Contact information

Affiliation Indian Institute of
	  Science   Associate Professor in Mathematics, Indian Institute of Science,
Member, Analysis and Probability Research Group (APRG);
currently also
SwarnaJayanti Fellow of DST (2020–25),
Ramanujan Fellow of SERB (2018–23)
                                Apoorva Khare
                    CVs:   [Abridged]   |   [Detailed]
Office Room X-26 (top floor, X-wing), Mathematics department
Postal address    
Department of Mathematics
Indian Institute of Science
Bangalore – 560012, INDIA
Phone +91–80–2293–2514         (Fax +91–80–2360–0146)
Email (no quantum)
Ph.D.   University of
	  Chicago University of Chicago (Mathematics)
	  University Research Associate in Mathematics and Statistics,
Stanford University

Selected international conference talks

Papers: Analysis (positivity, Schur polynomials, probability); also Combinatorics, algebra, FPSAC; also Representation theory.       [Publication list]

Selected publications – analysis and positivity (also features Schur polynomials, combinatorics)

  1. Matrix analysis and entrywise positivity preservers (book), 2022.
    Vol. 471 in London Mathematical Society Lecture Note Series, Cambridge University Press, xxii+292 pp.         [Front+back cover]
    and Vol. 82 in Texts and Readings In Mathematics (TRIM Series), Hindustan Book Agency, xxii+339 pp.         [Front+back cover]
  2. (With Alexander Belton, Dominique Guillot, Mihai Putinar.)   Totally positive kernels, Pólya frequency functions, and their transforms,
    Journal d'Analyse Mathématique (63 pp.), in press.   [arXiv]
    [Joint announcement of this + 2 other papers, in Mathematics Research Reports]
  3. (With Alexander Belton, Dominique Guillot, Mihai Putinar.)   Moment-sequence transforms,
    Journal of the European Mathematical Society 24 no. 9, pages 3109–3160, 2022.   [arXiv]
  4. Smooth entrywise positivity preservers, a Horn–Loewner master theorem, and symmetric function identities,
    Transactions of the AMS 375 no. 3, pages 2217–2236, 2022.   [arXiv]
  5. (With Alexander Belton, Dominique Guillot, Mihai Putinar.)   Hirschman–Widder densities,
    Applied and Computational Harmonic Analysis 60, pages 396–425, 2022.   [arXiv]
    [Joint announcement of this + 2 other papers, in Mathematics Research Reports]
  6. Matrix analysis, Pólya frequency functions, and preservers of (total) positivity (PDF, 320+ pp., Sections 31, 35–40 being updated. Last updated on 19 May, 2022.)
    Expanded lecture notes from a topics graduate course (or for advanced undergraduates) taught in Spring 2018 and Autumn 2019.
  7. (With Terence Tao.)   On the sign patterns of entrywise positivity preservers in fixed dimension,
    American Journal of Mathematics 143 no. 6, pages 1863–1929, 2021.   [arXiv]   [FPSAC]   [Poster]
  8. Sharp nonzero lower bounds for the Schur product theorem, Proceedings of the AMS 149 no. 12, pages 5049–5063, 2021.   [arXiv]
    (Summary: Refines results by Schur in Crelle 1911 and Vybíral in Adv. Math. 2020, with an application to numerical integration.)
  9. (With Projesh Nath Choudhury, M. Rajesh Kannan.)   Sign non-reversal property for totally non-negative and totally positive matrices, and testing total positivity of their interval hull, Bulletin of the London Mathematical Society 53 no. 4, pages 981–990, 2021.   [arXiv]
  10. (Survey article, with Alexander Belton, Dominique Guillot, Mihai Putinar.)   A panorama of positivity (83 pages)   [arXiv] – split for publication:
  11. (With Dominique Guillot, Bala Rajaratnam.)   Preserving positivity for rank-constrained matrices,
    Transactions of the AMS 369 no. 9, pages 6105–6145, 2017.   [arXiv]
  12. (With Dominique Guillot, Bala Rajaratnam.)   Preserving positivity for matrices with sparsity constraints,
    Transactions of the AMS 368 no. 12, pages 8929–8953, 2016.   [arXiv]
  13. (With Dominique Guillot, Bala Rajaratnam.)   Critical exponents of graphs,
    Journal of Combinatorial Theory Series A 139, pages 30–58, 2016.   [arXiv]   [FPSAC]
  14. (With Alexander Belton, Dominique Guillot, Mihai Putinar.)   Matrix positivity preservers in fixed dimension. I,
    Advances in Mathematics 298, pages 325–368, 2016.   [arXiv]   [FPSAC]   [Poster]
  15. Multiply positive functions, critical exponent phenomena, and the Jain–Karlin–Schoenberg kernel, preprint.   [arXiv]
    [Joint announcement of this + 2 other papers, in Mathematics Research Reports]

