Special Year Research Seminar

I'll talk about the o-minimal structures R_LN and R_{LN,exp} where one has an effective form of the finiteness property of o-minimality. Unlike the more classical structure of Pfaffian function, R_{LN,exp} contains the period integrals for aribtrary algebraic families. The existence of such a structure was conjectured (in slightly weaker form) by Khovanskii in the eighties.

I'll explain the definition of the structure and illustrate the main ideas from hyperbolic geometry, commutative algebra and differential equations that lead to effective bounds.