Books, monographs, and surveys

1 Texts and Readings in Mathematics / TRIM Cambridge University Press Matrix analysis and entrywise positivity preservers (Book), 2022.
Vol. 471 in London Mathematical Society Lecture Note Series, Cambridge University Press, xxii+292 pp.
and Vol. 82 in Texts and Readings In Mathematics (TRIM Series), Hindustan Book Agency, xxii+339 pp. 		Book-cover                               Book-cover    
(CUP book cover)             (TRIM book cover)
2 arXiv Positivity Survey A panorama of positivity (83 pp., with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Survey article, 2018 – split into two parts for publication:
Shimorin Volume Part 1: Dimension free – in Analysis of Operators on Function Spaces (the Serguei Shimorin Memorial Volume)
A. Aleman, H. Hedenmalm, D. Khavinson, M. Putinar, Eds., pages 117–165, Trends in Mathematics, Birkhauser, 2019.
CRM Proceedings Part 2: Fixed dimension – in Complex Analysis and Spectral Theory (in honor of Thomas Ransford)
Proceedings of the CRM Workshop held at Laval University, QC, May 21–25, 2018; G. Dales, D. Khavinson, J. Mashreghi, Eds.
CRM Proceedings, AMS Contemporary Mathematics 743, pages 109–150, 2020.
3 Lecture notes on Positivity Matrix analysis, Pólya frequency functions, and preservers of (total) positivity (PDF, 320+ pp.)
Expanded lecture notes from a topics graduate course (or for advanced undergraduates) taught in Spring 2018 and Autumn 2019.
Last updated on 27 Nov, 2023.
4 Yale University Press Beautiful, Simple, Exact, Crazy: Mathematics in the Real World (Book, 480 pp., with Anna Lachowska)
Yale University Press, 2015.
  • The book title is inspired from a talk by Israel Moiseevich Gelfand.
  • The course which I developed and taught in Spring 2011 at Yale, and which led to the book.
  • The announcements from Yale Mathematics and from Yale University.
5 arXiv TRIM Invited survey article: Representations of complex semi-simple Lie groups and Lie algebras
In Connected at Infinity II: A Selection of Mathematics by Indians (R. Bhatia, C.S. Rajan, and A.I. Singh, Eds.),
Texts and Readings in Mathematics (TRIM), Hindustan Book Agency, pages 85–129, 2013.

Publications by research interests; or list (PDF)

Representation theory Combinatorics, discrete mathematics, and algebra
(including FPSAC extended abstracts)
Analysis: positivity, Schur polynomials, probability, graph limits

Analysis: positivity, Schur polynomials, probability, graph limits     Back to top

(shares fifteen papers with "Combinatorics, discrete mathematics, and algebra" section)

