ISL Complex Analysis & Number Theory (2013)
| Venue: | CEMS, Almora. |
| Dates: | - |
| Convener(s) | Speakers, Syllabus and Time table | Applicants/Participants |
The oragnisers of the ISL on Complex Analysis and Number Theory, scheduled for the dates 24 June to 6 July 2013 at SSJ Campus, Kumaun University, Almora, regret to announce that this school is postponed on account of inclement weather. Fresh dates for the school will be notified in due course.
Overview:
Complex analysis and Number Theory are integral components of Post Graduate syllabi in Mathematics. The aim of the school is to describe how basic complex analysis intervenes in the study of fundamental properties of L-functions. These functions are central to modern number theory. To keep the background from number theory to a minimum, the lectures will illustrate the general theory using the Riemann Zeta function, Dirichlet L-functions and simple examples of Dedekind zeta functions. Preliminaries from complex analysis will be developed through short courses on the Gamma function, Mellin transform, Poisson-Jensen formula and its consequences and Subharmonic Functions. The short courses on L-functions will cover functional equations, convexity bound in the critical strip, asymptotics for zeros in the critical strip, zero-free regions, prime number theorem and the explicit formula.
| Name | Hoshiyar Singh Dhami |
Sanjay Kumar (Pant) | D. Surya Ramana |
| Mailing Address | Professor and Head, Department of Mathematics, SSJ Campus, Kumaun University Almora, Uttarakhand-263601 |
Associate Professor, Department of Mathematics, Deen Dayal Upadhyaya College,New Delhi-110015 |
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019. |
Speakers and Syllabus:
| Sr. | Speaker | Topic & Code | Lectures |
|
| 1 | Sanjay Kumar Pant DDU College, Delhi University. |
The Gamma Function.— The properties of this special function as a meromorphic function on the complex plane are basic to any coverage of analytic theory of L-functions. The course will begin with the definition of the Gamma function as an integral, discuss its functional equation, analytic continuation and characterisation by Wielandt. From this chracterisation various standard representations of the Gamma function will be obtained, as also Striling’s formula in the complex form. Thereafter, key bounds for the ratio of Gamma functions and the real part of the logarithmic derivative of the Gamma function will be derived. | P1 | 4 Lectures of 1.5 hours each. |
| 2 | D. Surya Ramana HRI Allahabad. |
The Mellin Transform.— The Mellin transform transforms the asymptotic behaviour, in neighbourhoods of 0 and +∞, of a function defined on the positive real numbers into polar data of the transformed function, which is a meromorphic function on a half plane. This property of the Mellin transform makes it the main tool for passing to asymptotics from analytic continuation and vice versa. The course will cover the basic properties of the Mellin transform and, in particular, the important Perron’s formula. A key application of this formula is the Prime Number Theorem, which is the theme of a separate course. We will see several other examples of the use Mellin transform and Perron’s formula in the present course. | P2 | 4 Lectures of 1.5 hours each |
| 3 | M. Ram Murty Queens University. |
Poisson-Jensen formula and its consequences.— Starting from a proof of the Poisson-Jensen formula, we shall obtain various frequently used results such as the real parts theorem of Borel-Caratheodory as well as Hadamard’s theorem on entire functions and discuss some applications. | P3 | 3 Lectures of 1.5 hours each |
| 4 | Ravi Raghunathan IITB, Mumbai. |
Subharmonic Functions.— The purpose of this set of lectures is to expose the version of the ¨Phragmen-Lidelof principle due to Rademacher. This is done by means of basic results on subharmonic functions and the solution of the Dirichlet problem for a strip. We shall apply these results to obtain convexity bounds for L-functions, and their central values, in their critical strips. | P4 | 4 Lectures of 1.5 hours each |
| 5 | R. Thangadurai HRI Allhabad. |
Preliminaries from Algebraic Number Theory.— This course will summarise the properties of Dirichlet characters and number fields required for the study of the zeta functions associated to them presented later. | P5 | 3 Lectures of 1.5 hours each |
| 6 | R. Balasubramanian IMSc Chennai. |
Analytic Continuation and Functional Equation.— We will show how the Riemann zeta function and Dirichlet L-functions may be analytically continued and also give two proofs of their functional equations : first by means of the Hankel contour and second the Poisson formula. The results for Dedekind zeta functions will then be stated. | M1 | 3 Lectures of 1.5 hours each |
| 7 | Ritabrata Munshi TIFR Mumbai. |
Zero-Free regions.— The classical zero-free regions for the Riemann zeta function and the Dirichlet L-functions shall be derived, keeping the results completely explicit. Then we will pass to the analogue for a general L-function following Iwaniec and Kowalski, Chapter 10. Finally, we will present a fairly up-to-date account of the developments on this topic. This course will depend on the contents of P3 and M1. | M2 | 5 Lectures of 1.5 hours each |
| 8 | Gyan Parakash HRI Allahabad |
Prime Number Theorem.— Using the results established in the courses P2, M1 and M2 we shall obtain a general prime number theorem as given by Iwaniec and Kowalski. We shall then cover the Sielgel-Walfisz theorem. | M3 | 4 Lectures of 1.5 hours each |
| 9 | Tim Browning Bristol University |
Density Bounds.—Using results established in the previous courses we shall describe the so-called density bounds for zero’s of L-functions. | M4 | 4 Lectures of 1.5 hours each |
| 10 | T.D. Browning University of Bristol. |
Course of 4 Lectures. Details to be announced later. |
M5 | 4 Lectures of 1.5 hours each |
Finally, we detail the special lectures.
