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 Lfunctions. 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 Lfunctions and simple examples of Dedekind zeta functions. Preliminaries from complex analysis will be developed through short courses on the Gamma function, Mellin transform, PoissonJensen formula and its consequences and Subharmonic Functions. The short courses on Lfunctions will cover functional equations, convexity bound in the critical strip, asymptotics for zeros in the critical strip, zerofree 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, Uttarakhand263601 
Associate Professor, Department of Mathematics, Deen Dayal Upadhyaya College,New Delhi110015 
HarishChandra 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 Lfunctions. 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. 
PoissonJensen formula and its consequences.— Starting from a proof of the PoissonJensen formula, we shall obtain various frequently used results such as the real parts theorem of BorelCaratheodory 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 ¨PhragmenLidelof 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 Lfunctions, 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 Lfunctions 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. 
ZeroFree regions.— The classical zerofree regions for the Riemann zeta function and the Dirichlet Lfunctions shall be derived, keeping the results completely explicit. Then we will pass to the analogue for a general Lfunction following Iwaniec and Kowalski, Chapter 10. Finally, we will present a fairly uptodate 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 SielgelWalﬁsz 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 socalled density bounds for zero’s of Lfunctions.  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 Lfunction associated to a modular form.
3. S3 and S4 by M. Ram Murty, Queen’s University, Canada introduce Artin Lfunctions.
Time Table for the first week:
Date  09:30  11:00  11:3013:00  14:3016:00  16:3017:30 
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  11:3013:00  14:3016:00  16:3017:30 
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 — PoissonJensen 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 —ZeroFree 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 Lfunctions. S5
and S6 Special Lectures by B. Ramakrishnan, HRI, Allahabad on Lfunctions 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.
 Pleasedoconfirmyourparticipationby20^{th}March,2013tothefollowingemail: sanjpant at gmail.com Incaseofnonconfirmationfromselectedcandidatesthewait listedcandidateswillbeinformedby25^{th}March,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) 
Postdoc 
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 
Postdoc 
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 