arXiv KRP Volume, IJPAM Probability inequalities for strongly left-invariant metric semigroups/monoids, including all Lie groups
Indian Journal of Pure and Applied Mathematics (tribute volume in memory of K.R. Parthasarathy), in press.
(Summary: Sequel to paper in [Ann. of Prob. 2017], extending those+other results to all left-/right-invariant metric monoids, e.g. all Lie groups.)
arXiv Canadian Journal of Mathematics The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra (with Projesh Nath Choudhury)
Canadian Journal of Mathematics, in press.
arXiv Journal d'Analyse Mathematique Totally positive kernels, Pólya frequency functions, and their transforms (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Journal d'Analyse Mathématique 150 no. 1, pages 83–158, 2023.
arXiv Journal of Mathematical Analysis and Applications The Khinchin–Kahane and Lévy inequalities for abelian metric groups, and transfer from normed (abelian semi)groups to Banach spaces (with Bala Rajaratnam)
Journal of Mathematical Analysis and Applications 582 no. 2, article 127545 (18 pp.), 2023.
arXiv Journal of the European Mathematical Society Moment-sequence transforms (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Journal of the European Mathematical Society 24 no. 9, pages 3109–3160, 2022.
arXiv Acta Scientiarum Mathematicarum (Szeged) Matrix compression along isogenic blocks (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Acta Scientiarum Mathematicarum (Szeged) 88 nos. 1–2, 100th anniversary special issue, pages 417–448, 2022.
arXiv Transactions of the AMS Smooth entrywise positivity preservers, a Horn–Loewner master theorem, and symmetric function identities
Transactions of the American Mathematical Society 375 no. 3, pages 2217–2236, 2022.
arXiv FPSAC 2022 Blowup polynomials and delta-matroids of graphs (with Projesh Nath Choudhury)
(Extended abstract of the paper The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra)
Talk at 34th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2022),
Séminaire Lotharingien de Combinatoire 86B, Article #42, 2022.
arXiv Applied and Computational Harmonic Analysis Hirschman–Widder densities (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Applied and Computational Harmonic Analysis 60, pages 396–425, 2022.
arXiv Mathematics Research Reports Preservers of totally positive kernels and Pólya frequency functions (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Mathematics Research Reports 3, pages 35–56, 2022.
[Joint announcement – via a unified, probability-themed narrative – of many results from the three works arXiv:2006.16213, arXiv:2008.05121, and arXiv:2101.02129.]
arXiv Linear and Multilinear Algebra The p-norm of circulant matrices (with Ludovick Bouthat, Javad Mashreghi, and Frédéric Morneau-Guérin)
Linear and Multilinear Algebra 70 no. 21, pages 7176–7188, 2022.
arXiv American Journal of Mathematics On the sign patterns of entrywise positivity preservers in fixed dimension (with Terence Tao)
American Journal of Mathematics 143 no. 6, pages 1863–1929, 2021.
arXiv Proceedings of the American Mathematical Society Sharp nonzero lower bounds for the Schur product theorem
Proceedings of the American Mathematical Society 149 no. 12, pages 5049–5063, 2021.
(Summary: Refines results by Schur in Crelle 1911 and Vybíral in Adv. Math. 2020, with an application to numerical integration.)
arXiv Bulletin of the London Mathematical Society Sign non-reversal property for totally non-negative and totally positive matrices, and testing total positivity of their interval hull
(with Projesh Nath Choudhury and M. Rajesh Kannan)
Bulletin of the London Mathematical Society 53 no. 4, pages 981–990, 2021.
(Summary: Novel characterization of total positivity, using only sign non-reversal; novel test for total positivity of interval hull.)
arXiv Bulletin des Sciences Mathématiques On an asymptotic characterisation of Griffiths semipositivity (with Vamsi Pritham Pingali)
Bulletin des Sciences Mathématiques 167, article 102956 (7 pp.), 2021.
arXiv Advances in Operator Theory Differential calculus on the space of countable labelled graphs (with Bala Rajaratnam)
Advances in Operator Theory 6 no. 1, article 12 (28 pp.), 2021.
arXiv Advances in Operator Theory Probability inequalities and tail estimates for metric semigroups (with Bala Rajaratnam)
Advances in Operator Theory 5 no. 3, pages 779–795, 2020.
(Special issue in honor of Rajendra Bhatia)
arXiv Linear Algebra and its Applications Simultaneous kernels of matrix Hadamard powers (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Linear Algebra and its Applications 576, pages 142–157, 2019.
arXiv Algebra & Number Theory Polymath14: Homogeneous length functions on groups
(as D.H.J. Polymath, with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao)
