Venue: |
BP and Pune University. |

Dates: |
1 - 28 June 2006 |

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

Name |
S. A. Katre |

Mailing Address |
University of Pune |

**AFS being organised in Pune in June 2006 is the Second of the 2nd Annual Foundation Schools being organised on behalf of NBHM.**

##### National Coordinating Committee

Director: R. S. Kulkarni

Secretary: J. K. Verma

Members : S. D. Adhikari, Satya Deo, Shobha Madan, I. B. S. Passi, R. A. Rao

#### Members of the Local Organising Committee

**Bhaskaracharya Pratishthana**: C. S. Inamdar (Custodian), R. V. Gurjar (Res. Director, Hon.)

**University of Pune**: B .N. Waphare (HOD, Maths.), S. A. Katre, H. Bhate

Lecture notes and Problems at AFS-II

Algebra

- S. R. Ghorpade
- Balwant Singh
- S. A. Katre

Complex Analysis

- Ravi Kulkarni
- R. R. Simha
- Kaushal Verma

Algebraic Topology

- Anandteertha Mangasuli
- Ravi Kulkarni
- A. Parameswaran

**(Note: lecture notes and problems are available as attachment at end of page)**

**Objectives of AFS**

Basic knowledge in algebra, analysis, discrete mathematics and topology forms the core of all advanced instructional schools the schools to be organized in this programme.

The objective of the Annual Foundation Schools, to be offered in Winter and Summer every year, is two fold:

- To bring up students with diverse backgrounds to a common level.
- To identify those who are fit for further training.

Any student who wishes to attend the advanced instructional schools is strongly encouraged to enroll in the Annual Foundation Schools.

**Format of AFS**

The topics listed in the syllabi will be quickly covered in the lectures. There will be intensive problem sessions in the afternoons. The objective will not be to cover the syllabus prescribed, but to inculcate the habit of problem solving. However, the participants will be asked to study all the topics in the syllabus at home since the syllabi of these schools will be assumed in all the advanced instructional schools devoted to individual subjects.

**Participants in AFS**

These schools will admit 40 students in their first and second years of Ph. D. programme, students of M. Sc. (II Year), university lecturers and college teachers who lack the knowledge of basic topics covered in these schools.

A participant who has attended AFS-I and II will never be allowed to attend these again.

### Syllabus for the Annual Foundation School (AFS)-II (Jun., 2006)

#### Algebra-II

(1) Basic commutative algebra-I: Prime ideals and maximal ideals, Zariski topology, Nil and Jacobson radicals, Localization of rings and modules, Noetherian rings, Hilbert Basis theorem, modules, primary decomposition, integral dependence, Noether normalization lemma, principal ideal theorem, Hilbert's Nullstellensatz, structure of artinian rings, Dedekind domains. (12 lectures)

(2) Introduction to Algebraic Number Theory: (6 lectures)

(3) Introduction to Algebraic Geometry: (6 lectures)

Text/References:

1. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra.

2. D. Eisenbud, Commutative algebra with a view towards algebraic geometry.

3. P. Samuel, Algebraic Number Theory.

#### Complex Analysis

(1) Euclidean similarity geometry, inversive geometry, hyperbolic geometry and complex analysis.

(2) Analytic functions, Path integrals, Winding number, Cauchy integral formula and consequences. Hadamard gap theorem, Isolated singularities, Residue theorem, Liouville theorem.4

(3) Casorati-Weierstrass theorem, Bloch-Landau theorem, Picard's theorems, Mobius transformations, Schwartz lemma, Extremal metrics, Riemann mapping theorem, Argument principle, Rouche's theorem.

(4) Runge's theorem, Infinite products, Weierstrass p-function, Mittag-Leffler expansion.

Text/References:

1. Murali Rao & H. Stetkaer, Complex Analysis, World Scientific, 1991

2. L. V. Ahlfors, Complex Analysis, McGraw-Hill, Inc., 1996

3. A. R. Shastri, Complex Analysis

4. Krantz

5. A. F. Beardon, Geometry of Discrete Groups, GTM Springer Verlag.

#### Algebraic Topology

(1) Basic notion of homotopy; contractibility, deformation etc. Some basic constructions such as cone, suspension, mapping cylinder etc. fundamental group; computation for the circle. Covering spaces and fundamental group. Simplicial Complexes, CW complexes.

