Theory, algorithms, and applications in a wide range of topics divided into the following four main areas:
(1) Mathematical Programming and Matrix Scaling
(2) Discrete and Combinatorial Optimization
(3) Polynomial Root-Finding and Related Problems
(5) Computational Geometry
Highlight of Topics of Research
Each of the the four areas consists of several different topics.
Topics in (1) include: Canonical formulation of linear, quadratic, or convex programming as homogeneous programming; complexity of algorithms; polynomial-time path-following and projective algorithms for homogeneous programming; diagonal scaling of a positive semidefinite matrix and connections with linear programming; complexity of solving the problems over rational, algebraic, and real numbers; diagonal matrix scaling; matrix scaling dualities in linear, quadratic, semideinite, and general self-concordant homogeneous programming; applications in path-following and potential reduction algorithms; multidimensional matrix scaling and generalizations; the RAS algorithm; complexity of doubly-stochastic matrix scaliing; complexity of matrix balancing; complexity of RAS algorithm for nonnegative matrix
scaling and entropy minimization.
Topics in (2) include: Approximation algorithms and complexity; fast heuristics with guaranteed bounds for combinatorial optimization problems such as minimum-weight perfect matching; minimum-weight constrained-forest; traveling salesman; weighted magic labeling; Euclidean optimization problems; approximation of diameter of finite sets; bounding schemes for NP-complete quadratic zero-one optimization; applications in branch and bound algorithms.
Topics in (3) include: Polynomial root-finding; iteration functions; characterizations of Basic Family, a fundamental class of iteration function; recurrence relations; determinantal generalization of Taylor's theorem; multipoint version of Basic Family; truncated Basic Family; analysis of local and global convergence; characterization of fixed-points of Basic Family; approximation of pi and zeros of analytic functions using Basic Family; lower bound on modulus of determinant; computing bounds on zeros of polynomials; general convergence.
Topics in (4) include: Polynomiography and its applications in art, education, mathematics and the science such as: novel visualization, theoretical applications of polynomiography in conjecturing theorems, applications in measuring performance of iteration functions, artistic and design applications of polynomiography, educational applications of polynomiography, scientific applications of polynomiography, applications in animation and visualization of mathematical properties, working toward national and international popularization of polynomiography and its development as a serious and effective medium of art and education and science at various levels.
Topics in (5) include: Algorithms for geometric problems, Voronoi diagrams, zone diagrams, mollified zone diagrams, approximation of diameter, approximation of convex hulls in higher dimensions, application of optimization techniques in computational geometry.
Topics in (6) include: : Development of educational material for K-16 education, Working with K-12 teachers and educators in order to develop lesson plans and modules.