IST Complex Analysis and Analytic Number Theory (2018)

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Venue: SRTM University,  Nanded, Maharashtra
Date:  29th, Jan 2018 to 10th, Feb 2018

 

School Convener(s)

Name K. Srinivas Usha Sangale
Mailing Address Institute of Mathematical Sciences,
4. CIT Campus, Taramani, Chennai 600113.
School of Mathematical Sciences,
SRTM University, Vishnupuri, NANDED, 431606, Maharashtra.

 

Speakers and Syllabus 

syllabus covered in the IST programme
There were two themes running parallel in this programme: Analytic Number Theory and Complex Analysis. The following is the actual syllabus covered in this school.

  • Analytic number theory course covered by Usha Sangale (4), Kasi Viswanad-ham (2).
    • The Möbius function μ(n), Euler totient function φ(n), Mangoldt function Λ(n), Lioville’s function λ(n) and a few other counting functions were defined, some important identities P satisfied by these functions were derived. Elementary ways of finding a sum formula (e.g., n≤x d(n) = x log x + O(x)) for these arithmetical functions were discussed. Abel summation formula, Euler’s summation formula were derived. Emphasis was given on the application of these formulas to obtain a better error term in the sum formulas for arithmetical functions. Chebyschev type estimates were derived. Prime counting functions and their equivalence were discussed. Riemann zeta-function, its analytic continuation. Dirichlet characters were introduced, their orthogonality relations were derived.
  • Complex analysis (I) course covered by K. Srinivas (6).
    • Power series, its radius of convergence, uniqueness of power series, differentiability of power series, analytic functions, Cauchy-Riemann conditions, complex integration,  auchy’s theorem (statement only), Cauchy’s integral formula, analytic functions expressible as power series (Taylor series), Laurent series expansion, discussion of zeros and singularities of analytic functions.
  • Complex analysis (II) course covered by Gautam Bharali (6).
    • The second week of lectures on Complex Analysis were devoted to the following broad themes: Applications of the Residue Theorem, with an emphasis on computing/estimating improper Riemann integrals via contour integration (which also supplements some of the material being presented in the Number Theory lectures). Geometric aspects of holomorphic functions.
    • A description of the topics covered follows: How to compute residues; the winding number; a few examples of improper Riemann integrals. The meaning of the logarithm; the significance of branch points and branch cuts; the holomorphicity of (local) inverses of a holomorphic function; the principal branch of the logarithm. Integration around a branch point; transformations of improper integrals involving integration around a branch point to equivalent forms. The stereographic projection and the Riemann sphere; Möbius transformations; the action of Möbius transformations on circles. Biholomorphic maps and conformality; the automatic holomorphicity of the inverse of an injective holomorphic map; examples. The (holomorphic) automorphism group of a domain; computation of the automorphic group for the open unit disc and the complex plane; the Riemann Mapping Theorem (statement only).
  • Analytic number theory (II) course covered by K. Srinivas (6).
    • The main theme of the course in the second week was to give a self contained proof of prime number theorem and Dirichlet’s theorem on primes in arithmetical progression.
    • The following topics were covered: Euclid’s proof of infinitude of primes, its extension to certain arithmetic progressions and failure to some other arithmetical progressions, Perron’s formula and its importance. Discussion on the zeros of Riemann zeta-function, its order in the critical strip, zero-free regions, non-vanishing of ζ(1 + it), proof of prime number theorem. Statement of Dirichlet theorem and its proof.
  • Tutorial sessions.
    • The tutors Kasi Viswanadham, Jaykumar and Karthick Babu proved to be indispensable. They were always there, in the lectures and in the tutorials. They prepared problem sheets for each day and ensured that each and every one does the problems given to them. The discussions went beyond the schedule. Participants showed no sign of leaving the lecture hall and going back to their room. The tutors did a splendid job.

