**KALYAN CHAKRABORTY**

**Director**

KSCSTE-KERALA SCHOOL OF MATHEMATICS

I am a pure mathematician and work in various areas in Number Theory. Over the years I have visited many countries and various places in India relating to my work and developed a kind of all weather personality. I am a fitness freak and strongly believe in the saying `A healthy mind lives in a healthy body'. I love to be amongst the younger people and over the years motivated many students and Post doctoral fellows to choose healthier life which have helped them immensely in their personal lives as well as in their career. I have developed a belief that I can help many younger students not only academically but also in their pursuit for a brighter future. I love to take new challenges which brings out the best in me.

**Education**

D. Phil. Mathematics, 1998, Harish-Chandra Research Institute (formerly MRI), Allahabad University (Allahabad).

**Current Position
**

**Director**

Kerala School of Mathematics, Kozhikode, Kerala, India.

**Recent Visit Positions**

- Research Scholar ( 1993 - 96 ) : HRI, Allahabad
- Postdoctoral Fellow (1997 - 99 ) : IMSC, Chennai
- Postdoctoral Fellow (1999 - 2001 ) : Queen's University, Canada
- Postdoctoral Fellow (2001 August to October) : HRI, Allahabad
- Visiting Professor (2004 May to June) : University of Paris VII, Paris.
- Visiting Professor (2004 October) : Tokyo Metrop. Univ., Tokyo.
- Visiting Professor (2005 June) : Waseda Univ., Tokyo.
- Visiting Professor (2005 December): Shandong Univ., China.
- Tribhuvan University, Kathmandu.
- Kathmandu University, Nepal.
- Università Roma Tre, Italy.
- Eötvös Loránd University, Budapest.
- University of Paris VI, Paris.
- University of Hong Kong, Hong Kong.
- Kinki University, Japan.
- Kyoto Sangyo University, Japan.
- North West University, Xian.
- Shangdong University, Shangdong.
- Shangdong University, Weihai.
- University of Calgary, Calgary, Canada.
- Queen's University, Kingston, Canada
- Universite Laval, Quebec, Canada.

**Participation in Conferences and Schools**

- RIMS conference in Number Theory, Kyoto University, 2004, Japan.
- Workshop in Algebraic Number Theory, Kochi University, Japan, 2004.
- Conference in Number Theory, Waseda University, March 2005, Japan.
- Workshop on "Number Theory and Applications", Shandong University, December 2005, China.
- India - UK number theory conference, Institute of Mathematical Sciences, Chennai, Sept. 18-23, 2006, India.
- Conference in Number Theory and Applications, Ramakrishna Mission Vivekananda College, Chennai, September 2007.
- Number Theory Session of the \National Symposium", University of Mumbai, Mumbai, India, January 2007.
- "Mathematics and Applications - Recent Trends (Mart - 2007)", University of Burdwan, Burdwan, India, January 2007.
- Third Noth West Number Theory Conference, Weinan Teacher's College and Institute of Nothwest University, Weinan, China, March 23-25, 2007.
- "Mathematics and Applications - Recent Trends (Mart - 2008)", University of Burdwan, Burdwan, India, November 2008.
- Fourth NothWest Number Theory Conference, Shengyang Teacher's College and Institute of Nothwest University, Xi'an, China, March 23- 25, 2008.
- Fifth North West Number Theory Conference, Sangluo University, Sangluo, China, March 16-18, 2009.
- "Two Day International Conference On Number Theory", Kinki University, Japan, October 17-18, 2009.
- "International Conference On Recent Trends in Mathematics and Applications", Burdwan University, Burdwan, India, January 13-15, 2010.
- "CIMPA School On Number Theory in Cryptography and Its Applications", Kathmandu, Nepal, July 19- 24.

**Other Professional Activities**

- Reviewer for AMS.
- Mentoring 'Plus 2' students at various DST
**'INSPIRE'**Camps. - Referee (Research Journals): Proceedings – Mathematical Sciences, JRMS, IJNT, JNT, SpringerPlus, Defence Research Journal etc.
- Referee (PhD Thesis): Most Recent Ones:
**(1)**‘Higher rank Artin’s conjecture with weight and average results’—Università Roma Tre, Italy.

**(2)**‘A study of Class Number Of Real Quadratic and Cubic Fields’—IIT Guwahati.

**(3)**‘Gap distribution among sequences arising in the theory of modular forms’ — IISER, PUNE. - Vice President SSFA INDIA.
- Chief Organiser ICCGNFRT Yearly International Conference.

