Here is an announcement of a past

Here are the pdf files of some talks I have given at assorted places:

1. Planar algebras and 1+1-dimensional TQFTs (lectures given at UBC, Vancouver in September 2006)
2. Hopf C^*-algebras and their quantum doubles - from the point of view of subfactors (lecture given at Max-Planck Institut in Bonn in August 2007)
3. $II_1$ factors and Ergodic Theory (lecture given at ISI, Bangalore in February 2008)
4. Non-crossing partition and 2-cabled Temperley-Lieb planar algebras (lecture given at BIRS, Banff in March 2008)
5. von Neumann algebras and free probability (lecture given at the Ramanujan Math. Society at Bangalore in May 2009)
6. Catalan numbers (lecture given at CESCI, Madurai to school students on August 25, 2009.)
7. Free probability: classical and free CLTs (lecture given at ISI, Bengaluru in November 2009)
8. On the GJS construction for graphs: I. Presentation of two categories} (first of two-part lecture given along with Vijay Kodiyalam at Vanderbilt University in May 2010)

1. Temperley-Lieb algebras and non-crossing partitions (lecture given at the Ramanujan Institute for Advanced Study in Mathematics)
2. When is a knot not the unknot? (lecture given at IMSc)
3. Braids (lecture given at Ethiraj College)
4. The Perron Frobenius Theorem (lecture given at Chennai Mathematical Institute)
5. Transfinite considerations (lecture given to finishing school students at IIT, Madras)

Here are the pdf files of a course of lectures I gave (at IIT, Kanpur, ISI, Bangalore and at IISc, Bangalore during 2008-'09) as part of the "Panorama of Mathematics" series organised by the National Board for Higher Mathematics. The course was titled "Operator algebras - Stage for non-commutativity", and consisted of five lectures, as follows:

1. $C^*$-algebras: the Gelfand Naimark Theorems
2. K-theory of $C^*$-algebras
3. von Neumann algebras
4. $II_1$ factors and subfactors
5. The Jones polynomial invariant of knots

Here are the pdf files of a course of lectures I gave at IIT, Bombay, in April 2007, on "Temperley-Lieb algebras"

1. Temperley-Lieb algebras
2. Wenzl's theorem
3. von Neumann algebras, II_1 factors and their subfactors
4. The standard invariant of a subfactor

Here are the pdf files of a course of lectures I gave on von Neumann algebras at IISc, Bangalore in March 2006.

Here is an announcement of some lectures some of us gave to beginning undergraduate students in the Chennai area:

More material on the linear algebra course which I taught.

Research Interests

List of publications

Monographs

• Bounded Integral operators on $L^2$-spaces, with P.R. Halmos, Ergeb. der Math., No. 96, Springer-Verlag, Berlin, 1978. (M.R. 80g:47036 and 87f:47041, Zbl 573.47020, 389.47001).
• An invitation to von Neumann algebras, Universitext, Springer-Verleg, New York, 1986. (M.R. 88f:46115, Zbl 616.46052).
• Introduction to Subfactors, with Vaughan Jones, London Math. Soc. Lect. Note Series, No. 234, Cambridge Univ. Press, Cambridge, 1997. (M.R.98h:46067 Zbl 903.46062).
• Functional Analysis: Spectral Theory, TRIM Series, No. 13, Hindustan Book Agency, Delhi, 1997, and Birkhauser Advanced Texts, Basel, 1997. (M.R. 99h:47001, Zbl 957.47001, 919.46002).
• Topological quantum field theories from subfactors, with Vijay Kodiyalam, CRC Research Notes in Mathematics Series, 423, 2000 (Zbl 992.07801).
• Measure and Probability, with Siva Athreya, Universities Press, Hyderabad, 2008. (ISBN:978-81-7371-613-3)