1. S1 and S2 by R. Balasubramanian, IMSC, Chennai shall describe the importance of the Riemann Hypothesis.
2. S5 and S6 by B. Ramakrishnan, HRI, Allahabad shall introduce the L-function associated to a modular form.
3. S3 and S4 by M. Ram Murty, Queen’s University, Canada introduce Artin L-functions.
Time Table for the first week:
| Date | 09:30 - 11:00 | 11:30-13:00 | 14:30-16:00 | 16:30-17:30 |
| 24 June | P1 | P2 | P5 | |
| 25 June | P1 | P2 | P5 | |
| 26 June | P1 | P2 | P5 | T |
| 27 June | P1 | P2 | M1 | T |
| 28 June | M3 | P4 | M1 | S1 |
| 29 June | M3 | P4 | M1 | S2 |
- 30 June 2013 Sunday.— Trip to Binsar.
- Tea Breaks.— 11:00 to 11:30 and 16:00 to 16:30.
- Lunch Break.— 13:00 to 14:30
Course Codes
1. P1 — The Gamma Function — Sanjay Kumar Pant, DDU College, Delhi University.
2. P2 — The Mellin Transform.— D. Surya Ramana, HRI, Allahabad.
3. P5 — Preliminaries from Algebraic Number Theory.— R. Thangadurai, HRI Allahabad.
4. P4 — Subharmonic Functions.— Ravi Raghunathan, IITB, Mumbai.
5. M1 — Analytic Continuation and Functional Equation — R. Balasubramanian, IMSc Chennai.
6. M3 — Prime Number Theorem — Gyan Prakash, HRI Allahabad.
7. T — Tutorial for P1 and P2 — P. Akhilesh, HRI, Allahabad.
8. S1 and S2 — Special Lectures by R. Balasubramanian, IMSC, Chennai on the importance of
the Riemann Hypothesis.
Time Table for the second week:
| Date | 09:30 - 11:00 | 11:30-13:00 | 14:30-16:00 | 16:30-17:30 |
| 1 July | M3 | P4 | S3 | S5 |
| 2 July | P3 | P4 | S4 | S6 |
| 3 July | P3 | M4 | M2 | T |
| 4 July | P3 | M4 | M2 | T |
| 5 July | M3 | M4 | M2 | |
| 6 July | M3 | M4 | M2 |
Tea Breaks.— 11:00 to 11:30 and 16:00 to 16:30.
Lunch Break.— 13:00 to 14:30
Course Codes
1. P3 — Poisson-Jensen formula and its consequences.— M. Ram Murty, Queen’s University.
2. P4 — Subharmonic Functions.— Ravi Raghunathan, IITB, Mumbai.
3. M3 — Prime Number Theorem — Gyan Prakash, HRI Allahabad.
4. M2 —Zero-Free regions.— Ritabrata Munshi, TIFR Mumbai.
5. M4 —Density Bounds.— Tim Browning, University of Bristol.
6. T —Tutorial for M3.— Kasi Viswanadham, HRI, Allahabad.
7. S3 and S4 Special Lectures by M. Ram Murty, Queens University on Artin L-functions. S5
and S6 Special Lectures by B. Ramakrishnan, HRI, Allahabad on L-functions associated to a
modular form.
The oragnisers of the ISL on Complex Analysis and Number Theory, scheduled for the dates 24 June to 6 July 2013 at SSJ Campus, Kumaun University, Almora, regret to announce that this school is postponed on account of inclement weather. Fresh dates for the school will be notified in due course.
- Starred (*) candidates are wait listed.
- Pleasedoconfirmyourparticipationby20thMarch,2013tothefollowinge-mail: sanjpant at gmail.com Incaseofnon-confirmationfromselectedcandidatesthewait listedcandidateswillbeinformedby25thMarch,2013.
- We suggest that you should book train ticket well in advance for journey between Delhi and Kathgodam.
