests broadly lie in Commutative Algebra, the first week will be devoted to reviewing foundational topics, including mapping cones, spectral sequences, injective resolutions, and scheme theory. The second week will include Matlis Duality, Gorenstein rings, divisors and cohomology. The final week will focus on advanced topics, including Local cohomology, Local duality theorems, Vanishing theorems (Kodaira-Akizuki-Nakano, Grauert-Riemenschneider) and Formal duality.
Dates:
Venue:
Venue Address:
Dr. Parangama Sarkar DepartmentofMathematics Sangraha, Office no. 4
IndianInstituteofTechnologyPalakkad Kanjikode, Palakkad Kerala – 678623
Venue State:
Venue City:
PIN:
Syllabus:
|
Name of the Speaker with affiliation |
No. of Lectures |
Detailed Syllabus |
|
Dr. Rohith Varma |
6 (1.5 hours) |
Presheaves, sheaves, sheafificaiton, ringed spaces, schemes, morphisms of schemes, sheaves of modules, constructions of sheaves. |
|
Prof. Manoj Kummini |
4 (1.5 hours) |
Homological tools (Mapping cones, Spectral sequences) |
|
Dr. Sanjay Amrutiya |
4 (1.5 hours) |
Separated and proper morphisms, divisors, differentials, invertible sheaves, Leray spectral sequence. |
|
Dr.Vivek Mukundan |
4 (1.5 hours) |
Injective resolutions and Matlis duality |
|
Prof. Jugal K Verma |
5 (1.5 hours) |
Gorenstein rings and canonical modules |
|
Prof.Krishna Hanumanthu |
4 (1.5 hours) |
Derived functors, sheaf cohomology, Grothendieck vanishing, cohomology of affine schemes, Serre’s affineness criterion, Cech cohomology, cohomology of projective spaces. |
|
Prof. Tony J. Puthenpurakal |
5 (1.5 hours) |
Local cohomology, local duality and some applications |
|
Prof. A J Parameswaran |
4 (1.5 hours) |
Serre’s vanishing, ext groups and sheaves, higher direct image sheaves, formal function theory, semi continuity, vanishing theorems (Kodaira-Akizuki -Nakano,Grauert-Riemenschneider), Formal duality. |
References:
1. W. Bruns and J. Herzog, Cohen-Macaulay rings, Revised edition 1998.
2. R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977.
3. Joseph J. Rotman. An introduction to homological algebra. Universitext. Springer, New York, second edition, 2009.
4. Lecture notes on Cohomology of coherent sheaves by K. H. Paranjape and V. Srinivas. (Lecture notes from the speakers, if available)
Pre-requisites for Commutative Algebra :H. Matsumura, Commutative Ring Theory, Chapters 1-7 and Appendices A and B (excluding the sections on Gorenstein rings and injective modules).
Tutorial Assistants:
|
S. No. |
Name |
Affiliation |
|
1 |
Samarendra Sahoo |
Indian Institute of Technology Bombay |
|
2 |
Meghana Bhat |
Indian Institute of Technology Dharwad |
|
3 |
Omkar Javadekar |
Chennai Mathematical Institute |
|
4 |
Sudeshna Roy |
Indian Institute of Technology Gandhinagar |
|
5 |
Suprajo Das |
Indian Institute of Technology Madras |
|
6 |
CyrilJ. Jacob |
Indian Institute of Technology Bombay |
|
7 |
Debojyoti Bhattacharya |
Chennai Mathematical Institute |
|
8 |
Pavan Adroja |
Indian Institute of Technology Gandhinagar |
Time Table:
| Day | Date | Lecture 1 (9.30–11.00) |
Tea (11.05 – 11.25) |
Lecture 2 (11.30–1.00) |
Lunch (1.05–2.25) |
Tutorial (2.30–3.30) |
Tea (3.35-3.55) |
Tutorial (4.00-5.00) |
Snacks 5.05-5.30 |
| (name of the speaker | (name of the speaker | (name of the speaker + tutors) | (name of the speaker + tutors) | ||||||
| Mon | 8th June | MK | MK | MK+SS+ MB | MK+SS+ MB | ||||
| Tues | 9th June | MK | VM | VM+SS+ MB | VM+SS+ MB | ||||
| Wed | 10th June | MK | VM | MK+SS+ MG | MK+SS+ MB | ||||
| Thu | 11th June | VM | RV | RV+CJ+ PA | RV+CJ+ PA | ||||
| Fri | 12th June | VM | RV | VM+SS+ MB | VM+SS+ MB | ||||
| Sat | 13th June | JKV | RV | RV+CJ+ PA | RV+CJ+ PA | ||||
| SUNDAY: OFF | |||||||||
| Mon | 15th June | JKV | RV | JKV+SS+ MB | JKV+SS+ MB | ||||
| Tues | 16th June | JKV | RV | RV+CJ+ PA | JKV+SS+ MB | ||||
| Wed | 17th June | JKV | RV | JKV+SS+ MB | RV+CJ+ PA | ||||
| Thu | 18th June | JKV | SA | JKV+OJ+ SR | SA+CJ+ PA | ||||
| Fri | 19th June | SA | KH | SA+CJ+ PA | KH+SD+ DB | ||||
| Sat | 20th June | SA | KH | SA+SD+ DB | KH+SD+ DB | ||||
| SUNDAY: OFF | |||||||||
| Mon | 22 nd June | SA | KH | SA+SD+ DB | KH+SD+ DB | ||||
| Tues | 23rd June | TJP | KH | TJP+OJ+ SR | KH+SD+ DB | ||||
| Wed | 24th June | TJP | AJP | TJP+OJ+ SR | TJP+OJ+ SR | ||||
| Thu | 25th June | TJP | AJP | AJP+SD+ DB | AJP+SD+ DB | ||||
| Fri | 26th June | TJP | AJP | TJP+OJ+ SR | AJP+SD+ DB | ||||
| Sat | 27th June | TJP | AJP | TJP+OJ+ SR | AJP+SD+ DB | ||||
Full forms for the abbreviations of speakers and tutors:
RV: Rohith Varma
VM: Vivek Mukundan
SA: Sanjay Amrutiya
MK Manoj Kummini
JKV: Jugal K Verma
KH: Krishna Hanumanthu
TJP: Tony J. Puthenpurakal
AJP: A J Parameswaran
SS: Samarendra Sahoo
MB: Meghana Bhat
OJ: Omkar Javadekar
SR: Sudeshna Roy
SD: Suprajo Das
PA: Pavan Adroja
CJ: Cyril J. Jacob
DB: Debojyoti Bhattacharya
Selected Applicants:
How to Reach:
TBA