Prof. Sethuraman: Research and Publications

Research Interests

Finite-dimensional division algebras. 

Matrix varieties: applications of algebraic geometry to problems in matrix theory. 

Application of division algebras to space-time coding (wireless communication).

Biography

      Ph.D 1991, University of California, San Diego, Mathematics.

      B.Tech 1980, Indian Institute of Technology (IIT) Madras, Mechanical Engineering.

       Positions held

• Professor, Krea University, India (2020 to present)
• Emeritus Professor, California State University Northridge (2016 to present)
• Professor, California State University Northridge (1999 to 2016)
• Visiting Professor, Indian Statistical Institute Bangalore, India (Spring 2012)
• Visiting Professor, Indian Institute of Technology Bombay, India (2010 to 2011).
• Visiting Professor, University of California at Santa Barbara, Spring 2000.
• Associate Professor, California State University Northridge (1994 to 1999).
• Visiting Associate Professor, University of Southern California, Spring 1998.
• Visiting Associate Professor, New Mexico State University, Spring 1996.
• Assistant Professor, California State University Northridge (1991 to 1994)
• Instructor, University of San Diego, 1990--91.
• Software Engineer, Calma, San Diego, 1984--1988.
• Engineer Trainee, Eicher, New Delhi, 1980--1981.

      Grants

• National Science Foundation, Division of Computing and Communication Foundations, 2013-2016, $249,975.
• National Science Foundation, Division of Mathematical Sciences, 2007-2012, $121,031.
• National Science Foundation, International Research, 2002--2005, $27,178
• K.B. Chandrashekhar Family Foundation Award, 2000-2001, $113,865.
• National Science Foundation, Division of Mathematical Sciences, 1994--97, $59,850.
• Miscellaneous internal CSU Northridge Research Awards, 1991--Present.

     Publications  

     Research Publications

• Mutually Orthogonal Matrices From Division Algebras, Journal of Pure and Applied Algebra. 222, 3538—3546, 2018.
• Counterexample to the Generalized Belfiore-Solé Secrecy Function Conjecture for l-modular lattices (with Anne-Maria Ernvall-Hytönen), IEEE Transactions on Information Theory. 62, 4514—4522, August 2016.
• Bounds on fast decodability of space-time block codes, skew-Hermitian matrices, and Azumaya algebras (with Gregory Berhuy and Nadya Markin), IEEE Transactions on Information Theory, 61, 1959--1970, April 2015.
• Hilbert series of certain jet schemes of determinantal varieties (with Boyan Jonov and Sudhir Ghorpade), Pacific J. Math., 272(1):147--175, 2014. 
• Quotients of Orders in Cyclic Algebras and Space-Time Codes (with Frederique Oggier), Advances in Mathematics of Communications 7, 441--461, November 2013.
• Division algebras and wireless communication, Notices of the AMS , 57 1432--1439, December 2010.
• Jet schemes of the commuting matrix pairs scheme (with Klemen Sivic), Proceedings of the AMS 137, 3953-3967, 2009.
• A Note on the Hankel Transform of the Central Binomial Coefficients (with Mario Garcia Armas), Journal of Integer Sequences, 11 Article 08.5.8, 2008.
• Perfect space-time codes for any number of antennas (with Petros Elia and P. Vijaykumar), IEEE Transactions on Information Theory, 53, 3853--3868, November 2007.
• Construction of Orthonormal Lattices and Quaternion Division Algebras for Totally Real Number Fields (with Frederique Oggier), Proceedings of Applied Algebra and Error Correcting Codes 17 (Lecture Notes in Computer Science), Bangalore, December 2007.
• Moving Zeros Among Matrices (appendix with Tomaz Kosir to a paper by John Holbrook and Jean-Pierre Schoch), Linear Algebra and Applications, Special Issue Honoring Roger Horn, 424, 83--95, 2007.
• Information-lossless space-time block codes from crossed product algebras (with B.Sundar Rajan and V.Shashidhar) IEEE Transactions on Information Theory, 52, 3913--3935, September 2006 .
• Valuation on tensor powers of a division algebra (with Pat Morandi), Journal of Algebra, 293, 385--394, 2005.
• A note on the special unitary group of a division algebra (with B. Sury), Proc. Amer. Math. Soc, 134, 351--354, 2005.
• A Groebner basis for the 2X2 determinantal ideal mod t^2 (with Tomaz Kosir), Journal of Algebra, 292, 138--153, 2005.
• Division algebras with an anti-automorphism but no involution (with Pat Morandi and Jean-Pierre Tignol), Advances in Geometry, 5, 485--495, 2005.
• Determinantal varieties over truncated polynomial rings (with Tomaz Kosir), Journal of Pure and Applied Algebra, 195, 75--95, 2005 .
• Full-diversity, high-rate space-time block codes from division algebras (with B. Sundar Rajan and V. Shashidhar), IEEE Transactions on Information Theory, Oct 2003, Special Issue on Space-Time Coding. 
• Decomposition of involutions on inertially split division algebras, (with Pat Morandi), Math. Zeitschrift , 235 ,  195--212, 2000. 
• Generalized cocycles with values in one-units of henselian division algebras, (with Pat Morandi), Journal of Algebra , 224 , 123--129, 2000.
• Commuting pairs and triples of matrices, and related varieties, (with Robert Guralnick), Linear Algebra and Applications , 310 , 139--148, 2000.
• Commuting pairs in the centralizers of 2-regular matrices, (with Michael Neubauer) Journal of Algebra 214 (1999) 174--181. 
• Divisors on division algebras and error correcting codes, (with Pat Morandi) Communications in Algebra 26 (1998), 3211--3221.  
• Degrees of high-dimensional subvarieties of determinantal varieties, Proc. Amer. Math. Soc. 126 (1998) 9--14.
• Indecomposable division algebras with a Baer ordering, (with Pat Morandi) Communications in Algebra, 22 (1994), 5401--5418. 
• Noncrossed product division algebras with a Baer ordering, (with Pat Morandi) Proc. Amer. Math. Soc. 123 (1995), 1995--2003. 
• Kummer subfields of tame division algebras, (with Pat Morandi) Journal of Algebra, 172 (1995), 554--583. 
• Division algebras whose p^ith powers have arbitrary indices, Journal of Algebra, 165 (1994), 476--482. 
• Dec groups of arbitrarily high exponent, Pacific J. Math., 162 (1994), 373--391. 
• Indecomposable division algebras, Proc. Amer. Math. Soc., 114 (1992), 661-665. 

