o-minimal method and generalized sum-product phenomena
I will discuss a joint work with Yifan Jing and Souktik Roy where we show that for two bivariate polynomials with coefficients in fields with char 0 to simultaneously exhibit small expansion, they must exploit the underlying additive or multiplicative structure of the field in nearly identical fashion. Within combinatorics, this generalizes a result of Shen and yields an Elekes-Ronyai type structural result for symmetric nonexpanders, resolving a question mentioned by de Zeeuw. In the context of model theory, our result belongs to a growing body of works investigating combinatorial ramification of modularity or its absence.