Papers

• Two characterisation theorems for integral operators, Integral Eqns. and Oper. Theory, 1(1978), 250-269. (M.R.80d:47079, Zbl 376.47024).
• Absolutely bounded matrices, Indiana Univ Math. J. 27, (1978),919-927. (M.R. 80d:47052, Zbl 419.47011).
• Algebra of kernels, J of the Indian Math. Soc., 44, (1979), 143-156. (M.R. 86j:47076, Zbl 621.47031).
• Unitary equivalence to integral operators, Pacific J, of Math. 92, (1981), 211-215 (M.R. 82f:47045, Zbl 508.47020).
• A non-commutative analogue of $|D X^k| = |k x^{k-1}|$, Lin. Alg. and Applicns, 44(1982), 87-95, (M.R. 83h:47004, Zbl 486.46050).
• Distance between normal operators, Proc. Amer. Math. Soc. 84(1982), 483-484, (M.R. 83c:47040, Zbl 486.47015).
• On permutations, convex hulls and normal operators, Lin Alg and Applicns., 48,(1982),403-411.(M.R.85b:15032, Zbl 537.15014).
• A spectrum modulo operators of rank less than n, Math. Today , 1,(1983), 81-88. (M.R.85a:47018, Zbl 532.47010).
• On ideals and duals of $C^*$-algebras, Indian J. of Pure and Appl. Math, 15,(1984),601-608.(M.R.85i:46079, Zbl 588.46039).
• On majorisation and Schur products, with R.B. Bapat, Lin. Alg and Applicns., 72,(1985), 107-117. (M.R.87c:15018, Zbl 577.15016).
• An extremal property of the determinant and the permanent, with R.B. Bapat, Lin. Alg. and Applicns, 76, (1986), 153-164.(M.R.87j:15012, Zbl 602.15008).
• Stochastic integration in Fock space, Pac. J. of Math., 122,(1986), 481-491. (M.R.88c:60126, Zbl 583.60051, 543.60063).
• A model for AF-algebras and a representation of the Jones projections, J. of Oper. Theory, 18,(1987),289-301.(M.R.89e:46079, Zbl 693.46054).
• N subspaces, Canadian J. of Math., Vol XL, No.1,(1988), 38-54. (M.R.89i:46023, Zbl 656.46017).
• Monomials in the Jones projections, Proc. Amer. math. Soc., 103, (1988),761-764. (M.R.89f:46119, Zbl 673.46040).
• Hypergroups and subfactors, Proc. of a symposium on Operator Theory and Functional Analysis, Cochin (1989) 49-65. (M.R. 91 m: 46096, Zbl 718.46035).
• The Laplacean spectrum of a graph, with R. Grone and R. Merris, SIAM J. of Matrix Anal. and Applns., 11 (1990), 218-238.(M.R.91c:05130, Zbl 733.05060).
• Pairs of $II_1$ factors, Proc. of the Indian Acad. of Sci. (Math. Sci.), Vol. 100, No.2, August 1990, pp. 357-377. (M.R.:92b:46098, Zbl 727.46039).
• The work of Vaughan Jones - From subfactors to knot invariants, Current Science,59,(1990),1285-1292.
• On hypergroups of matrices, with R.B. Bapat, J. of Lin. and Multilin. Alg., 29, (1991), 125-140.(M.R.92k:20137, Zbl 734.20033).
• On commuting squares and subfactors, J. of Functional Anal. 101(1991), 286-311.(M.R.92j:46111, Zbl 739.46042).
• $II_1$ factors, their bimodules and hypergroups, Trans. of the Amer. Math. Soc. 330(1992), 227-256. (M.R. 92f:46076, Zbl 757.46053).
• Unitary orbits of self-adjoints in some $C^*$- algebras, with Klaus Thomsen, Houston J. of Math. 18(1992). 127-138, (Special Volume dedicated to the memory of Domingo Herrero). (M.R. 93e:46071, Zbl 812.46057).
• From hypergroups to subfactors, Operator Algebras and Operator Theory, Ed. W. Arveson et al,Proceedings of OATE 2 conference, Romania 1989, Pitman Research Notes in Mathematics, Longman Scientific, Essex, UK, 1992, pp. 198-216. (M.R. 93m:46074, Zbl 792.46045).
• On the non-occurrence of the Coxeter graphs $\beta_{2n+1}, D_{2n+1}$ and $E_7$ as the principal graph invariant of an inclusion of $II_1$ factors, with A.K. Vijayarajan, Pacific J. of Math., 161,(1993),185-200. (M.R.94g:46067, Zbl 798.43005)
• On some convex sets and their extreme points, with A. Rajarama Bhat and V. Pati, Math. Annalen., 296,(1993),637-648.(M.R.94f:46076, Zbl 791.46006)
• On principal graphs and weak duality, with Uma Krishnan, Math. Scand., 74,(1994),49-63.(M.R.95i:46087, Zbl 809.05057).
• On the relation between subfactors and hypergroups, Applications of hypergroups and related measure algebras, Contemp. Math., 183, pp. 331-340, 1995. (M.R.96d:46087, Zbl 830.46053).
• On boundedness and $||~||_p$-continuity of second quantisation, with H. Araki and K.B. Sinha, Publ. R.I.M.S., 31, (1995), 941-952. (M.R.97j:81166, Zbl 848.47017).
• On some subfactors of integer index arising from vertex models, with Uma Krishnan and Cherian Varughese, J. Functional Analysis, 140, (1996), 449-471. (M.R.97j:46066, Zbl 899.46049).
• On biunitary permutation matrices and some subfactors of index $9$, with Uma Krishnan, Trans. of the Amer. Math. Soc., 348, (1996), 4691-4736. (M.R. 97c:46077, Zbl 862.46035).
• Notes on the imprimitivity theorem, Analysis, Geometry and Probability, an expository volume brought out in honour of K.R. Parthasarathy), 299-321, Hindustan Book Agency, Delhi, 1996. (M.R.98k:22027, Zbl 949.22007).
