In this talk we give the definition of logarithmic transseries with formal variable at zero. Furthermore, we introduce the differential algebra of logarithmic transseries equipped with three different topologies: the valuation topology, the product topology and the weak topology. We use the valuation and the product topology to define a composition of logarithmic transseries. In order to control the depth of a composition we observe logarithmic transseries without logarithms in their first terms. Among them we distinguish parabolic, hyperbolic and strongly hyperbolic logarithmic transseries. We discuss some fixed point techniques for finding normal forms of such logarithmic transseries and we solve the linearization problem. In particular, we prove the fixed point theorem motivated by the Krasnoselskii Fixed Point Theorem and we solve various nonlinear differential equations on the differential algebra of logarithmic transseries. This is joint work with M. Resman, J.-P. Rolin and T. Servi.

Bio: Dino Peran has earned his PhD in 2021 and works as postdoctoral researcher at Faculty of Science, University of Split, Croatia. His web-page can be found at https://www.pmfst.unist.hr/team/dino-peran/?lang=en

Scientific interests: local fixed point dynamics (complex and real), formal and analytic classification of Dulac germs, logarithmic transseries and asymptotic expansion, normal forms of logarithmic transseries