(2) Simplicial Complexes, CW complexes. Homology theory and applications: Simplicial homology, Singular homology, Cellular homology of CW-complexes, Jordan-Brouwer separation theorem, invariance of domain, Lefschetz fixed point theorem etc.

(3) Categories and functors; Axiomatic homology theory.

Texts/References:

1. E. H. Spanier, Algebraic Topology, Tata-McGraw-Hill

2. A. Hatcher, Algebraic topology, Cambridge University Press.

Lecture | 9.00 - 10.00 |

Lecture | 10.30 - 11.30 |

Lecture | 11.45 - 12.45 |

Tutorial | 2.15 - 4.15 |

UM Lecture |
4.30 - 5.30 |

#### Syllabus for Annual Foundation School-II (December 2004)

##### Algebra II

(1) **Homological algebra**: Derived functors, projective modules, injective modules, free and projective resolutions, tensor, exterior and symmetric algebras, injective resolutions, Ext and Tor. ( 12 lectures)

(2) **Basic commutative algebra**: Prime ideals and maximal ideals, Zariski topology, Nil and Jacobson radicals, Localization of rings and modules, Noetherian rings, Hilbert Basis theorem, modules, primary decomposition, integral dependence, Noether normalization lemma, principal ideal theorem, Hilbert's Nullstellensatz, structure of artinian rings, Dedekind domains. (12 lectures)

**Text/References**:

1. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra.

2. D. Eisenbud, Commutative algebra with a view towards algebraic geometry.

##### Complex analysis

(1) Analytic functions, Path integrals, Winding number, Cauchy integral formula and consequences. Hadamard gap theorem, Isolated singularities, Residue theorem, Liouville theorem.4

(2) Casorati-Weierstrass theorem, Bloch-Landau theorem, Picard's theorems, Mobius transformations, Schwartz lemma, Extremal metrics, Riemann mapping theorem, Argument principle, Rouche's theorem.

(3) Runge's theorem, Infinite products, Weierstrass p-function, Mittag-Leffler expansion.

**Text**: Complex analysis by Murali Rao & H. Stetkaer, World Scientific, 1991

##### Algebraic topology

(1) Basic notion of homotopy; contractibility, deformation etc. Some basic constructions such as cone, suspension, mapping cylinder etc. fundamental group; computation for the circle. Covering spaces and fundamental group. Simplicial Complexes, CW complexes.

(2) Simplicial Complexes, CW complexes. Homology theory and applications: Simplicial homology, Singular homology, Cellular homology of CW-complexes, Jordan-Brouwer separation theorem, invariance of domain, Lefschetz fixed point theorem etc.

(3) Categories and functors; Axiomatic homology theory.

**Texts/References**

1. E. H. Spanier, Algebraic Topology, Tata-McGraw-Hill

2. A. Hatcher, Algebraic topology, Cambridge University Press.

Number Theory Arithmetic functions, congruences, quadratic residues, quadratic forms, Diophantine approximations, quadratic fields, Diophantine equations.

**Text/References **:

1. A. Baker, Theory of numbers.

2. K. Ireland and M. Rosen, A classical introduction to modern number theory.

#### Algebra II

Balwant Singh (co-ordinator), e-mail: balwantbagga* at *gmail.com (from 12th to 20th June)

Sudhir Ghorpade, e-mail: srg* at *math.iitb.ac.in (from 5th to 14th June)

S. A. Katre, e-mail: sakatre* at *math.unipune.ernet.in (from 1st to 28th June)

#### Complex Analysis

R. R. Simha (co-ordinator), e-mail: simhahome* at *yahoo.com (from 12th to 20th June)

Ravi Kulkarni, e-mail: kulkarni* at *mri.ernet.in (from 3rd to 28th June)

Kaushal Verma , e-mail: kverma* at *math.iisc.ernet.in (from 20th to 28th June)

#### Algebraic Topology

Anandateertha Mangasuli, e-mail: anandateertha *at *gmail.com (from 1st to 17th June)

Ravi Kulkarni, e-mail: kulkarni* at *mri.ernet.in (from 3rd to 28th June)

A. J. Parameshwaran (co-ordinator), param* at *math.tifr.res.in (from 3rd-5th, 19th to 28th June )

#### Special Lecture Series (Unity of Mathematics Lectures)

DINESH THAKUR, e-mail: thakur* at *math.arizona.edu

**Associates:**

**Algebra II**

Anuradha Gadre (ganura* at *math.unipune.ernet.in)