     Textbooks 

• Rings, Fields, and Vector Spaces, published in Springer-Verlag's Undergraduate Texts in Mathematics series, 1997, ISBN 0-387-94848-1. See also this link. 
• A Gentle Introduction to Abstract Algebra available for free download under GNU Free Documentation License.  
• Proofs and Ideas: A Prelude to Advanced Mathematics, to appear in MAA Press (an imprint of the American Mathematical Society), November 2021.

     Other Mathematical Publications 

• The Algebra Generated by Three Commuting Matrices, Mathematics Newsletter, Ramanujam Math. Soc., 21 number 2, September 2011, 26-31. 
• Fermat Numbers: A False Conjecture Leads to Fun and Fascination, At Right Angles, A Resource for School Mathematics, published by Azimji Premji Foundation, November 2013. 
• The Wedderburn Factorization Theorem and Valued Division Algebras (with Abhay Soman), Mathematics Student, 88, 55--66, 2019.

     Refereed Engineering Conference Proceedings

• Counterexample to the Generalized Belfiore-Solé Secrecy Function Conjecture for l-modular lattices (with Anne-Maria Ernvall-Hytönen), International Symposium on Information Theory, Hong Kong, 2015. 
• Fast lattice decodability of space-time block codes (with Gregory Berhuy and Nadya Markin), International Symposium on Information Theory, Honolulu, 2014. 
• The Belfiore-Solé Conjecture and a Certain Technique for Verifying it for a Given Lattice (with Julia Pinchak), ITA 2014. 
• Construction of Orthonormal Lattices and Quaternion Division Algebras for Totally Real Number Fields (with Frederique Oggier), Proceedings of Applied Algebra and Error Correcting Codes 17 (Lecture Notes in Computer Science), Bangalore, December 2007. (Cross listed above under Research Publications.) 
• Codes Achieving The DMD Tradeoff Of The MIMO-ARQ Channel} (with Petros Elia, K. Raj Kumar, P. Vijay Kumar, and Sameer A. Pawar), Proceedings of IEEE International Symposium on Information Theory (ISIT 2005).
• Perfect Space-Time Codes with Minimum and Non-Minimum Delay for Any Number of Antennas} (with Petros Elia and P. Vijay Kumar), WirelessCom 2005, International Conference on Wireless Networks, Communications, and Mobile Computing, June 2005
• Information -Lossless STBCs from Crossed-Product Algebras} (with B. Sundar Rajan and V. Shashidhar), Proceedings of IEEE International Symposium on
  Information Theory (ISIT 2004), Chicago, 2004.
• STBCs using capacity achieving designs from crossed-product division algebras} (with B. Sundar Rajan and V. Shashidhar), Proceedings of IEEE International Conference on Communications (ICC 2004), Paris, France, 2004.
• Information-Lossless Space-Time Block Codes from Division Algebras} (with B. Sundar Rajan and V. Shashidhar),Oberwolfach Meeting on Kodierungstheorie held at Mathematisches Forschungsinstitut Oberwolfach, Germany, 2003.
• STBCs using Capacity Achieving Designs from Cyclic Division Algebras (with B. Sundar Rajan and V. Shashidhar), IEEE Globecomm, San Francisco, 2003. 
• High-Rate Full-Diversity STBCs from Field Extensions (with B. Sundar Rajan et al), IEEE ISIT conference, Yokohama, Japan 2003. 
• An Algebraic Description of Orthogonal Designs and the Uniqueness of the Alamouti Code (with B. Sundar Rajan), IEEE Globecomm, Taipei, Taiwan 2002.
• Full-Rank, Full-Rate STBCs from Division Algebras (with B. Sundar Rajan), IEEE Information Theory Workshop, Bangalore, India 2002.
• STBC from field extensions of the rational field (with B. Sundar Rajan).  IEEE ISIT conference, Lausanne, Switzerland 2002.
• Optimal STBC from PSK signal sets from cyclotomic field extensions (with B. Sundar Rajan).  IEEE ICC conference, New York, 2002.