 
Time Table

First Week : 29th Jan to 3rd Feb 2018

Day

9.30-11.00

11.00–11.30

11.30–1.00

1.00–2.30

2.30–3.30

3.30–3.50

3.45–4.45

4.45–5.00

Mon

US

Tea

KS

Lunch

US/KV/KB

Tea

KS/JK/KV

Snacks

Tue

KV

Tea

KS

Lunch

KV/US/KB

Tea

KS/JK/KV

Snacks

Wed

US

Tea

KS

Lunch

US/KV/KB

Tea

KS/JK/KV

Snacks

Thu

KS

Tea

US

Lunch

KS/JK/KB

Tea

US/KV/KB

Snacks

Fri

KV

Tea

KS

Lunch

KV/US/JK

Tea

KS/KB/JK

Snacks

Sat

US

Tea

KS

Lunch

US/KV/KB

Tea

KS/JK/KV

Snacks

 Second Week : 5th Feb to 10th Feb 2018

Day

9.30–11.00

11.00–11.30

11.30–1.00

1.00–2.30

2.30–3.30

3.30–3.50

3.45–4.45

4.45–5.00

Mon

GB

Tea

KS

Lunch

GB/JK/KB

Tea

KS/KB/US

Snacks

Tue

GB

Tea

KS

Lunch

GB/JK/KB

Tea

KS/US/JK

Snacks

Wed

GB

Tea

KS

Lunch

GB/JK/KB

Tea

KS/US/KB

Snacks

Thu

GB

Tea

KS

Lunch

GB/JK/KB

Tea

KS/US/JK

Snacks

Fri

GB

Tea

KS

Lunch

US/JK/KB

Tea

GB/JK/KB

Snacks

Sat

GB

Tea

KS

Lunch

KS/JK/KB

Tea

US/JK/KB

Snacks


 

Actual Participants 

 

