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, Bak's unitary groups, Diophantine equations, Applications of prime density theorems, Zero-sum problems in number theory, Class groups of cyclic extensions.

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.
I have been at the Indian Statistical Institute, Bangalore since May 1999.



Books :

  • 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  • Hecke algebras, Intersection cohomology of Schubert varieties and Representation theory (1984 vintage - unedited, unpublished !).

    Publications


  • (with M.Kulkarni & D.Majumdar) Class groups of cyclic extensions of prime degree over cyclotomic fields, Journal of the Ramanujan Mathematical Society, To appear.
  • (with Nikolay Nikolov), Bounded generation of wreath products, Journal of Group Theory, To appear.
  • (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.
  • 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

    Some Lecture Notes, Talks, expository articles :


  • Fifteen, two hundred and ninety and Bhargava, Three science Academies' lecture workshop at Christ University, November 2014.
  • ISI Mathematics Day, outreach programme, November 29, 2014.
  • Support problem, Kummer theory, nested radicals - 2nd talk in Shivaji College Delhi, Academy workshop, October 2014.
  • FLT - a very brief outline of ideas (in 1993).
  • Various decompositions in GL(n) - NBHM school on SL(2), TIFR Bombay (in 1992).
  • Matrices Elementary, My Dear Homs.
  • Mixed Motive, The Mathematical Intelligencer 1997.
  • The World of Diophantine Equations, Talk in UGC workshop in Pavanatma college, Idukki Kerala.
  • Steinberg's chapters 6,7; ATM workshop on Chevalley groups, IISER Pune, May 2013.
  • What is the Tits index and how to work with it.
  • Ramanujan's mathematics - some glimpses.
  • Howlett-Lehrer theorem.
  • Ramanujan's route to roots of roots - RMS Mathematics Newsletter.
  • Group theory and tiling problems.
  • Talk in St.Petersburg Mathematical Society
  • Some applications of Chebotarev's density theorem
  • A modern Indian method
  • The ubiquitous modular group
  • Basic group theory.
  • Congruence subgroup problem.
  • Free Groups - Basics.
  • Some applications of representations of finite groups to classical number theory.
  • Some exercises for the tutorials in the AIS above.
  • As easy as Pie I - At Right angles, April 2013.
  • As easy as Pie II - At Right angles, July 2013.
  • Groups - Beyond the undergraduate syllabi.
  • Lectures on commutative ring theory.
  • Very basic algebraic number theory
  • Hecke algebras, Intersection cohomology of Schubert varieties and Representation theory (1984 vintage).
  • Primes and Riemann Hypothesis.
  • Algorithms in algebraic number theory.
  • Bringing the inner product out
  • Primes, cryptography and elliptic curves.
  • Is e^{sqrt{163}} odd or even ?
  • Existence and uniqueness of groups for root data.
  • Springer's chapter 16.
  • (with A.Raghuram) Groups acting on trees.
  • More group-theoretic applications of geometric methods.
  • Explicit reciprocity laws.
  • Introduction to Number fields, Proc. Conference on cyclotomic fields, Bhaskaracharya Pratishthana, Poona, 1999.
  • Absolute values and completions in brief.
  • (with D.S.Nagaraj) A quick introduction to algebraic geometry and elliptic curves.
  • Elliptic curves over finite fields.
  • (with D.S.Nagaraj) Mordell-Weil theorem.
  • Subgroup growth.

  • Articles published in Resonance


     Some student-projects supervised :

  • How to count - an exposition of Polya's theory of enumeration.
  • A converse to the Cayley-Hamilton theorem.
  • Necklaces, periodic points and permutation representations.
  • How safe is Sam Lloyd's bet ?
  • A number-theoretic game.
  • The Amitsur-Levitzkii identity via graph theory.
  • Weyl's equidistribution theorem.
  • Sums of powers of primitive roots.
  • Theorema Aureum - I.
  • Theorema Aureum - II.
  • The support problem.
  • What is Hilbert's 17th problem?
  • Division rings and their theory of equations.