# Homepage of Apoorva Khare

### Contact information

 Affiliation Research Associate (joint in Mathematics and Statistics), Stanford University Office Littlefield Center, Room 334 Postal address Sequoia Hall, 390 Serra Mall, Stanford, CA - 94305 USA Phone 650-723-2957 (Office) Fax 650-725-8977 (Office) Email khare AT university-name DOT edu Ph.D. University of Chicago

### Recent talks (with links to abstracts)

[2016 Talks]     [2015 Talks]     [2014 Talks]     [2013 Talks]

### Books; invited chapters/monographs

 1 Beautiful, Simple, Exact, Crazy: Mathematics in the Real World (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. 2 Representations of complex semi-simple Lie groups and Lie algebras 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.

## Analysis: Positivity, combinatorics, and probability

(shares three papers with "Representation theory and algebra" section)

The Hoffmann-Jørgensen inequality in metric semigroups (with Bala Rajaratnam)
Annals of Probability, in press, 2017.
Preserving positivity for rank-constrained matrices (with Dominique Guillot and Bala Rajaratnam)
Transactions of the American Mathematical Society, available online, 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 '17),
Séminaire Lotharingien de Combinatoire, in press, 2017.
Critical exponents of graphs (with Dominique Guillot and Bala Rajaratnam)
Journal of Combinatorial Theory, Series A 139, pages 30–58, 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 '16), DMTCS Proceedings BC, pages 155–166, 2016. (Poster)
Matrix positivity preservers in fixed dimension. I (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Advances in Mathematics 298, pages 325–368, 2016.
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.
Matrix positivity preservers in fixed dimension (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Comptes Rendus Mathematique de l'Académie des Sciences 354 no. 2, pages 143–148, 2016.
Differential calculus on graphon space (with Peter Diao, Dominique Guillot, and Bala Rajaratnam)
Journal of Combinatorial Theory, Series A 133, pages 183–227, 2015.
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.
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.
Moment-sequence transforms (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Preprint, 2016.
The Khinchin–Kahane inequality and Banach space embeddings for metric groups (with Bala Rajaratnam)
Preprint, 2016.
Model-free consistency of graph partitioning (with Peter Diao, Dominique Guillot, and Bala Rajaratnam)
Preprint, 2016.
Probability inequalities and tail estimates for metric semigroups (with Bala Rajaratnam)
Preprint, 2015.
Integration and measures on the space of countable labelled graphs (with Bala Rajaratnam)
Preprint, 2015.
Differential calculus on the space of countable labelled graphs (with Bala Rajaratnam)
Preprint, 2014.
On fractional Hadamard powers of positive block matrices (with Dominique Guillot and Bala Rajaratnam)
Preprint, 2014.

## Representation theory and algebra

The Weyl–Kac weight formula (with Gurbir Dhillon)
(Extended abstract of the preprint "Characters of highest weight modules and integrability")
Invited talk at 29th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC '17),
Séminaire Lotharingien de Combinatoire, in press, 2017.
Generalized nil-Coxeter algebras, cocommutative algebras, and the PBW property
AMS Contemporary Mathematics 688, pages 139–168, 2017.
Standard parabolic subsets of highest weight modules
Transactions of the American Mathematical Society 369 no. 4, pages 2363–2394, 2017.
Faces and maximizer subsets of highest weight modules
Journal of Algebra 455, pages 32–76, 2016.
(Here is an older, related announcement of some of the results.)
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 '16), DMTCS Proceedings BC, pages 155–166, 2016. (Poster)
Matrix positivity preservers in fixed dimension. I (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Advances in Mathematics 298, pages 325–368, 2016.
On Category $\mathcal{O}$ over triangular generalized Weyl algebras (with Akaki Tikaradze)
Journal of Algebra 449, pages 687–729, 2016.
Matrix positivity preservers in fixed dimension (with Alexander Belton, Dominique Guillot, and Mihai Putinar)
Comptes Rendus Mathematique de l'Académie des Sciences 354 no. 2, pages 143–148, 2016.
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.
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.
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")
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.
Center and representations of infinitesimal Hecke algebras of $\mathfrak{sl}_2$ (with Akaki Tikaradze)
Communications in Algebra 38 no. 2, pages 405–439, 2010.
Functoriality of the BGG Category $\mathcal{O}$
Communications in Algebra 37 no. 12, pages 4431–4475, 2009.
Vector spaces as unions of proper subspaces
Linear Algebra and its Applications 431 no. 9, pages 1681–1686, 2009.
Quantized symplectic oscillator algebras of rank one (with Wee Liang Gan)
Journal of Algebra 310 no. 2, pages 671–707, 2007.
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.
Category $\mathcal{O}$ over the symplectic oscillator algebra
Ph.D. Thesis, University of Chicago, 2006.
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.
Divisibility tests
Furman University Electronic Journal of Undergraduate Mathematics (FUEJUM) 3, pages 1–5, 1997.
Faces of highest weight modules and the universal Weyl polyhedron (with Gurbir Dhillon)
Preprint, 2016.
Characters of highest weight modules and integrability (with Gurbir Dhillon)
Preprint, 2016.
A Carlitz–von Staudt type theorem for finite rings (with Akaki Tikaradze)
Preprint, 2016.
Generalized nil-Coxeter algebras over discrete complex reflection groups
Preprint, 2016.
Axiomatic framework for the BGG Category $\mathcal{O}$
Preprint, 2015 (significant additions and major revisions to an earlier version).