My research revolves around the numerical methods for solving systems of nonlinear equations (or simply nonlinear systems) and the applications of these techniques in physics, chemistry, biology, and engineering. In the wider context, my research involves many different aspects in the field of numerical algebraic geometry. My recent research has been focused on a major class of numerical methods known as the homotopy continuation methods, which deform the given nonlinear system into a closely related system that is trivial to solve. Then continuation methods are applied to track paths originating at the known solutions of the trivial system and ending at the solutions of the given system. In the last decades, these methods have been proven to be reliable, efficient, and highly parallel. It is now used as the basic building block for other numerical methods, such as numerical irreducible decomposition algorithms, opening up new possibilities. My work touches many different aspects of this subject including its theoretical foundation, the construction of new homotopy methods for solving specific problems, its application to many different areas of science, as well as its high performance parallel implementations.
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 nonzero 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 the Bezout number. MixedVol-3 is a software package that computes mixed volume and mixed cells using fast numerical algorithms.
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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.
A binomial system is a polynomial system having exactly two terms on each equation. Solving such systems is an important step in the polyhedral homotopy method, since our target system is generally deformed into binomial systems. In addition, they also have a great number of applications, relating to toric algebraic geometry, particle physics, statistics, and many other subjects. Basic tools for such computation are generally developed in the context of Laurent binomial ideals where the problem of solving a binomial system is reduced to the geometric problem of understanding a corresponding lattice polytope.
The spontaneous synchronization in complex networks is a ubiquitous phenomenon that natural arise in seemingly independent systems in biology, chemistry, physics, and engineering. The Kuramoto model is one of the most successful model in the mathematical analysis of this phenomenon. My research focuses on the algebraic aspect of the frequency synchronization problem for the Kuramoto model.