Algebra & Number Theory 12 no. 7, pages 1773–1786, 2018.     [Blogpost 1]
Mathematics Student 10-page student-targeted account in The Mathematics Student 92 nos. 1-2, pages 127–136, 2023.
(Summary: Provides novel characterizations from analysis and geometry, of abelian, torsion-free groups.)
FPSAC 2018
Schur polynomials, entrywise positivity preservers, and weak majorization (with Terence Tao)
Extended abstract of the paper On the sign patterns of entrywise positivity preservers in fixed dimension,
30th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2018),
Séminaire Lotharingien de Combinatoire 80B, Article #14, 2018. 		Poster
(Poster)
arXiv Annals of Probability The Hoffmann-Jørgensen inequality in metric semigroups (with Bala Rajaratnam)
Annals of Probability 45 no. 6A, pages 4101–4111, 2017.
(Here is a sequel paper showing these results in a more primitive framework.)
arXiv Transactions of the AMS Preserving positivity for rank-constrained matrices (with Dominique Guillot and Bala Rajaratnam)
Transactions of the American Mathematical Society 369 no. 9, pages 6105–6145, 2017.
(The arXiv version was updated post-publication, in 2024.)
arXiv FPSAC 2017 The critical exponent: a novel graph invariant (with Dominique Guillot and Bala Rajaratnam)
(Extended abstract of following paper below)
29th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2017),
Séminaire Lotharingien de Combinatoire 78B, Article #62, 2017.
arXiv Journal of Combinatorial Theory Series A Critical exponents of graphs (with Dominique Guillot and Bala Rajaratnam)
Journal of Combinatorial Theory, Series A 139, pages 30–58, 2016.
arXiv FPSAC 2016 Schur polynomials and matrix positivity preservers (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
(Extended abstract of following paper below)
28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016),
DMTCS Proceedings BC, pages 155–166, 2016. 		Poster
(Poster)
arXiv Advances in Mathematics Matrix positivity preservers in fixed dimension. I (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Advances in Mathematics 298, pages 325–368, 2016.
Final version Comptes Rendus Mathematique Matrix positivity preservers in fixed dimension (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
(Announcement of preceding paper above)
Comptes Rendus Mathematique de l'Académie des Sciences 354 no. 2, pages 143–148, 2016.
arXiv Transactions of the AMS Preserving positivity for matrices with sparsity constraints (with Dominique Guillot and Bala Rajaratnam)
Transactions of the American Mathematical Society 368 no. 12, pages 8929–8953, 2016.
arXiv Journal of Combinatorial Theory Series A Differential calculus on graphon space (with Peter Diao, Dominique Guillot, and Bala Rajaratnam)
Journal of Combinatorial Theory, Series A 133, pages 183–227, 2015.
arXiv Journal of Mathematical Analysis and Applications Complete characterization of Hadamard powers preserving Loewner positivity, monotonicity, and convexity (with Dominique Guillot and Bala Rajaratnam)
Journal of Mathematical Analysis and Applications 425 no. 1, pages 489–507, 2015.
arXiv Linear Algebra and its Applications The critical exponent conjecture for powers of doubly nonnegative matrices (with Dominique Guillot and Bala Rajaratnam)
Linear Algebra and its Applications 439 no. 8, pages 2422–2427, 2013.
arXiv Pramana Solutions of several coupled discrete models in terms of Lamé polynomials of arbitrary order (with Avinash Khare and Avadh Saxena)
Pramana 79 no. 3, pages 377–392, 2012.
arXiv Preprint A remark on characterizing inner product spaces via strong three-point homogeneity (with Sujit Sakharam Damase)
Preprint, 2023.
arXiv Preprint Negativity-preserving transforms of tuples of symmetric matrices (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Preprint, 2023.
arXiv Preprint Matrix positivity preservers in fixed dimension. II: positive definiteness and strict monotonicity of Schur function ratios (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Preprint, 2023.
arXiv Preprint Multivariate blowup-polynomials of graphs (with Projesh Nath Choudhury)
Preprint, 2021.
arXiv Preprint Multiply positive functions, critical exponent phenomena, and the Jain–Karlin–Schoenberg kernel
Preprint, 2020.
arXiv Preprint The non-compact normed space of norms on a finite-dimensional Banach space
Preprint, 2018.
arXiv Preprint Model-free consistency of graph partitioning (with Peter Diao, Dominique Guillot, and Bala Rajaratnam)
Preprint, 2016.
arXiv Preprint Integration and measures on the space of countable labelled graphs (with Bala Rajaratnam)
Preprint, 2015.
arXiv Preprint On fractional Hadamard powers of positive block matrices (with Dominique Guillot and Bala Rajaratnam)
Preprint, 2014.