**Past Scholars**

Dr. Jay Mehta,Currently Assistant Professor, S P University, Vadodara, India

Dr. Debika Banerjee,Currently Assistant Professor, IIIT, Delhi.

Dr. Bibekananda Maji,Assit. Professor at IIT, Indore.

Dr. Soumyarup Banerjee,Currently Postdoctoral Fellow at IIT, Gandhinagar

Dr. Abhishek Juyal,Currently Post Doctoral Fellow at Institute of MathematicalSciences, Chennai.

**Current Schloars**

Ms. Richa Sharma, 4th Year, Co-guide

Mr. Mohit Sharma, 3rd Year.

Mr. Rishabh Agnihotri, 3rd Year

Ms. Srijonee Shabnam Chaudhury, 2nd Year

Mr. Shubham Gupta, 2nd Year.

Mr. K. Krishnarjun,2nd Year.

**Past Fellows**

Dr. Prem Prakash Pandey, Assistant Professor at IISER, Berhampur.

Dr. Sudhansu Sekhar Raut, Assistant Professor at IMA, Bhubaneswar.

Dr. Raj Kumar Mistri, Asst. Prof. at IIT, Bhillai.

Dr. Sudhir Pujahari, Post Doctoral fellow at University of Hong Kong.

**Current Fellows**

Dr. Azizul Hoque, NPDF, SERB.

Dr. Chiranjit Roy.

Dr. Kalyan Banerjee.

**Journal Publications**

M. Mishra, R. Agnihotri and K. Chakraborty, Primary rank of class groups of real cyclotomic fields, Rocky Mountain J., 2020 (To appear).

K. Chakraborty, A. Hoque and R. Sharma, Complete solutions of certain Lebesque-Ramanujan-Nagell type of equations, Publicationes Mathematicae Debreceen, 2020 (To appear).

K. Chakraborty, A. Hoque and M. Mishra, On certain order $4$ class groups of $\mathbb{Q}(\sqrt{n^2 + 1})$, Annales mathematiques du Quebec, 2020, 1--10; DOI 0.1007/s40316-020-00139-1 .

K. Banerjee, K. Chakraborty and A. Hoque, Divisibility of Selmer groups and class groups, Hardy-Ramanujan Journal, (Dec. 2019), Vol. 42, 85--99.

K. Chakraborty, A. Hoque and M. Mishra, A note on certain real quadratic fields with class number upto three, Kyushu J. M., 2020 (To appear).

K. Chakraborty and A. Hoque, Exponents of class groups of certain imaginary quadratic fields, Czechoslivak Mathematical Journal, 2020 (To appear).

S. Gupta and K. Chakraborty, A problem related to prime numbers, The Mathematics student, Vol. 87, No. 1-2, (2018), 127--130.

K. Chakraborty and A. Hoque, Class groups of certain imaginary quadratic fields of $3$-rank atleast $2$, Annales UMCS Mathematica, Vol. 47 (2018), 179--183.

K. Chakraborty, S. Kanemitsu and T. Kuzumaki, Seeing the invincible: Kubert identities, Annales UCMS Mathematica, Vol. 47 (2018), 185--195.

S. Banerjee and K. Chakraborty, Asymptotic behaviour of a Lambert series à La Zagier: Maass case, Ramanujan J., Vol. 48 (2019), No. 3, 567--575.

K. Chakraborty, A. Hoque, Y. Kishi and P. P. Pandey, Divisibility of the class numbers of imaginary quadratic fields, J. Number Theory, 185 (2018), 339--348.

K. Chakraborty and A. Hoque, Pell-type equations and class number of the maximal real subfield of a cyclotomic field, Ramanujan J., Vol. 46 (2018), 727--742 DOI: 10.1007/s11139-017-9963-9

K. Chakraborty and A. Hoque, Divisibility of class numbers of certain families of quadratic fields, J. Ramanujan Math. Soc., Vol. 34, No. 3 (2019), 281--289.

K. Chakraborty, A. Juyal, S.D. Kumar and B. Maji, Asymptotic expansion of a Lambert series, Int. J. Number Theory, 14 (2018), no 1, 289--299.

D. Banerjee, K. Chakraborty, S. Kanemitsu and B. Maji, Abel-Tauber process and asymptotic formulas, Kyushu J. Math., 71 (2017), no. 2, 363--385.