Himanee Apte (himanee* at *math.bprim.org)

Habeeb Basha (habeeb* at *math.iitb.ac.in)

**Complex Analysis**

Sameer Chavan (chavansameer* at *gmail.com)

Vikram Aithal (vikram* at *mri.ernet.in),

Dr. Sanjay Kumar Pant (sanjpant* at *yahoo.co.in)

**Algebraic Topology**

Vikram Aithal (vikram* at *mri.ernet.in), and others

Sameer Chavan (chavansameer* at *gmail.com)

Dr. Sanjay Kumar Pant (sanjpant* at *yahoo.com)

**Syllabus**

#### Algebra-II

(1) Basic commutative algebra-I: Prime ideals and maximal ideals, Zariski topology, Nil and Jacobson radicals, Localization of rings and modules, Noetherian rings, Hilbert Basis theorem, modules, primary decomposition, integral dependence, Noether normalization lemma, principal ideal theorem, Hilbert's Nullstellensatz, structure of artinian rings, Dedekind domains. (12 lectures)

(2) Introduction to Algebraic Number Theory: (6 lectures)

(3) Introduction to Algebraic Geometry: (6 lectures)

Text/References:

1. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra.

2. D. Eisenbud, Commutative algebra with a view towards algebraic geometry.

3. P. Samuel, Algebraic Number Theory.

** Complex Analysis**

(1) Euclidean similarity geometry, inversive geometry, hyperbolic geometry and complex analysis.

(2) Analytic functions, Path integrals, Winding number, Cauchy integral formula and consequences. Hadamard gap theorem, Isolated singularities, Residue theorem, Liouville theorem.4

(3) Casorati-Weierstrass theorem, Bloch-Landau theorem, Picard's theorems, Mobius transformations, Schwartz lemma, Extremal metrics, Riemann mapping theorem, Argument principle, Rouche's theorem.

(4) Runge's theorem, Infinite products, Weierstrass p-function, Mittag-Leffler expansion.

Text/References:

1. Murali Rao & H. Stetkaer, Complex Analysis, World Scientific, 1991

2. L. V. Ahlfors, Complex Analysis, McGraw-Hill, Inc., 1996

3. A. R. Shastri, Complex Analysis

4. Krantz

5. A. F. Beardon, Geometry of Discrete Groups, GTM Springer Verlag.

** Algebraic Topology**

(1) Basic notion of homotopy; contractibility, deformation etc. Some basic constructions such as cone, suspension, mapping cylinder etc. fundamental group; computation for the circle. Covering spaces and fundamental group. Simplicial Complexes, CW complexes.

(2) Simplicial Complexes, CW complexes. Homology theory and applications: Simplicial homology, Singular homology, Cellular homology of CW-complexes, Jordan-Brouwer separation theorem, invariance of domain, Lefschetz fixed point theorem etc.

(3) Categories and functors; Axiomatic homology theory.

Texts/References:

1. E. H. Spanier, Algebraic Topology, Tata-McGraw-Hill

2. A. Hatcher, Algebraic topology, Cambridge University Press.**Schedule of Lectures**

Lecture | 9.00 - 10.00 |

Lecture | 10.30 - 11.30 |

Lecture | 11.45 - 12.45 |

Tutorial | 2.15 - 4.15 |

UM Lecture |
4.30 - 5.30 |

**Schedule of Lectures / Tutorials**

1st Jun., Thur | 2 nd Jun.,Fri | 3rd Jun.,sat | ||

9.00 - 10.00 | Registration | 9.30-11.00 | Algebra-II | Algebra-II |

10.00 - 10.30 | Inauguration | Tea | ||

10.45 - 11.45 | Algebra-II | 11.30- 1.00 | AlgTopo | AlgTopo |

12.00 - 1.00 | AlgTopo | |||

Lunch | ||||

2.30 - 3.30 | Alg (SAK) | Alg (SAK) | Alg (SAK) | |

3.45 - 4.45 | AlgTopo (AM) | AlgTopo (AM) | AlgTopo (AM) |

5th June, Mon | 6th,Tue | 7th, Wed | 8th, Thur | 9th, Fri | 10th, Sat | |

9.30 - 10.30 | SRG | SRG | SRG | SRG | SRG | SRG |

10.30 - 10.45 | Tea | |||||

10.45 - 11.45 | RK (Comp) | RK | RK | RK | RK | AM |

12.00 - 1.00 | AM | AM | AM | AM | RK | SRG |

Lunch | ||||||

2.30 - 4.30 | Problem session: ALG | Comp | TOP | ALG | Comp | TOP |

12th, June, Mon | 13th,Tue | 14th, Wed | 15th, Thur | 16th, Fri | 17th, Sat | |

9.30 - 10.30 | SRG | BS | BS | BS | BS | BS |

10.30 - 10.45 | Tea | |||||

10.45 - 11.45 | DT | RRS | RRS | RRS | RRS | RRS |

12.00 - 1.00 | BS | RK (ALG TOP) | RK | RK | RK | RK |

Lunch | ||||||

2.30 - 4.30 | Problem session: ALG | Comp | TOP | ALG | Comp | TOP |

4.45 - 5.45 | RRS | DT | Prof. Passi | Dr. Pant |

19th, June, Mon | 20th,Tue | 21th, Wed | 22th, Thur | 23th, Fri | 24th, Sat | |

9.30 - 10.30 | BS | BS | SAK | SAK | SAK | SAK |

10.30 - 10.45 | Tea | |||||

10.45 - 11.45 | RRS | RRS | KV | KV | KV | KV |

12.00 - 1.00 | RK | KV | AP | AP | AP | AP |

Lunch | ||||||

2.30 - 4.30 | Problem session: ALG | Comp | TOP | ALG | Comp | TOP |

4.45 - 5.45 | Dr. Pant | AP | DT | DT |

26th June, Mon | 27th,Tue | 28th, Wed | |

9.30 - 10.30 | SAK | DT | KV |

10.30 - 10.45 | Tea | ||

10.45 - 11.45 | KV | KV | AP |

12.00 - 1.00 | AP | AP | TOP |

Lunch | |||

2.30 - 4.30 | Problem session: ALG | Comp | FeedBack Session |

Valedictory Function | |||

4.45 - 5.45 | DT |

Algebra | BS- Balwant Singh | SRG- Sudhir Ghorpade | SAK- S. A. Katre |

Complex Analysis | RRS- R.R. Simha | RK- Ravi Kulkarni | KV- Kaushal Verma |

Algebraic Topology | AM- Anand Mangasuli | RK- Ravi Kulkarni | AP- Parameswaran |

UM-Lecture Series | DT-Dinesh Thakur | ||

Guest Lecture | Early History of Groups | Prof. Passi |

**Associates**

Algebra | Vijay Patankar | Anuradha Garge, | Himanee Apte | Habeeb Basha |

Complex Analysis | Vikram Aithal | Sanjay Pant | Sameer Chavan | |

Algebraic Topology | Vikram Aithal | Sanjay Pant | Sameer Chavan |

Selected Applicants |

Sr. No. |
Name of Participant |
AccommodationH- BP Guest House, F- BP Flat, R-Rooms at BP main building |
Registration |

1 | Debasis Sen, ISI, Kolkata | F-3 | yes |

2 | Vineet Kumar Singh, Banaras Hindu Univ.,Varanasi. | H-6 | yes |

3 | Moriya Bhavin K., HRI, Allahabad. | R-5 | yes |

4 | Amrutiya Sanjaykumar Hansraj, HRI, Allahabad. | R-5 | yes |

5 | Ms. Kamlesh Tiwari, Chouksey Engg. College Bilaspur (C.G.) | R-1 | yes |

6 | Sushil Gorai, IISc, Bangalore. | F-3 | yes |

7 | Dr. N.V. Ramana Murty, Andhra Loyola College, Vijaywada, A.P. | F-2 | yes |

8 | Ms. R. Lakshmi Lavanya, Ramanujan Institute, Univ. of Madras, Chennai. | R-2 | yes |

9 | Musavvir Ali, A.M.U., Aligarh (U.P.). | H-8 | yes |

10 | Saidur Rahman Barbhuiya, Hailakandi, Assam. | H-8 | yes |

11 | Gurpreet Singh, Univ. of Delhi, Delhi. | H-7 | yes |

12 | Varinder Kumar, Univ. of Delhi, Delhi. | H-7 | yes |

13 | Venketasubramanian C. G., Univ. of Hyderabad, Hyderabad. | H-5 | yes |

14 | Vadiraja Bhatta, National Institute of Technology, Karnataka, Surathkal | H-5 | yes |