-
Youwillbereimbursed3AC/BUSfare(Pleasedokeepyourticketsafely)Notaxi/Autofarewillbepaid.
-
FeelfreetocontactSanjayKumarPant(sanjpant @ gmail.comor9810528236)foranyqueryrelatedtothisschool.
List of selected Applicants
|
Sr. No. |
Name |
Affiliation |
Position |
|
1 |
Dr.Praveen Agarwal |
Anand International College of Engineering,Jaipur, Rajasthan |
Associate Professor |
|
2 |
Mr. Tumiki N C Raju |
Govt. Degree College, Jammikunta,A.P. |
Assitant Professor |
|
3 |
Dr.(Ms)Saumya Singh |
O.P. Jindal Institute of Tech., Raigarh, Chhatisgarh |
Assitant Professor |
|
4 |
Mr. Triloki Nath |
NIT, Mizoram |
Assitant Professor |
|
5 |
Dr. Ganga Upendra Reddy |
Mahatma Gandhi Intitute of Technology, Hyderabad |
Assitant Professorr |
|
6 |
Dr. Yuvraj |
Govt. P G College, Rishikesh, Uttarakhand |
Associate Professor |
|
7 |
Dr. Raghvendra Mishra |
Govt. P G College, Ranikhet, Uttarakhand |
Assitant Professor |
|
8 |
Dr.(Ms) Neelam Singh |
Bundelkhand P G Degree College, Jhansi (UP) |
Assitant Professor |
|
9 |
Dr.(Ms) Anjeli Garg |
Mahamaya Govt. Degree College, Sherkot , Bijnor (UP) |
Assitant Professor |
|
10 |
Dr. Bibhas Chandra Saha |
Chandidas Mahavidyalaya , Bolpur (WB) |
Assitant Professor |
|
11 |
Dr. Javid Iqbal |
BGSB University, Rajouri, J&K |
Assitant Professor |
|
12 |
Dr (Ms) Sujata Singh |
Govt. P G College, Rishikesh, Uttarakhand |
Assitant Professor |
|
13 |
Dr. Dhirender Bahadur Singh |
G K V, Haridwar, Uttarakhand |
Assitant Professor |
|
14 |
Ms. Deepti Lohiya |
Guru Nanak College for Girls, Muktsar, Punjab |
Assitant Professor |
|
15 |
Mr. Avadhoot Balasaheb Kadam |
A D College of Engg and Tech, Ashta, Maharashtra |
Assitant Professor |
|
16 |
Dr. (Ms) Somna Mishra |
IMS, Noida (UP) |
Assitant Professor |
|
17 |
Mr. Sunil Gangaram Purane |
Jamkhed Mahavidyalaya, Jamkhed, Ahemednagar(MS) |
Assitant Professor |
|
18 |
Dr. Bhagwati Prasad Joshi |
SIT, Pithoragarh, Uttarakhand |
Assitant Professor |
|
19 |
Dr. Manoj Kumar Patel |
GKV, Haridwar, Uttarakhand |
Assitant Professor |
|
20 |
Ms Ankita Chaturvedi |
IIT, Kharagpur (WB) |
Post-doc |
|
21 |
N K Arvindbhai Patel |
SVBIT, Gandhinagar, Gujarat |
Assitant Professor |
|
22 |
Dr. Prasantha Kumar Ray |
IIIT, Bhubaneshwar, Orissa |
Assitant Professor |
|
23 |
Ms. Akshaa Vatwani |
Queen's University, Kingston,CA |
Research Scholar |
|
24 |
Mr. Senthil Kumar |
HRI, Allahabad |
Research Scholar |
|
25 |
Mr.Pawan Kumar Tamta |
Kumaun University, Almora |
Assitant Professor |
|
26 |
Dr. Hemlata Pande |
Kumaun University |
Post-doc |
|
27 |
Dr. Vivek Kumar Khare |
L B S P G College, Gonda (UP) |
Assitant Professor |
|
28 |
Mr. Tapas Chatterjee |
IMSc, Chennai |
Research Scholar |
|
29 |
Mr. Keshav Aggarwal |
IISER, Mohali |
MS Student |
|
30 |
Ms. Debika Banerjee |
HRI, Allahabad |
Research Scholar |
|
31* |
Mr. Bibekanand Maji |
HRI, Allahabad |
Research Scholar |
|
32* |
Mr. Balesh Kumar |
HRI, Allahabad |
Research Scholar |
|
33* |
Mr. Mallesham K. |
HRI, Allahabad |
Research Scholar |
|
34* |
Mr. Karamdeo Shankardhar |
HRI, Allahabad |
Research Scholar |
|
35* |
Naveen Gupta |
Delhi University |
Research Scholar |
|
36* |
Ms. Manisha Saini |
Delhi University |
Research Scholar |