K. Chakraborty, J. J. Urroz and F. Pappalardi, Pairs of integers which are mutually squares, Sci. China Math., 60 (2017), no. 9, 1--14.

K. Chakraborty, S. Kanemitsu and B. Maji, Modular type relations associated to the Rankin-Selberg L-functions, Ramanujan J., 42 (2017), no. 2, 285--299.

K. Chakraborty, I. Kàtai and B. M. Phong, Additive functions on the greedy and lazy Fibonacci expansions, J. Integer Seq., 19 (2016), no. 4, 16.4.5, 12 pp.

K. Chakraborty, S. Kanemitsu and H. Tsukuda, Ewald expansions of a class of zeta functions, Springer Plus, (2016),5:99 DOI: 10.1186/s40064-016-1732-5.

K. Chakraborty, S. Kanemitsu and H. -L. Li, Quadratic reciprocity and Riemann's non-differentiable function, Res. Number Theory, 1 (2015), Art. 14, 8 pp.

K. Chakraborty, S. Kanemitsu and H. Tsukada, Applications of the Beta-Transform, Siauliai Math. Sem., 10 ( 2015), no. 18, 5--28.

T. Arai, K. Chakraborty and J. Ma, Applications of the Hurwitz-Lerch zeta-function, Pure Appl. Math. J., 4 (2015), no. 1,2, 30--35.

K. Chakraborty and M. Minamide, On power moments of the Hecke multiplicative functions, J. Aust. Math. Soc., 99 (2015), 334--340.

K. Chakraborty, S. Kanemitsu and T. Kuzumaki, On the Barnes multiple gamma functions, Siauliai Math. Sem., 9 (2014), no. 17, 27--41.

K. Chakraborty and J. Mehta, Preventing unknown key-share attack using cryptographic bilinear maps, J. Discr. Math. Sci. Crypt., 17 (2014), no. 2, 135--147.

K. Chakraborty, S. Kanemitsu and T. Kuzumaki, On the class number formula of certain real quadratic fields, Hardy-Ramanujan J., 36 (2013), 1--7.

K. Chakraborty and M. Minamide, On partial sums of a spectral analogue of the Möbius function, Proc. Indian Acad. Sci. (Math. Sci.), 123 (2013), no. 2, 193--201.

K. Chakraborty, I. Kàtai and B. M. Phong, On the values of arithmetic functions in short intervals, Ann. Univ. Sci. Budapest. Sect. Comput., 38 (2012), 13--21.

K. Chakraborty, S. Kanemitsu and H. Tsukada, Arithmetical Fourier series and the modular relation, Kyushu J. Math., 66 (2012), no. 2, 411--427.

K. Chakraborty, I. Kàtai and B.M. Phong, On real valued additive functions modulo $1$, Ann. Univ. Sci. Budapest., Sect. Comput., 36 (2012), 355--373.

K. Chakraborty and J. Mehta, A stamped blind signature scheme based on elliptic curve discrete logarithm problem, Int. J. Network Security, 14 (2012), no. 6, 316--319.

K. Chakraborty and J. Mehta, On completely multiplicative complex valued functions, Ann. Univ. Sci. Budapest, Sect. Comput., 38 (2012), 19--24.

K. Chakraborty, I. Kàtai and B. M. Phong, On additive functions satisfying some relations, Ann. Univ. Sci. Budapest., Sect. Comput. 38 (2012), 1--12.

K. Chakraborty, S. Kanemitsu and X. -H. Wang, The modular relation and the digamma function, Kyushu J. Math., 65 (2011), no. 1, 39--53.

K. Chakraborty, S. Kanemitsu, Y. Katayama and Y. Nitta, Sustainibility of ancient constructions and non-associated numbers, J. Sangluo Univ., 24 (2010), no. 2, 60--79.

K. Chakraborty, S. Kanemitsu and H.-L. Li, On the value of a class of Dirichlet series at rational arguments, Proc. Amer. Math. Soc., 138 (2010), no. 4, 1223--1230.

K. Chakraborty, On the Chowla-Selberg integral formula for non-holomorphic Eisenstein series, Integral Transforms Spec. Funct., 21 (2010), no. 12, 917--923.

K. Chakraborty, F. Luca, Perfect powers in solutions to Pell equations, Rev. Columbiana Mat., 43 (2009), no. 1, 71--86.

K. Chakraborty, S. Kanemitsu and J. -H. Li, Manifestations of the Parseval identity, Proc. Japan Acad. Ser. A, 85 (2009), no. 9, 149--154.

