The solution of large sparse linear systems is an essential building block in many science and engineering applications. It is also often the main computational bottleneck. For large problems, direct solvers (based on, e.g., LU or Cholesky factorizations) can require a significant amount of computing resources. By contrast, iterative solvers (e.g., CG and GMRES) can…

 Hyperbolic polynomials are coordinate free generalization to the notion of real rooted polynomials. A special class of hyperbolic polynomials are determinantal polynomials and they bound spectrahedra, feasible sets of semidefinite programming. I shall discuss some techniques of real algebraic geometry to deal with convex semialgebraic sets such as spectrahedra, and  demonstrate the applications of hyperbolic…

We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily…

Schubert calculus has its origins in enumerative questions asked by the geometers of the 19th century, such as “how many lines meet four fixed lines in three-space?”  These problems can be recast as questions about the structure of cohomology rings of geometric spaces such as flag varieties.  Borel’s isomorphism identifies the cohomology of the complete…

Many polynomials in combinatorics (and in other areas of mathematics) have nice properties such as having all of their roots being real numbers, or having all of their coefficients being nonnegative.  By surveying recent advances in the Hodge theory of matroids (namely, the nonnegativity of Kazhdan-Lusztig polynomials of matroids and Dowling and Wilson's top-heavy conjecture…