15 | Ms. Mamta D. Gondalia, M.S.University of Baroda, Baroda. | R-2 | yes |

16 | Chandrajeet Singh Yadav, Ujjain, M.P. | R-5 | yes |

17 | Ms. Mala Parihar, Ujjain, M.P. | R-1 | yes |

18 | Chintamani Mohan Namdev, HRI, Allahabad. | R-5 | yes |

19 | Dr. Bankteshwar Tiwari, B.R.D.P.G. College, Deoria, U.P. | F-1 | yes |

20 | Dhorajia Alpesh Madhubhai, M.S.University of Barodara, Varodara. | H-5 | yes |

21 | Ms. Soma Purkait, ISI-Bangalore, Bangalore. | R-2 | yes |

22 | Ms. Soma Purkait, ISI-Bangalore, Bangalore. | R-2 | yes |

23 | Surya Prasath, IITM, Chennai. | H-5 | yes |

24 | Ms. Parvisha, Univ. of Jammu, Jammu, J & K. | R-1 | yes |

25 | Pabitra Barik, ISIcal, Kolkata. | F-3 | yes |

26 | Jagmohan Tanti, CMI, Chennai. | R-5 | yes |

27 | Dr. Avanish Kumar, Bundelkhand University, Jhansi (UP) | F-1 | yes |

28 | Mr. P. Pradhan, IITM, Chennai. | F-3 | yes |

29 | Ms. Anuradha Namjoshi, Sathye College, Mumbai | Self accom. | yes |

30 | Mr. Dattatraya Patil, A.Nagar, (Mah.) | self Accom. | yes |

31 | Mr. Arun Kumar Patil, Dept. of Maths, IIT Powai, Mumbai | H-1 | yes |

32 | Dr. Naik Uday H., Dept of Maths, Willingdon College, Sangli (Mah.) | F-2 | yes |

33 | Rohit Joshi, IIT, Kanpur. | Self. Accom. | yes |

**List of Local participants for AFS-2.**

1. | Shirolkar Devendra Dinkar, Dept. Maths, Univ. Pune, Pune. | Local | yes |

2. | Ms. Kavita Sutar, BP, Pune | Local | yes |

3. | Priyavrat Charudatta Deshpande, Pune. | Local | yes |

4. | Vikas S. Jadhav, Pune. | Local | yes |

5. | Mrs. Hurratulmalika Juzer Siamwalla, Abida Inamdar College, Pune. | Local | yes |

6. | Ms. Manjusha Joshi, BP, Pune | Local | yes |

7. | Ms. Aditi Marathe, SP, Pune | Local | yes |

**List of Students selected but not able to attend AFS-2.**

1. | Avanish Kumar, Varanasi. | Not attended |

2. | Mr. Gourab Bhattacharya, West Bengal | Not attended |

3. | P. Paramanathan, Gandhigram, Tamilnadu | Not attended |

4. | Susovan Pal, TIFR, Bangalore. | Not attended |

5. | Nilakshi Goswami, Gauhati Univ., Guwahati. | Not attended |

6. | Devendra Kumar, K.U., Kurukshetra. | Not attended |

7. | Indu Pal, K.U., Kurukshetra. | Not attended |

8. | Rahul Garg, A.M.U., Aligarh (U.P.). | Not attended |

9. | Ms. Geetan Khurana, Univ. of Delhi, Delhi. | Not attended |

10. | A. Thangam, Gandhigram Rural Institute, Gandhigram, TamilNadu. | Not attended |

11. | Ratanesh Kumar Dikshit, HRI, Allahabad. | Not attended |

12. | Ms. Meena S. Atak, Maharashtra Academy of Engineering, Alandi, Pune. | Not attended |

13. | Manoj Kumar Savita, I.T. B.H.U., Varanasi. | Not attended |

14. | Amit Kumar Singh, Banaras Hindu University, Varanasi. | Not attended |

15. | Siraj Uddin, A.M.U. Aligarh. | Not attended |

16. | Dr. Trilok Mathur, Banasthali Vidyapith, Rajasthan. | Not attended |

17. | Pravin Garg, Banasthali Vidyapith, Rajasthan. | Not attended |

18. | Sanjay Kumar Singh, ISI, Delhi. | Not attended |

19. | Anoop T. V., HRI, Allahabad | Not attended |

20. | Mr. Advait Kulkarni, Dept. of Maths, IIT Powai, Mumbai | Not attended |

21. | Dr. T. Venkatesh, Karnatak University, Belgaum. | Not attended |

How to reach |