K. Chakraborty, S. Kanemitsu, H. Kumagai and Y. Kubara, Shapes of objects and the golden ratio, J. Sangluo Univ., 23 (2009), no. 4, 18--27.

K. Chakraborty, S. Kanemitsu and T. Kuzumaki, Finite expressions for higher derivatives of the Dirichlet L -function and the Deninger R- function, Hardy-Ramanujan J., 32 (2009), 38--53.

K. Chakraborty, F. Luca and A. Mukhopadyay, Exponents of class groups of real quadratic fields, Int. J. Number Theory, 4 (2008), 1--15.

K. Chakraborty, F. Luca and A. Mukhopadyay, Real quadratic fields with class numbers having many distinct prime factors, J. Number Theory, 128 (2008), no. 9, 2559--2572.

K. Chakraborty, On the Diophantine equation x+y+z = xyz = 1, Ann. Univ. Sci. Budapest. Sect. Comput., 27 (2007), 145--154.

K. Chakraborty and A. Mukhopadyay, Exponents of class groups of real quadratic function fields (II), Proc. Amer. Math. Soc., 134 (2006), no. 1, 51--54.

K. Chakraborty and A. Mukhopadyay, Exponents of class groups of real quadratic function fields, Proc. Amer. Math. Soc., 132 (2004), no. 7, 1951--1955.

K. Chakraborty and M. Ram Murty, On the number of real quadratic fields with class number divisible by 3, Proc. Amer. Math. Soc., 131 (2003), no. 1, 41--44.

S. Baba, K. Chakraborty and Y. N. Petridis, On the number of Fourier coefficients that determine a Hilbert modular form, Proc. Amer. Math. Soc., 130 (2002), no. 9, 2497--2502.

K. Chakraborty and M. V. Kulkarni, Solutions of cubic equations in quadratic fields, Acta Arith., 89 (1999), no. 1, 37--43.

K. Chakraborty, A. K. Lal and B. Ramakrishnan, Modular forms which behave like theta series, Math. Comp., 66 (1997), no. 219, 1169--1183.

K. Chakraborty, B. Ramakrishnan and T. C. Vasudevan, A Note on Jacobi forms of Higher degree, Abh. Math. Sem. Univ. Hamburg, 65 (1995), 89--93.

K. Chakraborty and B. Ramakrishnan, A note on Hecke eigen forms, Arch. Math., 63 (1994), no. 6, 509--517.

S. D. Adhikari and K. Chakraborty, On the average behaviour of an arithmetical function, Arch. Math., 62 (1994), no. 5, 411--417.

**Conference Proceedings/ Book Chapters**

K. Chakraborty, A. Hoque and R. Sharma, Divisibility of class numbers of quadratic fields: Qualitative aspects, To appear in 'Advances in Mathematical Inequalities and Applications, Trends in Mathematics, Birkhäuser, Singapore, 2018'.

K. Chakraborty and A. Hoque, Quadratic reciprocity and some "Non-differentiable" Functions, In: Ruzhansky M., Cho Y., Agarwal P., Area I. (eds), Advances in Real and Complex Analysis with Applications, Trends in Mathematics. Birkhäuser, Singapore, 2017, pp. 145--181.

K. Chakraborty, H. Kanemitsu and H. -L. Li, Special functions in number theory and in science, Proceedings of the 14-th Annual Conf. SSFA, 14 (2015), 11--43.

K. Chakraborty, On the number of Fourier coefficients that determine a form, Conference Proceedings, Institute of mathematics, Waseda University, 2005, 19--30.

K. Chakraborty, On the divisibility of class numbers of real quadratic fields, RIMS Conference Proceedings, Kyoto Univ., 2004.

T. Arai, K. Chakraborty and S. Kanemitsu, On modular relations, Number theory, 1--64, Ser. Number Theory Appl., 11, World Sci. Publ., Hackensack, NJ, 2015.

K. Chakraborty, S. Kanemitsu and H. -L. Li, Special functions in number theory and in science, Proceedings of the 14-th Annual Conf. SSFA, Vol. 14, 2016.

K. Chakraborty and A. Hoque, Quadratic reciprocity and some ``non-differentiable'' functions, Trends in Mathematics, Birkhauser,Singapore, 2017.

