My current research focuses on numerical solutions to systems of polynomial equations and the more general field of numerical algebraic geometry.
Mixed volume and mixed cells
Originally developed in the context of convex geometry, mixed volume is a way to assign a generalized non-negative volume to a list of convex bodies. It depends on not only the volume of each individual convex body, but also their relative positions. This concept has a wide range of applications in geometry and combinatorics. In the recent decades, it has found important applications in the study of system of polynomial equations (i.e. algebraic geometry): by the Bershtein’s theorem, the number of isolated solutions a system of polynomial equations has, counting multiplicity, is bounded by the mixed volume of Newton polytopes of the system. This bound is usually much tighter than another well studied bound, the Bezout number.
More detailed discussion can be found on my main page on mixed volume and mixed cells.
Homotopy continuation methods
Homotopy continuation methods form a large class of general methods suitable for solving hard problems like nonlinear systems of equations, in particular, systems of polynomial equations. The basic idea behind such methods is to deform a hard problem in to an easier problem or into a problem whose solutions are already known, and use the solutions of the easier problem to find those of the harder problem.
More detailed discussion can be found on my main page on homotopy continuation methods.