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Abstracts and Talk Materials

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.