K. Chakraborty, A. Hoque and R. Sharma, Divisibility of class number of quadratic fields: Qualitative aspects, Advances in Mathematical Inequalities and Applications, Trends in Mathematics, Birkhauser, Singapore, 2018.

**Books**

K. Chakraborty, S. Kanemitsu and H. Tsukada, Vistas of special functions II, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.

K. Chakraborty, S. Kanemitsu and T. Kuzumaki, A quick introduction to complex analysis, World Scientific Publishing Co. Pte, Letd, Singapore, 2016.

K. Chakraborty, S. Kanemitsu and T. Kuzumaki, Class groups of Number Fields and Related topics, Springer, 2019.

**Other Publications/ Expository Articles**

H. -L. Li, K. Chakraborty and S. Kanemitsu, Music as mathematics of senses, Advances in Pure Mathematics, Vol.8, No. 12, 845--862, Dec. 2018.

K. Chakraborty, S. Kanemitsu and Y. Sun, Codons and Codes, Pure Appl. Math. J., 4 (2015), no. 1, 2, 25--29.

K. Chakraborty, S. Kanemitsu, Y. Katayama and Y. Nitta, Sustainability of ancient constructions and non-associated numbers, J. Sangluo Univ., 24 (2010), no. 2, 60--79.

K. Chakraborty, S. Kanemitsu, H. Kumagai and Y. Kubara, Shapes of objects and the golden ratio, J. Sangluo Univ., 23 (2009), no. 4, 18--27.

**Submitted/ Preprint(s)**

K. Chakraborty and A. Hoque, On the plus parts of the class numbers of cyclotomic fields, Submitted.

K. Chakraborty, S. Kanemitsu and T. Kuzumaki, Seeing the invisible: Kubert indentities, Submitted.

K. Chakraborty and A. Hoque, Class groups of imaginary quadratic fields of rank at least two, Submitted.

K. Chakraborty, A. Hoque, S. Kenmitsu and A. Laurinčikas, Complex powers of L-functions and integers without large prime factors, Submitted.

S. Banerjee, K. Chakraborty and A. Hoque, An analogue of Wilton's formula and values of Dedekind zeta functions, Preprint.

K. Chakraborty and A. Hoque, Exponents of class groups of certain imaginary quadratic fields, Submitted.

K. Chakraborty, Exponent of class groups of a family of cyclic cubic fields, Preprint.

K. Banerjee and K. Chakraborty, Tate-Shafarevich group and Selmer group constructions for Chow group of an abelian variety, Submitted.

K. Banerjee and K. Chakraborty, On the Chow group of the self-product of a CM elliptic curve defined over a number field, Submitted.

K. Chakraborty and A. Hoque and R. Sharma, On the solutions of certain Lebesgue-Ramanujan-Nagell equations, Submitted.

K. Chakraborty and T. Komatsu, Generalized hypergeometric Bernoulli numbers, Submitted.

K. Chakraborty, A. Hoque and K. Srinivas, On the Diophantine equation $cx^2 + p^{2m} = 4y^n, Submitted.

K. Chakraborty and A. Hoque, Certain eta-quotient and $l$-regular overpartitions, Submitted.

**Current Projects**

Chow groups, class groups and Tate-Shafarevich groups: Aimed at studying the inter=relations between these three important groups. In particular it aims at transferring certain properties of one group to the other.

L-functions: Study the asymptotic behaviour of ‘Riesz sums’ associated to the Rankin-Selberg L-functions involving various automorphic foms ( e.g. harmonic Mass forms and Hasse-Weil zeta functions associated to the elliptic curves, Hilbert modular forms of degree 2. (With S. Banerjee and A. Juyal).

Modular relations: (i) Consider applications of the theory of zeta-functions to ‘Theoretical Computer Science’. We begin by studying the work of Anshell and Goldfeld in which they add one more condition on the coefficients of the Euler product, i.e. polynomial time evaluation algorithm for the values at prime powers. Then we study the work of Kirschenhoffer and Prodinger about the trie algorithm. In contract to Knuth’s work, they deduce the exact formula for the expectation and for the variance they used Ramanujan’s formulas for the Riemann zeta-values involving Lambert series. We believe there must be a disguised form of the modular relation.

(ii) Study Mertens’ formula for a class of zeta-functions as we have studied in one of our previous work. We also reveal the hidden structure of Vinogradov’s work on a general Mertens’ formula in number fields.

