AIS Schemes & Cohomology (2014)
Venue: | Kerala School of Mathematics, Kunnamangalam, Kozhikode |
Dates: | 1^{st} - 19^{th} Dec, 2014 |
Convener(s) | Speakers, Syllabus and Time table | Applicants/Participants |
Name | V. Balaji | D.S. Nagaraj |
Mailing Address | Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Chennai-603103. |
Institute of Mathematical Sciences, Taramani, Chennai |
A Brief Description of the Subject:-
The development of Algebraic Geometry in the 20th century went through a few salient phases; the first one was dominated by the Italian geometers. This phase was when algebro-geometric techniques were closely linked with topology and analytic geometry and were used in developing the theory of algebraic surfaces. But the shortcoming of this phase was that the intuitive aspects of the geometry was given more importance and there was a neglecting of the key aspects of proofs and rigour as well as analysis of special examples which were needed to support the intuition. The period 1930-1960 was dominated by the work of Zariski, Weil intended to set right these shortcomings; this culminated in the grand synthesis in the hands of Grothendieck. An immense programme was launched by Grothendieck which introduced tools from commutative algebra into algebraic geometry which afforded a uniform language to handle geometry over characteristic p and characteristic zero. The goal was to create a geometry which synthesized the formal arithmetic aspccts of geometry as well as classical projective geometry. Some of the spectacular successes achieved by algebraic geometry both in its geometric as well as its arithmetic aspects can be directly linked to this grand synthesis initiated by the work of Grothendieck. Just to name a few, Deligne's proof of Weil conjectures, Faltings proof of the finiteness of rational points, Wiles's proof of Fermat's last theorem rely indispensably on this super structure.
The AIS will be directed towards making young researchers in India familiar with the developments in Algebraic Geometry, with special emphasis on Grothendieck's programme. The three week programme is primarily aimed at early researchers in this field to become familiar and users of the tools and techniques in this subject. To acquire a good knowledge of modern algebraic geometry it is essential to see the “local” aspects coming from Commutative algebra in its interplay in geometry as well as the immense machinery of cohomology. These two themes will be the main ones for the workshop. To give a flavour of the manner in which these tools have lead to fundamental theorems in algebraic geometry, the final week will stress on some research themes. A team of active researchers in the field has been invited for this purpose. The basic reference for this course will be the book by Hartshorne supplemented by books by Mumford, Ueno and others.
Schemes: Sheaves, schemes, elementary properties, morphisms, invertible sheaves and bundles, differentials, valuative criterion, Bertini’s theorems, Lefschetz theorem. Derived functors, cohomology of sheaves, Cech cohomology, Cohomology of projective space, Serre duality, semicontinuity theoremes , Zariski’s main theorem and connected theorem, Fulton Lazarsfeld connectedness theorem.
Name | Affiliation | No.of Lecture | Topic |
Krishna Chaitanya (KC), | CMI | 6 | Derived functors, cohomology of sheaves, cohomology of projective space.(3 |
