My research interests are in algebraic groups over global and local fields and over their rings, Arithmetic groups, Congruence subgroup problem, Bounded generation, Division algebras, Central extensions of groups, Milnor K_1 and K_2, Bak's unitary groups, Diophantine equations, Applications of prime density theorems, Zerosum problems in number theory, Class groups of cyclic extensions
of number fields.
Ph.D. (TIFR, Bombay) January 1991
 Supervisor : M S Raghunathan, FRS
I was at the Tata Institute of Fundamental Research, Bombay from August 1981
until April 1999.
Since May 1999, I have been at the Indian Statistical Institute, Bangalore.
Books :
Number theory and Combinatorics, Indian Academy of Sciences 2017.
The congruence subgroup problem : an elementary approach aimed at
applications : TRIM Series, Vol. 24, 2003, Hindustan Book Agency, New Delhi,
India.
ISBN108185931380.
Distributed by the AMS except in India :
ISBN139788185931388.
Group theory : selected problems, Universities Press (India)
Private Ltd., 2004, Hyderabad, India.
Distributed by Orient Longman Pvt.Ltd.
ISBN 8173714916.
(www.universitiespress.com, www.orientlongman.com)
Hecke algebras, Intersection cohomology of Schubert varieties
and Representation theory (1984 vintage  unedited, unpublished !).
A mathematical review.
Publications
(with Aleksander V. Morgan \& Andrei S. Rapinchuk), Bounded generation of $SL_2$ over rings of
$S$integers with infinitely many units, Algebra \& Number Theory, Vol. 12, No.8 (2018) P. 19491974.
Lower bound for the least common multiple, Amer. Math. Monthly, To Appear.
A Ring Theoretic Approach to Bound the Totient Function, Filler, Amer. Math. Monthly, Vol. 126, No.2 (2019) P.167.
(with Vaibhav Pandey \& Sagar Shrivastava), A Dedekind Domain with Nontrivial Class Group,
Amer. Math. Monthly, Vol. 125 (2018) pp.356359.
Variants of Carmichael numbers and Cunningham Chains,
Math. Gaz. 102 (2018), no. 555, 498–501.
(with Kannappan Sampath) Norm or Exception?, Mathematics Student, Vol. 86,
JanuaryJune 2017 (pp.3950).
(with M.Kulkarni & D.Majumdar) lClass groups of cyclic extensions of prime degree l, J. Ramanujan Math. Soc.,
Vol. 30, No. 4 (2015) 413454.
(with Nikolay Nikolov), Bounded generation of wreath products,
Journal of Group Theory, Vol.18, No.6 (2015) 951959.
Matrix groups over rings, Hansraj Gupta Memorial Award talk at the
annual conference of the Indian Mathematical Society,
Mathematics Student, Vol. 85 (2016) 7996.
(with Manisha Kulkarni), Diophantine equations of the form F(x)=G(y)  an exposition,
Mathematics Student, Vol. 84 (2015) 135155.
(with S.K.Prajapati) On the total character of finite groups,
International Journal of Group Theory, Vol.3, No.1 (2014) pp.4767.
(with N.A.Vavilov & A.V.Smolensky) Gauss decomposition for Chevalley groups  Revisited,
International Journal of Group Theory, Vol.1, No.1 (2012) pp.316.
(with N.A.Vavilov & A.V.Smolensky) Unitriangular factorizations of Chevalley groups, Zapiski Nauchnyh Seminarov POMI, vol.388 (2011), p.1747(in Russian).
English translation: J. Math. Sci. (N.Y.) 183 (2012) No.5, 584599.
Read G as E in the main theorems!
Tits building and an application to central extensions of padic algebraic groups by
finite pgroups, Proc.Amer.Math.Soc., Vol.139, No.6 (2011) pp.20332044.
(with Alec W.Mason) Subgroups of algebraic groups which are clopen in the Scongruence topology, Journal of Group Theory, Vol.15 (2012) P.4755.
Macaulay expansion, The American Mathematical Monthly, Vol. 121 (2014) pp.359360.
Some applications of Representation Theory to Classical Number Theory, Mathematics Newsletter, Ramanujan Mathamtical Society, Vol. 25, September 2014, Page 130139.
Chord lengths, Discriminants of cyclotomic fields, and reducibility of cyclomic polynomials modulo primes, Mathematics Newsletter, Ramanujan Mathematical Society, Vol. 24, March 2014, Page 9497.
Riemann Hypothesis  The Prime Problem, Mathematics Newsletter, Ramanujan Mathematical Society, Vol. 21, September 2011, Page 1226.
Ramanujan's awesome sums, Mathematics Newsletter, Ramanujan Mathematical Society, Vol. 24, September 2013, Page 3137.
A polynomial parent to a FibonacciLucas relation,
The American Math. Monthly, Filler, Vol. 121, March 2014, P.236.
A heuristic argument for Hua's identity using geometric series,
The American Math. Monthly, Vol. 121, June 2014, P.522.