Combinatorics, discrete mathematics (including Schur functions), and algebra     (also FPSAC)     Back to top

arXiv Canadian Journal of Mathematics The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra (with Projesh Nath Choudhury)
Canadian Journal of Mathematics, in press.                 [FPSAC 2022 abstract]
arXiv European Journal of Combinatorics Distance matrices of a tree: two more invariants, and in a unified framework (with Projesh Nath Choudhury).
European Journal of Combinatorics 115, article #103787 (30 pages), 2024.
arXiv European Journal of Combinatorics From Cauchy's determinant formula to bosonic and fermionic immanant identities (with Siddhartha Sahi)
European Journal of Combinatorics 110, article #103683 (16 pages), 2023; erratum.
(Summary: Extends identities by Cauchy, Frobenius, and Loewner to (a) "any" ground ring $R$, (b) any power series over $R$, (c) any subgroup $G$ of signed permutations, and (d) any character – or even complex class function – of $G$.)
arXiv Journal of Algebraic Combinatorics Recovering affine-linearity of functions from their restrictions to affine lines (with Akaki Tikaradze)
Journal of Algebraic Combinatorics 58 no. 3, pages 761–773, 2023.
arXiv Transactions of the AMS Smooth entrywise positivity preservers, a Horn–Loewner master theorem, and symmetric function identities
Transactions of the American Mathematical Society 375 no. 3, pages 2217–2236, 2022.
arXiv Applied and Computational Harmonic Analysis Hirschman–Widder densities (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Applied and Computational Harmonic Analysis 60, pages 396–425, 2022.
arXiv Communications in Algebra Covering modules by proper submodules (with Akaki Tikaradze)
Communications in Algebra 50 no. 2, pages 498–507, 2022.
arXiv American Journal of Mathematics On the sign patterns of entrywise positivity preservers in fixed dimension (with Terence Tao)
American Journal of Mathematics 143 no. 6, pages 1863–1929, 2021.                 [FPSAC 2018 abstract]
arXiv Transactions of the AMS A combinatorial model for computing volumes of flow polytopes (with Carolina Benedetti, Rafael S. González D'León, Christopher R. H. Hanusa, Pamela E. Harris, Alejandro H. Morales, and Martha Yip).
Transactions of the American Mathematical Society 372 no. 5, pages 3369–3404, 2019.                 [FPSAC 2018 abstract]
arXiv Linear Algebra and its Applications A Carlitz–von Staudt type theorem for finite rings (with Akaki Tikaradze)
Linear Algebra and its Applications 568, pages 106–126, 2019.
arXiv Transactions of the AMS Generalized nil-Coxeter algebras over discrete complex reflection groups
Transactions of the American Mathematical Society 370 no. 4, pages 2971–2999, 2018.                 [FPSAC 2018 abstract]
arXiv Advances in Mathematics Faces of highest weight modules and the universal Weyl polyhedron (with Gurbir Dhillon)
Advances in Mathematics 319, pages 111–152, 2017.
arXiv Transactions of the AMS Standard parabolic subsets of highest weight modules
Transactions of the American Mathematical Society 369 no. 4, pages 2363–2394, 2017; erratum.
arXiv Final version Generalized nil-Coxeter algebras, cocommutative algebras, and the PBW property
AMS Contemporary Mathematics 688, pages 139–168, 2017.
(Proceedings of International Conference on Groups, Rings, Group Rings, and Hopf Algebras (in honor of Donald S. Passman))
arXiv Journal of Combinatorial Theory Series A Critical exponents of graphs (with Dominique Guillot and Bala Rajaratnam)
Journal of Combinatorial Theory, Series A 139, pages 30–58, 2016.                 [FPSAC 2017 abstract]
arXiv Advances in Mathematics Matrix positivity preservers in fixed dimension. I (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Advances in Mathematics 298, pages 325–368, 2016.                 [FPSAC 2016 abstract]
Final version Comptes Rendus Mathematique Matrix positivity preservers in fixed dimension (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
(Announcement of preceding paper above)
Comptes Rendus Mathematique de l'Académie des Sciences 354 no. 2, pages 143–148, 2016.
arXiv Journal of Combinatorial Theory Series A Differential calculus on graphon space (with Peter Diao, Dominique Guillot, and Bala Rajaratnam)
Journal of Combinatorial Theory, Series A 133, pages 183–227, 2015.
arXiv Algebras and Representation Theory Faces of weight polytopes and a generalization of a theorem of Vinberg (with Tim Ridenour)
Algebras and Representation Theory 15 no. 3, pages 593–611, 2012.
arXiv Axioms The sum of a finite group of weights of a Hopf algebra
Axioms 1 no. 3, pages 259–290, 2012.
(Special issue on: "Hopf Algebras, Quantum Groups and Yang-Baxter Equations")
arXiv Linear Algebra and its Applications Vector spaces as unions of proper subspaces
Linear Algebra and its Applications 431 no. 9, pages 1681–1686, 2009.
arXiv JPANTA Divisibility tests and recurring decimals in Euclidean domains (final section with Pieter Moree)
JP Journal of Algebra, Number Theory, and Applications (JPANTA) 7 no. 1, pages 1–32, 2007.
FUEJUM Divisibility tests
Furman University Electronic Journal of Undergraduate Mathematics (FUEJUM) 3, pages 1–5, 1997.
arXiv Preprint Matrix positivity preservers in fixed dimension. II: positive definiteness and strict monotonicity of Schur function ratios (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Preprint, 2023.
arXiv Preprint The lattice of nil-Hecke algebras over real and complex reflection groups (with Sutanay Bhattacharya)
Preprint, 2021.
arXiv Preprint Multivariate blowup-polynomials of graphs (with Projesh Nath Choudhury)
Preprint, 2021.
arXiv Preprint The additive-multiplicative distance matrix of a graph, and a novel third invariant (with Projesh Nath Choudhury)
Preprint, 2023.