KN Raghavan (KNR), | IMSc | 6 | Sheaves, schemes, first properties,Morphisms, properness crtierion(1) |
DS Nagaraj (DSN) | IMSc | 5 | Fulton-Lazarsfeld connectedness theorems, applications in Brill-Noether theory.(7) |
Manoj Kummini(MK) | CMI | 6 | Sheaves of modules, divisors, Projective morphism, differentials, Bertini theorems(2) |
AJ Parameswaran (AJP) | TIFR | 5 | Higher direct images, flatness, Hilbert polynomial, smooth morphisms.(5) |
V Balaji (VB) | CMI | 5 | Formal function theorem, semicontinuity theorem, connectedness(6) |
Krishna Chaitanya (KC) | CMI | 6 | Ext groups, Serre duality.(4) |
References:
1. R. Hartshorne, Algebraic Geometry
2. K. Ueno, Algebraic Geometry I, II, III
3. Lazarsfeld, Positivity in Algebraic Geometry
4. Griffiths - Harris, Principles of Algebraic Geometry
Time-Table :
Tea: 11:00 – 11:30 and 03:30 – 04:00 Lunch: 01:00 – 2:30 Snacks: 05:00 – 05:30 | |||||
Day | Date | Lecture1 (09:30-11:00) |
Lecture1 |
02:30-3:30 | 04:00 - 05:00 |
Mon | 1/12/2014 | KC | KNR |
Tutorial / Guest Talk |
Tutorial / Guest Talk |
Tues | 2/12/2014 | KC | KNR | -do- | -do- |
Wed | 3/12/2014 | KC | KNR | -do- | -do- |
Thu | 4/12/2014 | KC | KNR | -do- | -do- |
Fri | 5/12/2014 | KC | KNR | -do- | -do- |
Sat | 6/12/2014 | KC | KNR | -do- | -do- |
Mon | 8/12/2014 | KC | MK | -do- | -do- |
Tues | 9/12/2014 | KC | MK | -do- | -do- |
Wed | 10/12/2014 | KC | MK | -do- | -do- |
Thus | 11/12/2014 | KC | MK | -do- | -do- |
Fri | 12/12/2014 | KC | MK | -do- | -do- |
Sat | 13/12/2014 | KC | MK | -do- | -do- |
Mon | 15/12/2014 | AJP | VB | -do- | DSN |
Tues | 16/12/2014 | AJP | VB | -do- | DSN |
Wed | 17/12/2014 | AJP | VB | -do- | DSN |
Thus | 18/12/2014 | AJP | VB | -do- | DSN |
Fri | 19/12/2014 | AJP | VB | -do- | DSN |
Tutorial Assistants:
S. No. | Name | Affiliation |
1 | Narasimha Chari | CMI |
2 | Pabitra Barik | IMSc |
3 | Krishanu Dan | IMSc |
4 | Rohith Varma | CMI |
5 | Suratno Basu | CMI |
Selected Applicants |
Serial | SID | Full Name | Gender | Affiliation | City,State | Position | University/Institute M.Sc. / M.A. | Year of Passing M.Sc. / M.A |
1 | 6420 | Mr Bharath Kumar Ethamakula | Male | Indian Institute of Science, Bangalore | Bangalore,Karnataka | PhD | IITKanpur | 2011 |
2 | 6466 | Mr. Rajiv Kumar | Male | IIT Bombay | Mumbai,Maharashtra | Ph.D | M.SC. | 2009 |
3 | 6477 | Mr Saurabh Singh | Male | Indian Statistical Institute, Bangalore. | Bangalore,Karnataka | Ph.D Student | University of Hyderabad | 2012 |
4 | 6539 | Ms. Parangama Sarkar | Female | IIT BOMBAY | Mumbai,Maharashtra | PhD | IIT MADRAS | 2011 |
5 | 6552 | Mr. Sumit Kumar Upadhyay | Male | The Institute of Mathematical Sciences | Chennai,Tamil Nadu | Post Doctoral Fellow | University of Allahabad | 2010 |
6 | 6635 | Mr Pratik Girishkumar Mehta | Male | Indian Statistical Institute, Bangalore | Bangalore,Karnataka | Senior Research Fellow | Indian Statistical Institute, Bangalore | 2011 |
7 | 6670 | Ms Samarpita Ray | Female | Indian Institute of Science | Bangalore,Karnataka | Integrated PhD student | Indian Institute of Science | 2013 |
8 | 6673 | Mr Prasant Singh | Male | Indian Institute of Technology, Bombay | Mumbai,Maharashtra | PhD Student | Banaras Hindu University | 2010 |
9 | 6684 | Ms Raisa Dsouza | Female | IIT Madras | Chennai,Tamil Nadu | Phd Student | Mount Carmel College | 2012 |
10 | 6741 | Mr. Abhinav Sahani | Male | Indian Institute of Science Education and Reserch, Pune | Pune,Maharashtra | BS-MS Student | Indian Institute of Science Education and Research, Pune | Appeared/ Awaiting Result |