Composition of polynomials, The Mathematical Gazette, March 2013.
An equation involving Euler's $\phi$, The Mathematical Gazette, 98.03, March 2014.
(with Hacene Belbachir and Mourad Rahmani)
Alternating Sums of the reciprocals of binomial coefficients,
J.Integer Sequences, Vol.15 (2012) Article 12.2.8.
(with Tewodros Amdeberhan, Valerio De Angelis, Minghua Lin & Victor H.Moll) A pretty binomial identity, Elemente der Mathematik, Vol.67 (2012) pp.1825.
(with Hacene Belbachir and Mourad Rahmani)
Sums involving moments of reciprocals of binomial coefficients,
Journal of Integer Seq., Vol.14 (2011) Article 11.6.6.
(with Manisha Kulkarni)
On Diophantine equations of the form
(xa_1)(xa_2)
...(xa_k) + r = y^n, Proc.Indian Acad.Sci., Vol.121, No.3
(2011) pp.245247.
(with S.D.Adhikari & A.A.Ambily)
Zerosum problems with subgroup weights, Proc.Indian Acad.Sci.,
Vol.120 (2010) pp.259266.
A generalisation of a converse to Schur's theorem, Arch.Math., Vol.95, No.4 (2010)
317318.
(with A.W.Mason, A.A.Premet & P.A.Zalesski) Congruence kernel of an arithmetic
lattice in a rank one algebraic group over a local field, J. Reine Angew. Math.,
623 (2008) P. 4372.
On SU(1,D)/[U(1,D),U(1,D)] for quaternion algebras, Arch. Math., Vol.90 (2008) 493500.
Generalized Catalan numbers : Linear recursion and divisibility, Journal of Integer Sequences, Vol.12 (2009) Article 09.7.5.
Comment on `On the energy of unitary Cayley graph' by
H.N.Ramaswamy \& C.R.Veena, Electronic Journal of Combinatorics, Vol.16 (2009) No.24.
(with P.Moree) Primes in prescribed arithmetic progressions dividing the sequence
{a^k+b^k}, International Journal of Number Theory, Vol.5, No.4 (2009) P.641665.
Wreath products, Sylow's theorem and Fermat's little theorem,
European Journal of Pure and Applied Mathematics, Vol. 3, No.1 (2010) P.1315.
Nothing lucky about 13 !, Mathematics Magazine, Vol.83, No.4 (2010) pp.289293.
Trigonometric expressions for Fibonacci and Lucas numbers,
Acta Math. Univ. Comenianae, Vol.LXXIX, 2(2010) P.199208.
Uncountably generated ideals of functions,
College Mathematics Journal, Vol.42:5 (2011) P.404406.
(with Manisha Kulkarni & Peter Muller)
Quadratic factors of f(x)g(y), Indagationes Math., Vol. 18 (2007) P.233243.
Some numbertheoretic identities from group actions, Rendiconti del circolo matematico di Palermo, Vol. 58 (2009) 99108.
(with I.Erovenko) Commutativity degree of wreath products of finite
abelian groups, Bulletin of the Australian Math. Soc., Vol.77, No.1
(2008) 3136.
An interesting consequence of the Heisenberg construction, Elemente der Mathematik, Vol.63 (2008) 184188.
(with Soma Purkait)
Some vanishing sums involving binomial coefficients in the denominator, Albanian Journal of Mathematics, Vol.2 (2008) 2732.
(with Manisha Kulkarni) On the Diophantine equation 1+x+x^2/2!+....+x^n/n! = g(y),
Proceedings of the International Conference on Diophantine equations
on the occasion of the 60th birthday of T.N.Shorey, Narosa Publishing House, New Delhi, Jan.2008, P.121134.
(with J.McLaughlin) Some observations on Khovanskii's matrix methods
for extracting roots of polynomials, Integers, Electronic Journal of Combinatorial Number Theory, Vol.7 (2007) # A48.
(Expository) The Congruence subgroup problem, Journal of the I.I.Sc.,Vol.84, No.4, 2007.
(with B.A.Sethuraman) A note on the special unitary group of a division algebra,
Proc. Amer.Math. Soc., Vol. 134 (2006) 351354.
(with M. Kulkarni) Diophantine equations with Bernoulli polynomials,
Acta Arithmetica, Vol. 116 (2005) 2534.
(with M. Kulkarni) A class of Diophantine equations involving Bernoulli polynomials,
Indagationes Math, Vol. 16 (2005) 5165.
(with James McLaughlin) Powers of a matrix and combinatorial
identities, Integers, Journal of Combinatorial number theory, Vol. 5 (2005) article A 13.
(with Ankit Gupta) Decimal expansion of 1/p and subgroup sums, Integers, Journal of
combinatorial number theory, Vol. 5 (2005) article A 19.
