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, Zero-sum 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. ISBN-10-81-85931-38-0. Distributed by the AMS except in India : ISBN-13-978-81-85931-38-8.
  • Group theory : selected problems, Universities Press (India) Private Ltd., 2004, Hyderabad, India. Distributed by Orient Longman Pvt.Ltd. ISBN 81-7371-491-6. (www.universitiespress.com, www.orientlongman.com)
  • Hecke algebras, Intersection cohomology of Schubert varieties and Representation theory (1984 vintage - unedited, unpublished !).


  • A special mathematical review.

    Publications


  • (with Aleksander V. Morgan \& Andrei S. Rapinchuk), Bounded generation of $SL_2$ over rings of $S$-integers with infinitely many units, Submitted.
  • (with Vaibhav Pandey \& Sagar Shrivastava), A Dedekind Domain with Nontrivial Class Group, Amer. Math. Monthly, To appear.
  • A Ring Theoretic Approach to Bound the Totient Function, Filler, Amer. Math. Monthly, To appear.
  • (with Kannappan Sampath) Norm or Exception?, Mathematics Student, Vol. 86, January-June 2017 (pp.39-50).
  • (with M.Kulkarni & D.Majumdar) l-Class groups of cyclic extensions of prime degree l, J. Ramanujan Math. Soc., Vol. 30, No. 4 (2015) 413-454.
  • (with Nikolay Nikolov), Bounded generation of wreath products, Journal of Group Theory, Vol.18, No.6 (2015) 951-959.
  • Matrix groups over rings, Hansraj Gupta Memorial Award talk at the annual conference of the Indian Mathematical Society, Mathematics Student, Vol. 85 (2016) 79-96.
  • (with Manisha Kulkarni), Diophantine equations of the form F(x)=G(y) - an exposition, Mathematics Student, Vol. 84 (2015) 135-155.
  • (with S.K.Prajapati) On the total character of finite groups, International Journal of Group Theory, Vol.3, No.1 (2014) pp.47-67.
  • (with N.A.Vavilov & A.V.Smolensky) Gauss decomposition for Chevalley groups - Revisited, International Journal of Group Theory, Vol.1, No.1 (2012) pp.3-16.
  • (with N.A.Vavilov & A.V.Smolensky) Unitriangular factorizations of Chevalley groups, Zapiski Nauchnyh Seminarov POMI, vol.388 (2011), p.17-47(in Russian).
  • English translation: J. Math. Sci. (N.Y.) 183 (2012) No.5, 584-599. Read G as E in the main theorems!
  • Tits building and an application to central extensions of p-adic algebraic groups by finite p-groups, Proc.Amer.Math.Soc., Vol.139, No.6 (2011) pp.2033-2044.
  • (with Alec W.Mason) Subgroups of algebraic groups which are clopen in the S-congruence topology, Journal of Group Theory, Vol.15 (2012) P.47-55.
  • Macaulay expansion, The American Mathematical Monthly, Vol. 121 (2014) pp.359-360.
  • Some applications of Representation Theory to Classical Number Theory, Mathematics Newsletter, Ramanujan Mathamtical Society, Vol. 25, September 2014, Page 130-139.
  • Chord lengths, Discriminants of cyclotomic fields, and reducibility of cyclomic polynomials modulo primes, Mathematics Newsletter, Ramanujan Mathematical Society, Vol. 24, March 2014, Page 94-97.
  • Riemann Hypothesis - The Prime Problem, Mathematics Newsletter, Ramanujan Mathematical Society, Vol. 21, September 2011, Page 12-26.
  • Ramanujan's awesome sums, Mathematics Newsletter, Ramanujan Mathematical Society, Vol. 24, September 2013, Page 31-37.
  • A polynomial parent to a Fibonacci-Lucas 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.18-25.
  • (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 (x-a_1)(x-a_2) ...(x-a_k) + r = y^n, Proc.Indian Acad.Sci., Vol.121, No.3 (2011) pp.245-247.
  • (with S.D.Adhikari & A.A.Ambily) Zero-sum problems with subgroup weights, Proc.Indian Acad.Sci., Vol.120 (2010) pp.259-266.
  • A generalisation of a converse to Schur's theorem, Arch.Math., Vol.95, No.4 (2010) 317-318.
  • (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. 43-72.
  • On SU(1,D)/[U(1,D),U(1,D)] for quaternion algebras, Arch. Math., Vol.90 (2008) 493-500.
  • 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.641-665.
  • Wreath products, Sylow's theorem and Fermat's little theorem, European Journal of Pure and Applied Mathematics, Vol. 3, No.1 (2010) P.13-15.
  • Nothing lucky about 13 !, Mathematics Magazine, Vol.83, No.4 (2010) pp.289-293.
  • Trigonometric expressions for Fibonacci and Lucas numbers, Acta Math. Univ. Comenianae, Vol.LXXIX, 2(2010) P.199-208.
  • Uncountably generated ideals of functions, College Mathematics Journal, Vol.42:5 (2011) P.404-406.
  • (with Manisha Kulkarni & Peter Muller) Quadratic factors of f(x)-g(y), Indagationes Math., Vol. 18 (2007) P.233-243.
  • Some number-theoretic identities from group actions, Rendiconti del circolo matematico di Palermo, Vol. 58 (2009) 99-108.