Sr. Name of the Participant SID Status Affiliation
1 Dr.Nintu Mandal 16289 Assistant Professor Chandernagore College strand road, Chandernagore, Hooghly, PIN-712136,West Bengal
2 Dr. Pulak Sahoo 16235 Assistant Professor Department of Mathematics
University of Kalyani
Kalyani, Nadia
WB-741235
3 Dr. Samten Tamang 16271 Assistant Professor The University of Mathematics,
Golapbag, Burdwan,
West Bengal, India
4 Mr. Mohd. Shadab 16312 PhD Scholar Jamia Millia Islamia Maulana Mohammad Ali Jauhar Marg, New Delhi, 110025
5 Ms. Pooja Raut 16406 Student Swami Ramanand Teerth Marathawada University
6 Ms. Radhika Baldawa 16581 Assistant Professor Dayanand Science College,Near Shivaji Chowk Khadgaon Road,LATUR-413512
,Maharashtra
7 Ms. Jita Parida 16515 Lecturer Department of Science and Humanity
College of Engineering Bhubaneswar-751024
Patia, Orissa
8 Mr. Mayavel P 16244 Assistant Professor GKM College of Engineering And Technology New perungalathur, Chennai , Tamil Nadu , India , PIN: 600063
9 Mr. Chaitanya G.K. 16593 PhD Student National Institute of Technology, Karnataka, P. O. Srinivas Nagar, Surathkal Mangalore-575025 Karnataka
10 Ms. Sehra Sahu 16567 PhD Student National Institute of Technology, Karnataka, P. O. Srinivas Nagar, Surathkal Mangalore-575025 Karnataka
11 Mr. Sumukha S. 16568 PhD Student National Institute of Technology, Karnataka, P. O. Srinivas Nagar, Surathkal Mangalore-575025 Karnataka
12 Mr. Muhammad Ismail Ansari 16613 Lecturer NES Science College, Nanded
13 Dr. Halle Dattu 16333 Assistant Professor Rajiv Gandhi University of knowledge technologies Basar-504107
Nirmal Dist Telangana
14 Mr. Mane Parshuram S. 16527 Mphil Student Annasaheb Dange college of engineering and technology, ashta Sangli- Islampur road,
Tal-walwa Dist-sangli.
Pin-416301
15 Dr. Metkar R.G. 16526 Assistant Professor Indira Gandhi (Sr.) College Cidco, New Nanded -431603,Maharashtra
16 Mr. Patil Mallikarjun 16548 Assistant Professor Indira Gandhi (Sr.) College Cidco, New Nanded -431603,Maharashtra
17 Mr. Subhasis Panda 16228 PhD Student Indian Institute of Technology Madras,
Sardar Patel Road, Opposite to C.L.R.I. Chennai-600036
18 Mr. Amit Sharma 162242 Assistant Professor Pratap Institute Of Technology And Science
Akhaipura , Rajasthan , India , PIN: 332402
19 Mr. Karande Yogesh N. 16578 Assistant Professor School Of Mathematical Sciences S.R.T.M.U. Nanded-431606, Maharashtra
20 Mr Ramdas V. Madan 16537 Assistant Professor moreshor college bhokardan-431203 Jalna, Maharashtra
21 Mr. Akshay Abhayrao Chede 16639 Assistant Professor Ankushrao Tope College Jalna-431203,Maharashtra
22 Mr. Biradar Mahesh 01 MPhil Student SRTM University, Nanded-431606, Maharashtra
23 Mr. Navghare Tukaram 02 Student SRTM University, Nanded-431606, Maharashtra
24 Mr. Sagar Sunil Agrawal 16561 Assistant Professor MGM College Of Computer Science And IT, P-13,MIDC Area, Parbhani-431401 Maharashtra
25 Mr. Shaikh Wajid Majid 16610 PhD Student Swami Ramanand Teerth Marathwada University 431606,Maharashtra
26 Ms. Asha R. Meshram 16367 Assistant Professor Netaji Subhash Chandra Bose college Tarasingh market, Nanded-431601,MH
27 Ms. Rupali R. Meshram 16459 Student AT Subhash Ward Bela PO Bela Tah DIST Bhandara
28 Mr. Malegaonkar Swapnil D. 16580 Assistant Professor Department of mathematics Rajarshi Shahu Mahavidyalay Hanuman Chauk Latur-413512,Maharashtra
29 Mr. Aziz Ahmed 03 Assistant Professor School of Engineering,MPGI Nanded
30 Ms. Agrawal Poonam K. 16577 PhD Student Swami Ramanand Teerth Marathwada University 431606,Maharashtra
31 Dr. Archana Bhosle 04 Assistant Professor Nagnath ACS College, Aundha Naghnath, Nanded, Maharashtra
32 Mr. Sayyed Jalil S Yousouf 16460 Assistant Professor Hutatma Jaywantrao Patil Mahavidyalaya Himayatnagar, Dist:Nanded. 431802,MH
33 Mr. Sharad B. Kadam 16629 Assistant Professor Sunderrao Solanke Mahavidyalaya Majalgaon431131 Maharashtra
34 Mr. Shaikh Javed Shafee 16623 Lecturer Government Polytechnic Hingoli-431513
35 Mr. N. S. Darkunde 16477 Assistant Professor School of Mathematical Sciences, SRTM University, Nanded-431606,MH.
36 Mr. Bindumadhaba Mohanty 16601 Lecturer Bhima Bhoi College, Rairakhol, sambalpur 751003,Orissa
37 Mr. Jayakumar R. 05 PhD Student Institute of Mathematical Sciences, Chennai
38 Mr. Karthick Babu 06 PhD Student Institute of Mathematical Sciences, Chennai
39 Ms. More Rohini B. 07 MSc Student SRTM University, Nanded-431606,Maharashtra
40. Mr. Akshay P. Sawant 08 MSc Student SRTM University, Nanded-431606,Maharashtra
41. Mr. Sandip U. Shinde 09 MSc Student SRTM University, Nanded-431606,Maharashtra
42 Mr. Shaikh Feroz Kalam 10 MSc Student SRTM University, Nanded-431606,Maharashtra
43. Mr. Gajanan N. Chavan 11 MSc Student SRTM University, Nanded-431606,Maharashtra
44. Mr. Satish Thakare 12 MSc Student SRTM University, Nanded-431606,Maharashtra
45. Mr. Khose Shubham 13 MSc Student SRTM University, Nanded-431606,Maharashtra
46. Mr. Indrajeet Gunjal 14 MSc Student SRTM University, Nanded-431606,Maharashtra
47. Ms. Snehal Sable 15 MSc Student SRTM University, Nanded-431606,Maharashtra
48. Ms. Shinde Indutai 16 MSc Student SRTM University, Nanded-431606,Maharashtra
49. Mr. Nilkanth Rajamwad 17 MSc Student SRTM University, Nanded-431606,Maharashtra
50. Ms. Rakhi P. Parve 18 MSc Student SRTM University, Nanded-431606,Maharashtra
51. Ms. Rajani Patil 19 MSc Student SRTM University, Nanded-431606,Maharashtra
52. Ms. Sneha Patil 20 MSc Student SRTM University, Nanded-431606,Maharashtra

 


How to reach

Swami Ramanand Teerth Marathwada University (SRTM) is located at Vishnupuri ,Nanded, Maharashtra (www.srtmun.ac.in).

  • By Air:  Shri Guru Gobind Singh Ji Airport, Nanded
    • Nanded is connected by air service to Hyderabad. Daily one flight(Trujet) is available from Hyderbad – Nanded - Hyderabad.
  • By Train: Hazur Sahib Nanded  Railway  Station
    • SRTM University  is about 10-12 kms away from Nanded railway station. This station has two sides namely `Gokul Nagar Side' and `Main Side'. Main side is just beside to Platform no.1. You ask for Main side (do not go to Gokul Nagar side). If you come out of railway station to the main side, you will get auto rickshaw stand. From there one can take an  auto for university.
  • By Road:
    • Nanded is well connected via road to rest of India.

Google map:
https://goo.gl/maps/gEayegWpG792