We use the methods of model-theoretic nonstandard analysis, in particular enlargements, to study the properties of regular and good ultrafilters and their role within Keisler's order on countable complete first-order theories. This understanding is used to produce alternative proofs of several key theorems in the study of Keisler's order, such as the well-definedness of Keisler's order and the maximality of theories with SOP_2 (Strong Order Property 2). Furthermore, we provide an analysis of the logical structure of the types used in the proof that theories with SOP_2 are Keisler maximal and use this analysis to give an easy-to-state set-theoretic characterization of ultrafilters that are both regular and good.

A Nonstandard Approach to Keisler's Order

Thu, Apr. 6 3:35pm (MATH 3…

Probability

Hoi Nguyen (Ohio State University)

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In this talk we discuss the number of real roots of a wide class of random orthogonal polynomials with gaussian coefficients. Using the method of Wiener Chaos we will show that the fluctuation in the bulk is asymptotically gaussian, even when the local correlations are different. We will also discuss partial progress toward the case of non-gaussian coefficients.

Central Limit Theorem for the number of real roots of gaussian random polynomials