FPSAC publications (International conference on Formal Power Series and Algebraic Combinatorics)

arXiv FPSAC 2022 Blowup polynomials and delta-matroids of graphs (with Projesh Nath Choudhury)
Extended abstract of the paper The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra
Talk at FPSAC 2022;
Séminaire Lotharingien de Combinatoire 86B, Article #42, 2022.
FPSAC 2018
Schur polynomials, entrywise positivity preservers, and weak majorization (with Terence Tao)
Extended abstract of the paper On the sign patterns of entrywise positivity preservers in fixed dimension,
Séminaire Lotharingien de Combinatoire 80B, Article #14, 2018. 		Poster
(Poster)
arXiv FPSAC 2017
Generalized nil-Coxeter algebras
Extended abstract of the paper Generalized nil-Coxeter algebras over discrete complex reflection groups,
Séminaire Lotharingien de Combinatoire 80B, Article #29, 2018. 		Poster
(Poster)
FPSAC 2018
The volume of the caracol polytope (with Carolina Benedetti, Rafael S. González D'León,
Christopher R. H. Hanusa, Pamela E. Harris, Alejandro H. Morales, and Martha Yip)
Extended abstract of the paper A combinatorial model for computing volumes of flow polytopes,
Séminaire Lotharingien de Combinatoire 80B, Article #87, 2018. 		Poster
(Poster)
arXiv FPSAC 2017 The Weyl–Kac weight formula (with Gurbir Dhillon)
Extended abstract of the paper The weights of simple modules in Category $\mathcal{O}$ for Kac–Moody algebras,
Talk at FPSAC 2017;
Séminaire Lotharingien de Combinatoire 78B, Article #77, 2017.
arXiv FPSAC 2017 The critical exponent: a novel graph invariant (with Dominique Guillot and Bala Rajaratnam)
Extended abstract of the paper Critical exponents of graphs,
Séminaire Lotharingien de Combinatoire 78B, Article #62, 2017.
arXiv FPSAC 2016
Schur polynomials and matrix positivity preservers (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Extended abstract of the paper Matrix positivity preservers in fixed dimension. I,
DMTCS Proceedings BC, pages 155–166, 2016. 		Poster
(Poster)

