Mathematics Colloquia and Seminars

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Algebraic operations on sets of complex numbers produce remarkably rich geometrical structures, with diverse applications and connections to science and engineering. For "simple" operands, such as circular disks, precise descriptions of their algebraic combinations are available in terms of the Cartesian and Cassini ovals, and higher-order generalizations. Algorithms can be formulated to approximate algebraic operations on complex sets with general (piecewise-smooth) boundaries to a given precision. This "Minkowski algebra of complex sets" is the natural extension of (real) interval arithmetic to sets of complex numbers. It provides a versatile two-dimensional "shape operator" language, with connections to mathematical morphology, geometrical optics, and stability analysis of dynamic systems.