(with Y. Bilu & M. Kulkarni) The Diophantine equation
x(x+1) ..... (x+m1) + r = y^n,
Acta Arithmetica, Vol. 113 (2004) 303308, Special volume in honour of R.Tijdeman's 60th birthday.
(with M. Kulkarni) On the Diophantine equation x(x+1)....(x+m1) = g(y), Indagationes Math., Vol. 14 (2003) 3544.
(with Tianming Wang & FengZhen Zhao) Identities involving
reciprocals of binomial coefficients, Journal of Integer Sequences
(electronic) Vol. 7 (2004) article 04.2.8, 12 pages.
On grandaunts and Italian cousins, The Mathematical Gazette, Article 92.01,
March 2008,pp.6364.
A generalisation of the irrationality of e, The Mathematical Gazette, Article 92.76,
November 2008, pp.535.
Vandermonde for cyclicity, The Mathematical Gazette, Article 93.24, July 2009, pp.3032.
A rectangular occurrence of pi, The Mathematical Gazette,
Article 93.53, November 2009, pp.3638.
Page 2
Page 3
Group theory lends a hand to number theory, The Mathematical Gazette, Article 94.13, July 2010, pp.1416.
A ring sine function  Integers, Electronic Journal of Combinatorial Number Theory,
Vol.4, Article A 17, 2004.
Central extensions of division algebras over padic fields,
Proc. Inst. Math. Sci., Ramanujan Mathematical Society, Vol. 1, 2004, p.111.
(with D.S.Nagaraj) On the commutativity of rings  Journal of the Ramanujan Math. Soc.,Vol. 8 (2003) 175180.
A parent of Binet's formula ?  Mathematics Magazine, Vol.77 (2004) P.308310.
(with R. Thangadurai) Gao's conjecture on zerosum sequences  Proc. Indian Acad. Sci., Vol.112 (2002) 399  414.
(with M. Kulkarni) On the vanishing of cubic recurrences  C.R.Math.Acad.Sci. Soc.R.Canada, Vol.24 (2002) 7276.
Values of Euler Polynomials  C.R. Math. Acad. Sci. Soc. R. Canada, Vol.23 (2001) P.1215.
An elementary proof of the HilbertMumford criterion  Electronic J. of Linear Algebra, Vol.7 (2000) P.174177.
On the Diophantine equation x^2 + 2 = y^n  Archiv der Math., Vol.74 (2000) P.350355.
On triangular representations of Aut (Fr)  J. of Algebra, Vol.222 (1999) P.164173.
(Ignore Corollary 2 which is false !)
On an example of Jacobson, Amer. Math. Monthly, Vol.106 (1999) P.7576.
On a conjecture of Chowla et al.  J. Number Theory, Vol.72 (1998) P.137139.
On permutation representations of polyhedral groups  Proc. Amer. Math. Soc., Vol.127 (1999) P.19731974.
Arithmetic of subgroup counting in some free products  Linear and Multilin. Algebra, Vol.44 (1998) P.347355.
The ChevalleyWarning theorem and a combinatorial question on finite groups  Proc. Amer. Math. Soc., Vol.127 (1999) P.951953.
(With V.P. Platonov) Adelic profinite groups  Jour. of Algebra, Vol.193 (1997) P.757763.
Bounded generation does not imply finite presentation  Comm. Alg., Vol.25 (1997) P.16731683.
Abel's theorem and a Gaussian Determinant  Expositiones Mathematicae, Vol.14 (1996) P.8591.
An integral polynomial  Mathematics Magazine, Vol.68 (1995) P.134135.
(With T.N. Venkataramana) Generators for all principal congruence subgroups of SL(n,Z)
with n ( 3  Proc. Amer Math. Soc., Vol.122 (1994) P.355358.
A variant of Moore's uniqueness of reciprocity laws  Mathematika, Vol.40 (1993) P.246255.
Central extensions of groups over local fields; a Theorem of Tate  Communications in Algebra, Vol.21 (1993) P.12031213.
The value of Bernoulli polynomials at rational numbers  Bulletin of the London Math. Soc., Vol.25 (1993) P.327329.
Fractional parts of log p and a digit function  Expositiones Math., Vol.11 (1993) P.381384.
Sum of the reciprocals of the binomial coefficients  Euro. J. Combinatorics, Vol.14 (1993) P.351353.
A curious polynomial identity  Nieuw Arch Wisk, Vol.11 (1993) P.9396.
(With U. Balakrishnan) Sum of digits of divisors  Nieuw Arch Wisk., Vol.11 (1993) P.169170.
Crossjump numbers  Fibonacci Quarterly, Vol.32 (1994) P.17.
Arithmetic groups and Salem numbers  Manuscripta Math., Vol.75 (1992) P.97102.
Congruence subgroup problem for anisotropic groups over semilocal rings  Proc. Indian Acad. Sci., Vol.101 (1991) P.87110.
American Math. Monthly, Filler, December 2008
American Math. Monthly, Filler, April 2009
(with Dinesh Khurana) A problem from the Monthly
