Advanced Training School in Mathematics for Lecturers (ATML) in Complex Analysis

Venue: BIM, Pune
Dates: 14 - 26 May 2007

 

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

 

School Convener(s)

Name A. R. Shastri (Academic)
S. A. Katre (Local)
Mailing Address

Dept of Mathematics,
IIT, Bombay, Mumbai-400076

Dept. of Mathematics,
University of Pune

Advanced Training School in Mathematics for Lecturers (ATML) in Complex Analysis is being organised in Pune at Bhaskaracharya Pratishthana in May 2007 on behalf of NBHM.

 National Coordinating Committee
Director: Ravi S. Kulkarni
Secretary: J. K. Verma
Members : S. D. Adhikari, Satya Deo, S. A. Katre, Shobha Madan, I. B. S. Passi, Ravi A. Rao

Members of the Local Organising Committee

A. R. Shastri, IIT Bombay (Academic Convener)
S. A. Katre, Univ. Pune (Local Coordinator)
C. S. Inamdar, Pune (Custodian, BP)
R. V. Gurjar, TIFR Hon. Res. Director, BP
Manjusha Joshi, BP  

 

Speakers and Syllabus 

The topics to be covered (numbers in the bracket indicate the the number of lectures allotted):

I- week
1.1 Basic properties of complex numbers:historical remarks, why the name complex etc.(1)
2.1 Review of Diff. Calculus of 2-variables.(1)
1.2 Geometry of complex numbers: rigid motions of the plane.(1)
2.2 Complex differentiability; CR equations.(1)
1.3 Sequences and series:uniform convergence, Weierstrass' M-test.(1)
3.1 Line integrals, M-L inequality; diff. under integral sign(1)
1.4 Formal power series, convergent power series, analytic functions; Exponential, trigonometric, and logarithmic functions; complex exponents.(1)
3.2 Path-independence; exactness, existence of primitives.(1)
1.5 Extended complex plane: FLTs.(1)
3.3 Cauchy-Goursat for triangles; Cauchy's theorem, Integral formula, estimates, Liouville's theorem. (1)
1.6 Conformality: applications.(1)
3.4 FTA, Taylor's theorem and analyticity, Zeros of holomorphic functions; Identity theorem.(1)

 

II-week
3.5 Isolated singularities.(1)
4.1 Winding number.(1)
3.6 Laurent series and residues.(1)
4.2 Argument principle, local solutions, open mapping theorem.(1)
2.3+2.4 Simple connectivity, homology and homotopy form of Cauchy's theorem.(2) 4.3 Inverse function theorem, branch theorem, Maximum Modulus Principle, Schwartz lemma.(1)
4.4 Residues, GAP, Rouche's theorem.(1)
4.5+4.6 Applications to improper integrals.(2)
2.5+2.6 Convergence in function theory, Runge's theorem with some applications. (2)

Special Lectures:
1. Application of Complex Analysis in Fluid Dynamics. (VDS)
2. Gamma function, Riemann zeta function. (GKS)
3. Gaussian Sums. (MN)

 

I-week:14th May to 19th May
KDJ: 1.1-1.6; ARS: 2.1,2.2; SAK: 3.1-3.4
Day Mon Tue Wed Thu Fri Sat
9.30-11.00 KDJ KDJ KDJ KDJ KDJ KDJ
11.30-1.00 ARS ARS SAK SAK SAK SAK
1.00-2.30 Lunch Break
2.30-4-30 Tutorial Tutorial Tutorial Tutorial Tutorial Tutorial
II-week:21th May to 26th May
SAK: 3.3-3.6; HB: 4.1-4.6; ARS: 2.3-2.6
Day Mon Tue Wed Thu Fri Sat
9.30-11.00 HB HB HB HB HB HB
11.30-1.00 SAK SAK ARS ARS ARS ARS
1.00-2.30 Lunch Break
2.30-4-30 Tutorial Tutorial Tutorial Tutorial Tutorial Tutorial
5.00-6.00   VDS GKS GKS MN  

 