Class number of number fields: (i) Class numbers of number fields are mysterious objects and not much is known about these numbers. One of the many questions is to show the existence of infinitely many number fields (for quadratic field case many results are known!) whose class number is divisible by a given integer. Quantifying these fields is also an interesting problem. The Cohen-Lenstra heuristics related to these questions is yet to be established. In the direction of indivisibility not much is known though. Its worth looking into these direction relating the arithmetic of some kind of modular forms and that of elliptic curves.

(ii) There are few well known conjectures (e.g. Vandiver conjecture) relating to the class number of maximal real subfields of cyclotomic fields. The above problems of divisibility and indivisibility are worth exploring here in this set up. This will give us information about the class numbers of the corresponding cyclotomic fields too. These numbers are not easy to calculate by using the class number formula.

**Teaching Experience**

1st Year Undergraduate Calculus, fall 1999, Queen's University.

Graduate course jointly with Yiannis N. Petridis on \Analytic Number Theory", winter 1999, Queen's University.

1st year Undergraduate Calculus, fall 2000, Queen's University.

1st year undergraduate Linear algebra, winter 2000, Queen's University.

Organised Visiting Student Summer Programme, 2002, HRI and taught basic Algebraic Number Theory.

One semester "Measure Theory" course for 1st year Graduate students, 2002-03, HRI.

One semester "Galois Theory" course for 1st year graduate students, 2004-05, at HRI.

Half a semester "Measure Theory" course for 1st year graduate students at HRI, 2005.

One semester "Algebra I" course for 1st year graduate students at HRI, 2006.

One semester "Introduction to elliptic curves" course to the 2nd year graduate students at HRI, 2007.

Given a course every year in "Summer Programme In Mathematics (SPIM), HRI".

Taught in "National School In Number theory and Cryptography", Kathmandu university, Dec. 26th-5th Jan., 09-10.

Taught a 3 credit course on "Number theory" to the M.Phil students at Kathmandu University, Jan. 6th - 11th, 2010.

Taught at CIMPA School, Kathmandu, July 19th to July 24th, 2010. . . Lecture1Lecture 1 Lecture 2Lecture 2 Lecture 3 lecture 3

Taken a course on `Arithmetic of Elliptic Curves' for the 2nd year graduate students from August to December 2010.

Given a course on `Introduction to Theory of Numbers' from Nov. 15th to Nov. 25th at Dept. of Mathematics, Tribhuvan University, Kirtipur, Nepal.

**Recent Talks and Seminars**

50 minutes plenary talk at RIMS Conference, Kyoto, Japan, 2004.

50 minutes talk at University of Paris, VII, 2004.

45 minutes talk at Waseda University Conference, 2005.

50 minutes plenary Talk at "Workshop on Number Theory and Applications", Shandong University, China, 2005.

40 minutes invited talk at India - UK number theory conference, IMSC, Chennai, October 2006.

45 minutes invited talk at the Number Theory Session of the "National Symposium", University of Mumbai, Mumbai, India, January 2007.

45 minutes invited talk at "Mathematics and Applications - Recent Trends (Mart - 2007)", University of Burdwan, Burdwan, India, January 2007.

60 minutes plenary talk at "North - West Number Theory Conference", Weinan Teachers University, Xian, China, March 2007.

45 minutes invited talk at "Mathematics and Applications - Recent Trends (Mart - 2008)", University of Burdwan, Burdwan, India, January 2007.

60 minutes seminar at Kinki University, Fukuoka, Japan, Oct.- Nov. 2007.

60 minutes seminar at Nagoya University, Nagoya, Japan, Oct.- Nov. 2007.

75 minutes seminar at Waseda University, Tokyo, Japan, Oct.- Nov. 2007.

60 minutes plenary talk at "Fourth North - West Number Theory Conference", Shengyang, Shaanxi Province, China, March 2008.

60 minutes seminar at "The University of Hong Kong", Hong Kong, March, 2008.

60 minutes seminar, University of Paris VI, Paris, May, 2009.

60 minutes talk at "Two Day International Conference on Number Theory", Kinki University, Fukuoka, Japan, 19-20 October, 2009.

Invited talk at "Recent Trends in Mathematics and Application", Burdwan University, Burdwan, January 13-15, 2010.

**Director**

KSCSTE-Kerala School of Mathematics

P.O. Kunnamangalam

Kozhikode, Kerala, India

PIN: 673571

**Phone Number :** +91 495 2809001

**Mobile Number : ** +91 9956567372

Email: kalychak@ksom.res.in