• Exponentials of indicator functions are total in the Boson Fock space $\Gamma (L^2[0,1])$, with K.R. Parthasarathy, Quantum Probability Communications, Vol. X, (1998), 281-284. (M.R.2000j:81136)
• Schrodinger operators with fairly arbitrary spectral features, with M. Krishna, Rev. Math. Phys., 19, (1997), 343-360. (M.R.98d:47100, Zbl 879.34081).
• Commuting Squares, The Mathematics Student, 67, (1998), 65-72. (M.R.2001b:46101).
• Fusion Rule Algebras and Walks on Graphs, with N.J. Wildberger, The Proceedings of the Fifth Ramanujan Symposium on Harmonic Analysis, Ed. K. Parthasarathy, Publ. of the Ramanujan Inst., No.6, (1999), 53-80.
• The subgroup subfactor, with Vijay Kodiyalam, Math. Scand., 86, (2000), 45-74. (M.R.2001b:46103, Zbl 991.60290).
• The algebra of $G$-relations, with Vijay Kodiyalam and R. Srinivasan, Proc. of the Indian Acad. Sci. (Math. Sci.), 110, (2000), 263-292. (MR. 2001k:16019, Zbl 992.06392).
• Flatness and fusion coefficients, with Vijay Kodiyalam, Pac. J. Math., 201, (2001), 177-204. (M.R. 2002k:46162).
• Spectra of Principal Graphs, with Vijay Kodiyalam, Internat. J. Math., 12, (2001), 203-210. (M.R.2002b:46101).
• Discrete series of fusion algebras, with R. Srinivasan and Norman Wildberger, J. of Aust. Math. Soc., 72, (2002), 419-425. (M.R. 2003j:46096).
• Planar depth and planar subalgebras, with Zeph Landau, J. of Functional Analysis, 195, (2002), 71-88.(M.R. 2003d:46080).
• On trace zero matrices, Resonance, 7, No.6, (2002), 14-26.
• Actions of finite hypergroups, with Norman Wildberger, J. of Algebraic Combinatorics, 18(2), (2003), 135-151.. (M.R. 2004g:20092).
• The planar algebra associated to a Kac algebra, with Vijay Kodiyalam and Zeph Landau, Proc. of the Indian Acad. Sci. (Math. Sci.), 113, (2003), 15-51. (M.R. 2004d:46075)
• Complete invariants for complex semisimple Hopf algebras, with Sumant Datt and Vijay Kodiyalam, Math. Res. Lett., 10, No. 5-6, (2003), 571-587.
• On Jones' planar algebras, with Vijay Kodiyalam, J. of Knot Theory and its Ramifications, 13, No. 2(March), (2004), 219-248.
• A complete set of numerical invariants for a subfactor, with Vijay Kodiyalam, J. of Functional Analysis, 212, (2004), 1-27.
• Hamel bases and measurability, with M.G. Nadkarni, Mathematics Newsletter, vol. 14, No. 3, (2004).
• The planar algebra of a semisimple and cosemisimple Hopf algebra, with Vijay Kodiyalam, Ind. Acad. of Sci., vol. 116 no. 4, (2006), 443-458; and e-print arXiv math.QA/0506153.
• von Neumann Algebras: Introduction, modular theory and classification theory, Encyclopedia of Math. Physics, eds. J.-P. Francoise, G.L. Naber and Tsou S.T., Oxford: Elsevier, 2006 (ISBN 978-0-1251-2666-3), vol 5 pp. 379-385.
• Subfactors and 1+1-dimensional TQFTs, with Vijay Kodiyalam and Vishwambhar Pati, International Journal of Mathematics, Vol. 18, No. 1 (2007) 69-112, e-print arXiv math.QA/0507050.
• Paul Halmos - Expositor par excellence, Resonance, (2007), 17-28.
• Temperley-Lieb and Non-crossing Partition planar algebras , with Vijay Kodiyalam, Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications Edited by: S. K. Jain, Ohio University, Athens, OH, and S. Parvathi, University of Madras, Chennai, India, Contemp. Math., Vol. 456, (2008).
• The Riesz representation theorem , Indian J. of Pure and Appl. Math., 39(6), December 2008, 467-481.
• von Neumann Algebras and Ergodic Theory , in `Perspectives in Mathematical Sciences II, Pure Mathematics', 199-208, World Scientific, June 2009.
• Kac algebras, quantum doubles and planar algebras , with S. Jijo, Contemporary Math. Vol. 490, 97-104, 2009.
• von Neumann algebras and free probability, Mathematics Newsletter of the Ramanujan Mathematical Society, Vol. 19, No. 1, pp. 7-9, June 2009.
• Planar algebras and Kuperberg's 3-manifold invariant, with Vijay Kodiyalam, Journal of Operator Theory (to appear), and e-print arXiv math.OA/0807.3714.
• From subfactor planar algebras to planar algebras, with Vijay Kodiyalam, Internat. J. Math. vol. {\bf 20}, No. 10, pp. 1207-1231, 2009, and e-print arXiv math.OA/0807.3714.
• Guionnet-Jones-Shlyakhtenko subfactors associated to finite-dimensional Kac algebras, Journal of Functional Analysis, Vol. 257, (2009), 3930-3948 and e-print arXiv math.OA/0901.3180.
• On the Guionnet-Jones-Shlyakhtenko construction for graphs, with Vijay Kodiyalam, e-print arXiv (math. OA) 0911.2047
• Double cones are intervals , Operator Theory: Advances and Applications, Vol. 207, 359-362, Birkhauser Verlag, Basel/Switzerland, 2010.

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