    Collaborations with Students and Postdocs 

• Beverly Hernandez: Masters student in pure mathematics at CSU Northridge. Wrote her masters thesis with me. She studied the l-modular secrecy function conjecture and proved that the lattice C^8 satisfies the conjecture. She made a series of very pretty conjectures on certain ratios of theta functions, and showed that if these conjectures are true, then the lattices C^l will satisfy the l-modular secrecy function conjecture for all l. She also studied the original Belfiore-Sole ́ conjecture on unimodular lattices, and showed that all lattices arising from certain self-dual codes of dimension up to 72 in a paper of Conway and Sloane satisfy this conjecture. 
• Julia Pinchak: Masters student in pure mathematics at CSU Northridge. Wrote her masters thesis with me. She studied the Belfiore-Sole ́ conjecture on unimodular lattices, and showed that infinitely many lattices satisfy the conjecture. She also showed that all lattices obtained by Construction A from binary, doubly even, self-dual codes of lengths up to 40 satisfy the conjecture. Her paper Wiretap Codes: Families of Lattices Satisfying the Belfiore-Sole ́ Secrecy Function Conjecture was presented at the International Symposium on Information Theory in 2013 and appeared in the Proceedings of IEEE ISIT 2013, pp. 2617--2620. Together, we wrote the paper The Belfiore-Sole ́ Conjecture and a Certain Technique for Verifying it for a Given Lattice for ITA 2014. She is currently working in the aero-space industry. 
• Sonia Chauhan: Masters student in pure mathematics at CSU Northridge. Wrote her masters thesis with me. She studied the principal component of the algebraic tangent bundle over the variety of m X (m+1) matrices of rank at most m-1 by associating a simplicial complex to a proposed Groebner Basis for the defining ideal. She is currently working on extending her thesis results for publication. 
• Boyan Jonov: Masters student in pure mathematics at CSU Northridge.  Wrote his masters thesis with me. His thesis resulted in two papers. His first paper from his MS thesis Intial complex associated to a jet scheme of a determinantal variety, a solo effort, has already appeared in Journal of Pure and Applied Algebra. His second paper from his MS thesis Hilbert series of certain jet schemes of determinantal varieties, joint with Sudhir Ghorpade any myself, will appear in the Pacific Journal of Mathematics. He is currently working on his Ph.D. at UCSB. 
• Klemen Sivic: Ph.D. student at the University of Ljubljana, Slovenia. We collaborated on the paper Jet schemes of the commuting matrix pairs scheme . His contributions were substantial and critical.
• Mario Garcia Armes: An undergraduate student at the Universidad de La Habana, Cuba.  We collaborated on the paper A Note on the Hankel Transform of the Central Binomial Coefficients ---he contributed very substantially to the underlying computations.
• Frederique Oggier: A former postdoc at Caltech. We collaborated on the paper Construction of Orthonormal Lattices and Quaternion Division Algebras for Totally Real Number Fields.
• Petros Elia: A former Ph.D student at USC. We collaborated on the paper Perfect space-time codes for any number of antennas.
• V. Shashidhar: A former Ph.D student at Indian Institute of Science.  We collaborated on two papers: Information-lossless space-time block codes from crossed product algebras and Full-diversity, high-rate space-time block codes from division algebras.

    Other Publications

• Op-Ed piece,  April 1999, published in the Los Angeles Times, also in the Sacramento Bee, and in a slightly modified version, joint with Beth Newell, in the Fresno Bee.

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Last modified October 19th 2021.