This two part tutorial shall introduce you to algorithmic algebraic geometry methods of manipulating algebraic (polynomial) splines necessary for the solution of multivariate integral and partial differential equations emanating from science and engineering applications. In the first part, we shall discuss well known (classical) algorithms for modeling the structure and energetics of physical domains (molecules to airplanes to oil fields), at multiple scales, with implicit algebraic splines (A-patches), and their rational parametric splines (NURBS) approximations. Algebraic geometry methods include computing global and local parameterizations using Newton factorization and Hensel lifting, computation of adjoints by interpolating through singularities, and low degree curve and surface intersections via Resultants and/or Groebner Basis calculations. The second part shall focus on more current research and attempts at using algebraic geometry methods for faster and more accurate calculations of multivariate definite integrals of algebraic functions arising from the solution of polarization energetics and forces (Generalized Born, Poisson Boltzmann) and Green function solutions of two-phase Stokesian flows. These algebraic geometry methods include the use of ideal theory and solutions of polynomial systems of equations for more accurate cubature formulas and their faster evaluation.

This talk will discuss how many problems in control theory may be formulated as problems of algebraic geometry. Topics covered include realization theory, homotopy methods for pole placement, construction of Schur functions, stabilization via coprime factorization, systems over rings, controllability, and multidimensional systems. We also discuss more recent research in the area of decentralized control.

This talk will discuss how many of the basic questions in mechanism science are naturally formulated as systems of polynomial equations. These questions include both analysis problems (how does this mechanism move?) and synthesis problems (which mechanisms will move the way I require?). Once such a problem has been formulated as a system polynomials, the irreducible decomposition of the solution set of the system becomes a powerful tool for describing the answer. We will discuss how some questions can be approached in stages by computing irreducible decompositions of subsets of the original equations and then intersecting solution components. Finally, we will see how some questions about the existence of special mechanisms are nicely formulated as fiber products of algebraic sets.

This talk will discuss how numerical polynomial continuation can be used to solve the problems formulated in the first lecture. In doing so, we will describe the basic constructs and algorithms of Numerical Algebraic Geometry. Foremost among these is the notion of a witness set, a numerical approximation to a linear section of an algebraic set. We will describe how witness sets are computed, how they are used in finding numerical irreducible decompositions, and how the witness set for the intersection of two algebraic sets, say A and B, can be found from the witness sets for A and B, via a diagonal homotopy. Some recent avenues of research, such as how to find the real solutions inside a complex curve, will be mentioned briefly.

Suggested reading: A.J. Sommese and C.W. Wampler, The numerical solution of systems of polynomials arising in engineering and science, World Scientific, 2005.