Research Projects

(1) Factorization and sampling in non-positive matrix cones: For over a hundred years, the Cholesky decomposition has been an indispensable tool in numerical analysis. It applies to all real symmetric positive definite matrices, which are synonymous with nonsingular (i.e., full rank) covariance matrices. Similarly, the Wishart and inverse Wishart distributions allow one to sample from the cone of positive definite matrices. It is now natural to ask from a theoretical perspective, if one can extend this factorization to non positive definite matrices; and similarly, if a counterpart of the Wishart density can be defined over other matrix cones. There is motivation from applications too: the embedding of discrete structures in hyperbolic manifolds is today a very lively topic, attracting researchers in areas such as image and language processing, finance, social networks, and geographic routing. Now if one takes Gram matrices of vectors in Euclidean space, one obtains positive (semi)definite Gram matrices. But the Riemannian metric is different on hyperbolic manifolds, and so Gram matrices here would have negative eigenvalues as well. One therefore needs tools to analyze, as well as to sample from, cones of non positive-definite matrices with specified inertia. The above provides strong theoretical and applied motivation to extend the Cholesky decomposition beyond the positive definite cone – and also to define the (inverse) Wishart density on such cones. This is one of the proposed goals in the project, and achieving it would open up new vistas for researchers in science and engineering, including providing tools and techniques to analyze data living on negatively curved manifolds. It would also provide useful factorizations for larger matrix cones in each dimension. (2) Symmetric functions and (weak) majorization: Majorization inequalities go back to Newton, and later by Muirhead, Gantmacher, and Schur among others. In this millennium, there was concrete progress along these lines by Cuttler-Greene-Skandera, who showed that if a ratio of two Schur polynomials is minimized at (1,...,1) in the entire positive orthant, then the partition in the numerator majorizes that in the denominator. This work sparked off much activity: first, Sra proved the converse of their result. Then the PI and T. Tao extended this to (a) separately characterize weak majorization via Schur polynomials, and (b) to extend all of these results to non-integer powers, aka ratios of generalized Vandermonde determinants. This was further extended to other Cartan/Lie types by McSwiggen-Novak under the name “W-majorization”. That said, these inequalities relating (weak) majorization to Schur polynomials are rather “classical”. As are Schur polynomials themselves, since cutting-edge work in symmetric function theory now involves one- and two-parameter generalizations of these: Hall-Littlewood, Jack, and (most generally) Macdonald polynomials. Thus, one would like to merge the old with the new, and ask if majorization inequalities can be formulated and proved for these more general families of polynomials – both in general, and for special values of the parameters (which would yield Jack / Schur polynomials). This would significantly increase our understanding of the dependence between multi-parameter families of symmetric functions, and real inequalities. (3) Continuous and discrete log-concavity of parabolic Verma module characters: Schur polynomials are characters of finite-dimensional simple modules over sl(n). It is thus of interest to study their coefficients. Only recently did June Huh and his collaborators prove that normalized Schur polynomials are in fact Lorentzian (a prominent cutting-edge tool introduced by Branden-Huh). Similarly, Huh et al showed: normalized Verma module characters are also Lorentzian. It is natural to ask if these hold over all Dynkin types. If true, this would reveal a universal truth in Lie algebra representations.