AIS Schemes & Cohomology (2014)

Venue: Kerala School of Mathematics, Kunnamangalam, Kozhikode
Dates: 1st - 19th Dec, 2014


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name V. Balaji D.S. Nagaraj
Mailing Address Chennai Mathematical Institute,
H1, SIPCOT IT Park, Siruseri,
Institute of Mathematical Sciences,
Taramani, Chennai


Speakers and Syllabus 

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)

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


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).