  • Resource Persons:
  1. Speakers
    Prof. K. D. Joshi kdj at math.iitb.ac.in
    Prof. S. A. Katre sakatre at math.unipune.ernet.in
    Dr. H. Bhate hbhate at math.unipune.ernet.in
    Prof. A. R. Shastri ars at math.iitb.ac.in
  2. Invited Speakers
    V. D. Sharma, IIT Bombay:  Application of Complex Analysis in Fluid Dynamics
     Gopal Srinivasan, IIT Bombay: Gamma function, Riemann zeta function
      Mangala Narlikar, BIM: Gaussian Sums
  3. Course Associates
    1. Sameer Chavan, IISER, Pune. sameer_chavan95 at yahoo.co.in
    2. Pratul Gadagkar, Univ. of Pune, Pune pratul1980 at math.unipune.ernet.in
    3. Vinay Wagh, IIT, Bombay, Mumbai vinay at math.iitb.ac.in
  • Syllabus
  • Basic algebraic and geometric properties of complex numbers, rigid motions. Open sets, Connectivity and path connectivity.
  • Complex differentiability, CR-equations, holomorphic anti-holomorphic functions, operators  harmonic functions.
  • Review of sequences and series, Uniform convergence, Weierstrass' M-test, power series, Analytic functions, exponential and trigonometric functions logarithmic function, complex exponents of complex numbers.
  • Contour integration, existence of primitives, Cauchy-Goursat theorem, winding number, Cauchy's theorem, Cauchy's integral formulae, Morera's theorem.
  • Local behaviour of Analytic functions, Taylor series. zeros of analytic functons, identity theorem, Analysis of isolated singularities. Poles and residues. Laurent Expansion, Casorati Weierstrass' theorem. Open mapping theorem. Maximum modulus principle. Schwarz' lemma.
  • Runge's Approximation Theorem, Generalization of Cauchy's theorem, Argument principle, simple connectivity, Residue theorem, Integral formula for Inverse function, Holomorphicity of Symmetric functions.
  • Computation of real integrals.
  • Convergene in function Theory, Hurwitz theorem, Mittag-Leffler theorem, partial fraction development, infinite products, Weierstrass's theorem, Gamma-function and Riemann zeta-function.
  • Normal families, Equicontinuity, Arzela-Ascoli, Montels theorem. Riemann mapping theorem.
  • Harmonic functions, Poisson Integral, Schwarz's reflection principle, Harnack' principle, Sub-harmonicity, Dirichlet's problem, Greens's function, another proof of Riemann mapping theorem.
  • References:
  1. Ahlfors, V., Complex Analysis MacGraw-Hill,
  2. Remmert, R., Theory of Complex Functions, GTM-122 Springer.
  3. Rudin, W., Real And Complex Analysis, MacGraw-Hill 2nd ed.
  4. Shastri A. R., An introduction to Complex Analysis, MacMillan India, 1999.

 
 
Selected Applicants
Sr. No. Name Place Accommodation: H-Rooms at Guest House of BP, R-Rooms at main building of BP, Flat-is flat near BP.
1. Dr. Arshad Khan Jamia Millia Islamia, New Delhi H-3
2. Priyabrat Gochhayat Berhampur University, Orissa. R-5
3. S. Balaji The M.D.T. Hindu College, Tirunelveli H-2
4. Phatangare Nanasaheb M. New Arts, Comm. Sci. College, Ahmednagar, Maha. Flat
5. Aher Bhalchandra Yadav Arts, Comm. Sci. College, Ozar Mig, Nashik, Maha. Flat
6. Ratan Kumar Dutta Dept. of Math. Visva-Bharati, Santiniketan R-5
7. Saiful Rahman Mondal IIT Roorkee H-3
8. Ms. Hemlata Pawar Dr. D. Y. Patil Institute of Engineering and Technology, Pune Local
9. Ms. Vandna Agnihotri C.S.J.M. University, Kanpur R-1
10. Dr. Uday Hanmant Naik Willingdon College, Sangli, Maha. H-4
11. Shivaji Ashok Tarate N.A.C. Sc. College, Ahmednagar, Maha. Flat
12. Bibhas Chandra Mondal Surendranath College, Kolkata R-5
13. Trilok Mathur Banasthali Vidyapith, Jaipur H-2
14. Faizan Ahmad Khan Women's College, A.M.U. Aligarh H-3
15. Vinod I. Sharma Sardar Patel College, Engineering, Mumbai, Maha. R-5
16. Pravin Garg Banasthali Vidyapith, Newai Rajasthan H-2
17. Umesh Rajdhani College, University of Delhi, Delhi R-5
18. Sachin D. Patil Jaysingpur College, Jaysingpur Flat
19. Ms. Sandhya Singhal Deen Dayal Upadhaya College, New Delhi R-1
20. Dr. M. Marudai The M.D.T. Hindu College, Tirunelveli H-4
21. S. Mathugnaniah St. Xavier's College, Palayamkottai H-4
22. Patil Dattatraya Sangamner college, Sangamner, Maha. Flat
23. Dr. P. Sam Johnson SSN College of Engineering, Chennai Flat
24. Ms. Shinde Akanksha K. T. H. M. College, Nasik, Maha. R-2
25. J. Maria Joseph Arun Anandar College, Tiruvanna Malai Flat
26. Ms. Asha L. Mangalani C.H.N. College, Ulhas Nagar, Dist. Thane, Mah. R-1
27. Sanjay Kumar IIT Bombay, Mumbai R-5
28. Karamjit Singh B.P.R. College, Kuruksketra, Hariyana R-5
29. Javid Ali A.M.U. Aligarh H-3
30. Jayanta Borah Itanagar, Arunachal Pradesh H-2
31. Vinay Arora S.S.H. Panjab U.R.C., Hoshiyarpur, Panjab R-5
32. Kuldeep Sharma S.S.H. Panjab U.R.C., Hoshiyarpur, Panjab R-5
33. Ms. Manju Prasad AISS M S women college of Engineering, Pune, Maha. Local
34. Ms. Hurratulmalika Juzer Siamwalla Abeda Inamdar Sr. College, Pune, Maha. Local
35. Dr. Krishna Kumar JSPM's Imperial College of Engg. Research, Wagholi, Pune, Maha. Local

 

How to reach

http://www.bprim.org/

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