This research is to be directed along two closely related avenues, namely, the solution of equations and paths of equations with homotopies, and the solution of parametric or short paths of equations with infinitesimals. The essence of the first avenue is path following the preimage of a point,perhaps, in subdivisions. The essence of the second is doing arithmetric with infinitesimals and, transferring algorithms, problems and solutions from one ordered field to another. The parametric problem is lifted to an ordered field containing an infinitesimal to obtain a single problem, the original algorithm is run in the new ordered field, and the solution is lowered to the original field to obtain a short path of solutions to the parametric problems; this sequence is referred to as a lift, solve, and lower. To see that the two avenues are related we note that piecewise affine homotopy methods regularly employ infinitesimals and can be used to solve parametric problems, and the lift, solve, and lower sequence applies to piecewise affine homotopy methods. Specific tasks and projects to be studied under these headings include variable dimension algorithms for infinite dimensional problems, parametric linear programs, average analysis of homotopy and simplex methods, matrix scaling, design centering, computing dynamic curves, various aspects of computing in ordered fields and problem characterization, transfer principles in ordered fields, a compiler for algorithms computing with an infinitesimal, solution and combinatorial analysis of economic models, and parametric production problems.