Selected publications – representation theory; algebraic combinatorics (in addition to some papers above)

  1. (With Gurbir Dhillon.)   The weights of simple modules in Category $\mathcal{O}$ for Kac–Moody algebras, Journal of Algebra 603, pages 164–200, 2022.   [arXiv]   [FPSAC]
  2. (With Carolina Benedetti, Rafael S. González D'León, Christopher R. H. Hanusa, Pamela E. Harris, Alejandro H. Morales, Martha Yip.)  
    A combinatorial model for computing volumes of flow polytopes, Transactions of the AMS 372 no. 5, pages 3369–3404, 2019.   [arXiv]   [FPSAC]   [Poster]
  3. Generalized nil-Coxeter algebras over discrete complex reflection groups, Transactions of the AMS 370 no. 4, pages 2971–2999, 2018.   [arXiv]   [FPSAC]   [Poster]
  4. (With Gurbir Dhillon.)   Faces of highest weight modules and the universal Weyl polyhedron, Advances in Mathematics 319, pages 111–152, 2017.   [arXiv]
  5. Standard parabolic subsets of highest weight modules, Transactions of the AMS 369 no. 4, pages 2363–2394, 2017.   [arXiv]
  6. Faces and maximizer subsets of highest weight modules, Journal of Algebra 455, pages 32–76, 2016.   [arXiv]
  7. (With Akaki Tikaradze.)   On Category $\mathcal{O}$ over triangular generalized Weyl algebras, Journal of Algebra 449, pages 687–729, 2016.   [arXiv]
    (Summary: Blocks of $\mathcal{O}$ categorify Young tableaux.)
  8. 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.), pages 85–129,
    Texts and Readings in Mathematics (TRIM), Hindustan Book Agency, 2013.   [arXiv]
  9. (With Wee Liang Gan.)   Quantized symplectic oscillator algebras of rank one, Journal of Algebra 310 no. 2, pages 671–707, 2007.   [arXiv]
  10. (With Projesh Nath Choudhury.)   Distance matrices of a tree: two more invariants, and in a unified framework, preprint.   [arXiv]
  11. (With Projesh Nath Choudhury.)   The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra, preprint.   [arXiv]   [FPSAC]
  12. (With G. Krishna Teja.)   A weight-formula for all highest weight modules, and a higher order parabolic category $\mathcal{O}$, preprint.   [arXiv]

Selected publications – other (book, probability, graphons, Polymath(geometric group theory))

  1. As D.H.J. Polymath (With Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, Terence Tao.)
    Polymath14: Homogeneous length functions on groups, Algebra & Number Theory 12 no. 7, pages 1773–1786, 2018.   [arXiv]     [Blogpost 1]     [Polymath14 wiki]
    (Summary: Provides novel characterizations from analysis and geometry, of abelian torsion-free groups.)
  2. (With Bala Rajaratnam.)   The Hoffmann-Jørgensen inequality in metric semigroups, Annals of Probability 45 no. 6A, pages 4101–4111, 2017.   [arXiv]
  3. (With Peter Diao, Dominique Guillot, Bala Rajaratnam.)   Differential calculus on graphon space, Journal of Combinatorial Theory Series A 133, pages 183–227, 2015.   [arXiv]
  4. (With Anna Lachowska.)   Beautiful, Simple, Exact, Crazy: Mathematics in the Real World (480 pp., book), Yale University Press, 2015.

Support (selected)

SERB, India DST, India Indian
Institute of Science International Centre for Mathematical Sciences
(SERB, Govt. of India) (DST, Govt. of India) (IISc, Bangalore, India) (ICMS, Edinburgh, UK)
Foundation Tata
Trusts International Linear Algebra Society Taylor & Francis
(Infosys Foundation, India) (Tata Trusts, India) (ILAS, USA) (Taylor & Francis, UK)
DARPA Simons Foundation American Institute of
Mathematics National Science Foundation
(DARPA, USA) (Simons Foundation, USA) (AIM and NSF, USA)