Representation theory     Back to top

(shares five papers with "Combinatorics, discrete mathematics, and algebra" section)

arXiv European Journal of Combinatorics From Cauchy's determinant formula to bosonic and fermionic immanant identities (with Siddhartha Sahi)
European Journal of Combinatorics 110, article #103683 (16 pages), 2023; erratum.
(Summary: Extends identities by Cauchy, Frobenius, and Loewner to (a) "any" ground ring $R$, (b) any power series over $R$, (c) any subgroup $G$ of signed permutations, and (d) any character – or even complex class function – of $G$.)
arXiv Journal of Algebra The weights of simple modules in Category $\mathcal{O}$ for Kac–Moody algebras (with Gurbir Dhillon)
Journal of Algebra 603, pages 164–200, 2022.                 [FPSAC 2017 abstract]
arXiv FPSAC 2017 The Weyl–Kac weight formula (with Gurbir Dhillon)
(Extended abstract of preceding paper above)
Talk at 29th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2017);
Séminaire Lotharingien de Combinatoire 78B, Article #77, 2017.
arXiv Advances in Mathematics Faces of highest weight modules and the universal Weyl polyhedron (with Gurbir Dhillon)
Advances in Mathematics 319, pages 111–152, 2017.
arXiv Transactions of the AMS Standard parabolic subsets of highest weight modules
Transactions of the American Mathematical Society 369 no. 4, pages 2363–2394, 2017; erratum.
arXiv Journal of Algebra Faces and maximizer subsets of highest weight modules
Journal of Algebra 455, pages 32–76, 2016.
arXiv Journal of Algebra On Category $\mathcal{O}$ over triangular generalized Weyl algebras (with Akaki Tikaradze)
Journal of Algebra 449, pages 687–729, 2016.
(Summary: Blocks of $\mathcal{O}$ categorify Young tableaux.)
arXiv Algebras and Representation Theory Faces of weight polytopes and a generalization of a theorem of Vinberg (with Tim Ridenour)
Algebras and Representation Theory 15 no. 3, pages 593–611, 2012.
arXiv Journal of Pure and Applied Algebra Faces of polytopes and Koszul algebras (with Vyjayanthi Chari and Tim Ridenour)
Journal of Pure and Applied Algebra 216 no. 7, pages 1611–1625, 2012.
arXiv Communications in Algebra Center and representations of infinitesimal Hecke algebras of $\mathfrak{sl}_2$ (with Akaki Tikaradze)
Communications in Algebra 38 no. 2, pages 405–439, 2010.
arXiv Communications in Algebra Functoriality of the BGG Category $\mathcal{O}$
Communications in Algebra 37 no. 12, pages 4431–4475, 2009.
arXiv Journal of Algebra Quantized symplectic oscillator algebras of rank one (with Wee Liang Gan)
Journal of Algebra 310 no. 2, pages 671–707, 2007.
Thesis Category $\mathcal{O}$ over the symplectic oscillator algebra,
Ph.D. Thesis, University of Chicago, 2006.
arXiv Journal of Pure and Applied Algebra Category $\mathcal{O}$ over a deformation of the symplectic oscillator algebra
Journal of Pure and Applied Algebra 195 no. 2, pages 131–166, 2005; erratum.
arXiv Preprint A weight-formula for all highest weight modules, and a higher order parabolic category $\mathcal{O}$ (with G. Krishna Teja)
Preprint, 2022.
arXiv Preprint Axiomatic framework for the BGG Category $\mathcal{O}$
Preprint, 2015.