Mathematical Sciences: Model Theory and Its Geometric Applications

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9626856 Marker In studying the real numbers with exponentiation, van den Dries, Macintyre and Marker gave an algebraic construction of a nonstandard model. This model has proved useful in understanding the asymptotic behavior of functions. Marker's work concentrates on the relationship between formally solving differential equations in this model and the problem of finding actual solutions at infinity. Marker will also work on problems in the model theory of differential fields. This is a fascinating area, requiring a sophisticated mixture of ideas from model theory, differential algebra and algebraic geometry. (Through the work of Buium and Hrushovski, differential algebra has had significant number theoretic applications.) Marker studies the geometry of solutions to algebraic differential equations, concentrating on the use of model theoretic dimensions to develop intersection theory. Model theory is a branch of mathematical logic where one studies mathematical structures by looking at their properties expressed in formal languages. For example, in the real numbers one can say that every number is a cube, but one cannot say that every bounded set has a least upper bound. Nonstandard models of the real numbers share all their formal properties. Studying nonstandard models often leads to fresh insights into the real numbers themselves. Recently, model theorists have been studying the exponential function. These investigations have led to important new insights into the geometry of solution sets of equations involving exponentials. This work has already found applications in analysis, control theory and neural networks. ***