NSF/CBMS Regional Conference in the Mathematical Sciences -- Solving Polynomial Equations

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One of the most fundamental problems in mathematics is that of solving polynomial equations. Such systems are ubiquitous in applied mathematics, arising in robotics, coding theory, optimization, mathematical biology, computer vision, statistics, and numerous other areas. Of course, the study of polynomialsystems is a beautiful topic in its own right and is the subject of algebraic geometry, which istraditionally regarded as a deep and difficult subject in pure mathematics.In recent years, an explosive development in explicit algorithms and practical software for geometric calculations has revolutionized the area, making many formerly inaccessable problems tractable, and providing a fertile ground for experimentation and conjecture. These algorithms have also generated a surprising interest in algebraic geometry among scientists, engineers, and applied mathematicians. Thetools of the trade employed in this field span the spectrum of mathematics, ranging from numericalmethods and differential equations to algebraic geometry to combinatorics. The purpose of these lectures is toprovide an overview of recent results and the ``state of the art''; and to discuss the wealth of open questions raised by these results and suggest new directions ripe for exploration.