Mathematical Sciences: Mathematical Logic and Applications

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9400889 Keisler Many existence theorems in analysis have standard proofs using convergence arguments and nonstandard proofs yielding stronger results using lifting arguments. The investigator has recently introduced a new model-theoretic method of proving existence theorems, where one can often make an easy computation to obtain an approximate solution to a problem, and then invoke a general forcing theorem to get a harder result giving an exact solution. The present project will continue development of this method. For a variety of problems in both pure and applied mathematics, it is easy to show that there are approximate solutions but very difficult to show that there are exact solutions. This is typical for stochastic differential equations, optimization problems, and equilibrium problems. Keisler's new approach to such problems, which is justified by a theorem in mathematical logic, can often dramatically simplify the construction of exact solutions. This method has already produced several new results and has the potential of becoming a valuable and broadly applicable tool. ***