Pablo Cubides Kovacsics

In the last twenty years, non-archimedean geometry has shown to be at the cornerstone of severalquestions in mathematics, providing new insides to problems which were rediscovered through a newangle. The employment of model-theoretic tools in its study is quite recent and opened yet new pathsand possibilities of applications. The present project takes place precisely in the development of thisfruitful interaction. It compounds the 4 branches of investigation listed in Section 2 which we furtherdescribe now:(B1) Spaces of types as geometric spaces: extend and generalize the analytification functor definedby E. Hrushovski and F. Loeser [11] to other geometric contexts such as real algebraic geometry,adic spaces and tropical geometry.(B2) Cohomological/homological methods in non-archemedian geometry: develop both cohomologicaland homological theories using the spaces studied in the previous point. In the case of valuedfields, further develop the corresponding cohomology introduced in [3] and show expected resultssuch as duality or the Grothendieck 6 operations. Develop homology in this context via homologyin divisible ordered abelian groups.(B3) t-Stratifications and singularity theory: further study the use of t-stratifications (non-archimedeanvariants of the classical notion of stratification, see [10] and [4]) to approach different problemsin singularity theory.(B4) Interactions with tropical and differential tropical geometry: apply model-theoretic methods tostudy generalizations of the fundamental theorem of differential tropical geometry as presentedin [1].Dissemination of the obtained results will be a key part of the project as well as bringing studentsof the Mathematics Department at Los Andes University closer to the subjects of study.

In the last twenty years, non-archimedean geometry has shown to be at the cornerstone of severalquestions in mathematics, providing new insides to problems which were rediscovered through a newangle. The employment of model-theoretic tools in its study is quite recent and opened yet new pathsand possibilities of applications. The present project takes place precisely in the development of thisfruitful interaction. It compounds the 4 branches of investigation listed in Section 2 which we furtherdescribe now:(B1) Spaces of types as geometric spaces: extend and generalize the analytification functor definedby E. Hrushovski and F. Loeser [11] to other geometric contexts such as real algebraic geometry,adic spaces and tropical geometry.(B2) Cohomological/homological methods in non-archemedian geometry: develop both cohomologicaland homological theories using the spaces studied in the previous point. In the case of valuedfields, further develop the corresponding cohomology introduced in [3] and show expected resultssuch as duality or the Grothendieck 6 operations. Develop homology in this context via homologyin divisible ordered abelian groups.(B3) t-Stratifications and singularity theory: further study the use of t-stratifications (non-archimedeanvariants of the classical notion of stratification, see [10] and [4]) to approach different problemsin singularity theory.(B4) Interactions with tropical and differential tropical geometry: apply model-theoretic methods tostudy generalizations of the fundamental theorem of differential tropical geometry as presentedin [1].Dissemination of the obtained results will be a key part of the project as well as bringing studentsof the Mathematics Department at Los Andes University closer to the subjects of study.