IST Complex Analysis and Analytic Number Theory (2018)
Venue: | SRTM University, Nanded, Maharashtra |
Date: | 29th, Jan 2018 to 10th, Feb 2018 |
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. |
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