# 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 Phone 650-723-2957 (Office) Fax 650-725-8977 (Office) Email khare AT university-name DOT edu Ph.D. University of Chicago

### Recent and upcoming talks (with links to abstracts)

[Upcoming]     [2015]     [2014]

### Books; invited chapters/monographs

 1 Beautiful, Simple, Exact, Crazy: Mathematics in the Real World (480 pp., with Anna Lachowska) Yale University Press, 2015. See here for more information on the course which I developed and taught in Spring 2011 at Yale, and which led to the book. (The book title is inspired from a talk by Israel Moiseevich Gelfand.) 2 Entry on the BGG Category O, for the Encyclopedia of Mathematics (Michiel Hazewinkel, Editor), Kluwer Academic Publishers.

## Papers in analysis: positivity and graph limits

 30 Matrix positivity preservers in fixed dimension (with Alexander Belton, Dominique Guillot, and Mihai Putinar) Comptes Rendus Mathematique de l'Académie des Sciences, in press. 29 Critical exponents of graphs (with Dominique Guillot and Bala Rajaratnam) Journal of Combinatorial Theory, Series A 139 (2016), pages 30-58. 28 Preserving positivity for rank-constrained matrices (with Dominique Guillot and Bala Rajaratnam) Transactions of the American Mathematical Society, available online. 27 Preserving positivity for matrices with sparsity constraints (with Dominique Guillot and Bala Rajaratnam) Transactions of the American Mathematical Society, available online. 26 Differential calculus on graphon space (with Peter Diao, Dominique Guillot, and Bala Rajaratnam) Journal of Combinatorial Theory, Series A 133 (2015), pages 183-227. 25 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 (2015), pages 489-507. 24 The critical exponent conjecture for powers of doubly nonnegative matrices (with Dominique Guillot and Bala Rajaratnam) Linear Algebra and its Applications 439 No. 8 (2013), pages 2422-2427. 23 Probability inequalities and tail estimates for metric semigroups (with Bala Rajaratnam) Preprint, 2015. 22 Integration and measures on the space of countable labelled graphs (with Bala Rajaratnam) Preprint, 2015. 21 Matrix positivity preservers in fixed dimension. I (with Alexander Belton, Dominique Guillot, and Mihai Putinar) Preprint, 2015. 20 Differential calculus on the space of countable labelled graphs (with Bala Rajaratnam) Preprint, 2014. 19 On fractional Hadamard powers of positive block matrices (with Dominique Guillot and Bala Rajaratnam) Preprint, 2014.

## Papers in representation theory and algebra

 18 On Category $\mathcal{O}$ over triangular Generalized Weyl Algebras (with Akaki Tikaradze) Journal of Algebra, to appear. 17 Standard parabolic subsets of highest weight modules Transactions of the American Mathematical Society, available online. 16 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 (2013), pages 85-129. 15 Faces of polytopes and Koszul algebras (with Vyjayanthi Chari and Tim Ridenour) Journal of Pure and Applied Algebra 216 No. 7 (2012), pages 1611-1625. 14 Faces of weight polytopes and a generalization of a theorem of Vinberg (with Tim Ridenour) Algebras and Representation Theory 15 No. 3 (2012), pages 593-611. 13 The sum of a finite group of weights of a Hopf algebra Axioms 1 No. 3 (2012), pages 259-290 (Special issue on: "Hopf Algebras, Quantum Groups and Yang-Baxter Equations"). 12 Solutions of several coupled discrete models in terms of Lamé polynomials of arbitrary order (with Avinash Khare and Avadh Saxena) Pramana 79 No. 3 (2012), pages 377-392. 11 Center and representations of infinitesimal Hecke algebras of $\mathfrak{sl}_2$ (with Akaki Tikaradze) Communications in Algebra 38 No. 2 (2010), pages 405-439. 10 Functoriality of the BGG Category $\mathcal{O}$ Communications in Algebra 37 No. 12 (2009), pages 4431-4475. 9 Vector spaces as unions of proper subspaces Linear Algebra and its Applications 431 No. 9 (2009), pages 1681-1686. 8 Quantized symplectic oscillator algebras of rank one (with Wee Liang Gan) Journal of Algebra 310 No. 2 (2007), pages 671-707. 7 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 (2007), pages 1-32. 6 Category $\mathcal{O}$ over a deformation of the symplectic oscillator algebra Journal of Pure and Applied Algebra 195 No. 2 (2005), pages 131-166; erratum. 5 Divisibility tests Furman University Electronic Journal of Undergraduate Mathematics (FUEJUM) 3 (1997), pages 1-5. 4 Axiomatic framework for the BGG Category $\mathcal{O}$ Preprint, 2015 (updated with major revisions from an earlier version). 3 Faces and maximizer subsets of highest weight modules Preprint, 2013. Here is a related announcement, which computes the weights of all simple highest weight modules (and others) over a complex semisimple Lie algebra: Weights of simple highest weight modules over a complex semisimple Lie algebra, math.RT/1305.4104. 2 The covering problem for Chinese rings Preprint (under revision), 2009. 1 Drinfeld-Hecke algebras over cocommutative algebras Preprint (under revision), 2007.

### Support

• My research at Stanford has been supported by a DARPA grant (DARPA YFA N66001-11-1-4131).