11 | 6768 | Mr Pradeep Das | Male | Harish-Chandra Research Institute | Allahabad,Uttar Pradesh | PhD Student | IIT Kanpur | 2011 |
12 | 6805 | Mr. Arpan Kabiraj | Male | INDIAN INSTITUTE OF SCIENCE | Bangalore,Karnataka | Student | Indian Institute Of Science M.Sc. | 2011 |
13 | 6806 | Mr Krishanu Roy | Male | The Institute of Mathematical Sciences | Chennai,Tamil Nadu | Integrated ph D student | Appeared/ Awaiting Result | |
14 | 6897 | Mr Manikandan S | Male | Harish-Chandra Research Institute | Allahabad,Uttar Pradesh | PhD Student | Bharathidasan University | 2011 |
15 | 6898 | Ms. Poornapushkala Narayanan | Female | IIT Madras | Chennai,Tamil Nadu | PhD student | IIT Madras | 2012 |
16 | 6924 | Mr. Dipankar Ghosh | Male | IIT Bombay | Mumbai,Maharashtra | PhD | IIT Guwahati | 2011 |
17 | 6946 | Mr. Amit Kumar Singh | Male | Indian Institute of Technology , Madras | Chennai,Tamil Nadu | Ph.D | University of Allahabad | 2011 |
18 | 6952 | Ms Jai Laxmi | Female | Indian Institute of Technology,Mumbai | Mumbai,Maharashtra | Research Scholar | University of Hyderabad | 2012 |
19 | 6990 | Mr. Bibekananda Mishra | Male | Indian Statistical Institute, Kolkata | Kolkata,West Bengal | Junior Research Fellow | The university of Warwick, UK | 2013 |
20 | 6999 | Mr. Navaneeth C C | Male | Chennai Mathematical Institute | Chennai,Tamil Nadu | MSc Student | Chennai Mathematical Institute | Appeared/ Awaiting Result |
21 | 7001 | Mr Suhas B N | Male | IIT MADRAS | Chennai,Tamil Nadu | Research Scholar | BANGALORE UNIVERSITY | 2010 |
22 | 7014 | Mr. Naveen Kumar | Male | Chennai Mathematical Institute (CMI-Chennai) | Chennai,Tamil Nadu | PhD | UM-DAE-Centre for Excellence in Basic Sciences, Mumbai | 2014 |
23 | 7030 | Mr Mohan R | Male | Indian Statistical Institute Bangalore Cenater | Bangalore,Karnataka | PhD Student | Indian Institute of Technology Madras | 2012 |
24 | 7043 | Mr. Abhishek Gupta | Male | I.I.T. Kanpur | Kanpur,Uttar Pradesh | M.Sc. Student | I.I.T. Kanpur | Appeared/ Awaiting Result |
25 | 7131 | Mr. Sourav Das | Male | Chennai Mathematical Institute | Chennai,Tamil Nadu | PhD | Indian Institute of Technology Guwahati | 2012 |
26 | 7138 | Mr Samir Kumar Hazra | Male | Chennai Mathematical Institute | Chennai,Tamil Nadu | PhD | IIT BOMBAY | 2011 |
27 | 7153 | Mr Praphulla Kumar Koushik | Male | IIT, Bombay | Mumbai,Maharashtra | PhD | University Of Hyderabad | 2013 |
28 | 7156 | Mr. Girish Murlidhar Kulkarni | Male | Indian Institute of Science Education and Research Pune | Pune,Maharashtra | PhD | IIT Madras | 2010 |
29 | 7057 | Mr. Debaditya Raychaudhury | Male | Chennai Mathematical Institute | Chennai,Tamil Nadu | MSc Student | Chennai Mathematical Institute M.Sc | Appeared/ Awaiting Result |
30 | 7058 | Mr Jayan Mukherjee | Male | Chennai Mathematical Institute | Chennai,Tamil Nadu | Msc Student | Chennai Mathematical Institute | Appeared/ Awaiting Result |
31 | 7075 | Mr. Sujoy Chakraborty | Male | Chennai Mathematical Institute | Chennai,Tamil Nadu | MSc Student | Chennai Mathematical Institute | Appeared/ Awaiting Result |
How to reach
Kerala School of Mathematics (KSOM) is located at a distance of about 15 Kms from the city centre (train/bus stations) situated off the Kozhikode-Medical College-Kunnamangalam bus route.Prepaid auto counter is available at Railway Station itself
(Near 1st number Platform).