  • (with I.Erovenko) Commutativity degree of wreath products of finite abelian groups, Bulletin of the Australian Math. Soc., Vol.77, No.1 (2008) 31-36.
  • An interesting consequence of the Heisenberg construction, Elemente der Mathematik, Vol.63 (2008) 184-188.
  • (with Soma Purkait) Some vanishing sums involving binomial coefficients in the denominator, Albanian Journal of Mathematics, Vol.2 (2008) 27-32.
  • (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.121-134.
  • (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) 351-354.
  • (with M. Kulkarni) Diophantine equations with Bernoulli polynomials, Acta Arithmetica, Vol. 116 (2005) 25-34.
  • (with M. Kulkarni) A class of Diophantine equations involving Bernoulli polynomials, Indagationes Math, Vol. 16 (2005) 51-65.
  • (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+m-1) + r = y^n, Acta Arithmetica, Vol. 113 (2004) 303-308, Special volume in honour of R.Tijdeman's 60th birthday.
  • (with M. Kulkarni) On the Diophantine equation x(x+1)....(x+m-1) = g(y), Indagationes Math., Vol. 14 (2003) 35-44.
  • (with Tianming Wang & Feng-Zhen 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.63-64.
  • 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.30-32.
  • A rectangular occurrence of pi, The Mathematical Gazette, Article 93.53, November 2009, pp.36-38.
  • Page 2
  • Page 3
  • Group theory lends a hand to number theory, The Mathematical Gazette, Article 94.13, July 2010, pp.14-16.
  • A ring sine function - Integers, Electronic Journal of Combinatorial Number Theory, Vol.4, Article A 17, 2004.
  • Central extensions of division algebras over p-adic fields, Proc. Inst. Math. Sci., Ramanujan Mathematical Society, Vol. 1, 2004, p.1-11.
  • (with D.S.Nagaraj) On the commutativity of rings - Journal of the Ramanujan Math. Soc.,Vol. 8 (2003) 175-180.
  • A parent of Binet's formula ? - Mathematics Magazine, Vol.77 (2004) P.308-310.
  • (with R. Thangadurai) Gao's conjecture on zero-sum 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) 72-76.
  • Values of Euler Polynomials - C.R. Math. Acad. Sci. Soc. R. Canada, Vol.23 (2001) P.12-15.
  • An elementary proof of the Hilbert-Mumford criterion - Electronic J. of Linear Algebra, Vol.7 (2000) P.174-177.
  • On the Diophantine equation x^2 + 2 = y^n - Archiv der Math., Vol.74 (2000) P.350-355.
  • On triangular representations of Aut (Fr) - J. of Algebra, Vol.222 (1999) P.164-173. (Ignore Corollary 2 which is false !)
  • On an example of Jacobson, Amer. Math. Monthly, Vol.106 (1999) P.75-76.
  • On a conjecture of Chowla et al. - J. Number Theory, Vol.72 (1998) P.137-139.
  • On permutation representations of polyhedral groups - Proc. Amer. Math. Soc., Vol.127 (1999) P.1973-1974.
  • Arithmetic of subgroup counting in some free products - Linear and Multilin. Algebra, Vol.44 (1998) P.347-355.
  • The Chevalley-Warning theorem and a combinatorial question on finite groups - Proc. Amer. Math. Soc., Vol.127 (1999) P.951-953.
  • (With V.P. Platonov) Adelic profinite groups - Jour. of Algebra, Vol.193 (1997) P.757-763.
  • Bounded generation does not imply finite presentation - Comm. Alg., Vol.25 (1997) P.1673-1683.
  • Abel's theorem and a Gaussian Determinant - Expositiones Mathematicae, Vol.14 (1996) P.85-91.
  • An integral polynomial - Mathematics Magazine, Vol.68 (1995) P.134-135.
  • (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.355-358.
  • A variant of Moore's uniqueness of reciprocity laws - Mathematika, Vol.40 (1993) P.246-255.
  • Central extensions of groups over local fields; a Theorem of Tate - Communications in Algebra, Vol.21 (1993) P.1203-1213.
  • The value of Bernoulli polynomials at rational numbers - Bulletin of the London Math. Soc., Vol.25 (1993) P.327-329.
  • Fractional parts of log p and a digit function - Expositiones Math., Vol.11 (1993) P.381-384.
  • Sum of the reciprocals of the binomial coefficients - Euro. J. Combinatorics, Vol.14 (1993) P.351-353.
  • A curious polynomial identity - Nieuw Arch Wisk, Vol.11 (1993) P.93-96.
  • (With U. Balakrishnan) Sum of digits of divisors - Nieuw Arch Wisk., Vol.11 (1993) P.169-170.
  • Cross-jump numbers - Fibonacci Quarterly, Vol.32 (1994) P.17.
  • Arithmetic groups and Salem numbers - Manuscripta Math., Vol.75 (1992) P.97-102.
  • Congruence subgroup problem for anisotropic groups over semilocal rings - Proc. Indian Acad. Sci., Vol.101 (1991) P.87-110.
  • American Math. Monthly, Filler, December 2008
  • American Math. Monthly, Filler, April 2009
  • (with Dinesh Khurana) A problem from the Monthly