Function valued valuations

The applicant Jin Li received his Ph.D. from Shanghai University in 2016 and is currently working at the Institute of Discrete Mathematics and Geometry, TU Wien, as a post-doctoral researcher. The co-applicant Monika Ludwig is a Professor at the same institute. The focus of the proposed research project is the affine theory of function valued valuations. The concept of function valued valuations is new, but many classical and fundamental operators in mathematics and physics are function valued valuations. Our horizons are broadened by this concept and many interesting questions arise. In the proposed project, the applicant expects to characterize some fundamental operators in convex geometric analysis by their invariance under geometric transformations and the valuation property. Related classification problems are also parts of the project. Since these operators are very influential in many subjects, their characterization might have wide applications. Also, the proposed classification problems might not only help to better understand certain developing theories but also to define new and useful operators. The proposed problems will be investigated based on the applicant and co-applicants` experience in the affine theory of valuations.

With the support of the grant, I (Jin Li) worked at the Institute of Discrete Mathematics and Geometry of TU Wien, as a post-doctoral researcher with the co-applicant Monika Ludwig, a professor at the same institute. The originally planned project duration was from 1.2.2019 to 31.1.2021. Due to the corona pandemic, the project was extended to 30.8.2021 (part-time job since 1.2.2021). In the project, we successfully established several beautiful classification theorems in the theory of valuations. They unify many previous results and may lead to applications in convex geometry. We also characterize two important transforms of functions: the Legendre transform and the Laplace transform. With the support of this grant, I participated in conferences, workshops, and seminars in Austria, Canada, China, Germany, Hungary, Russia, Spain, and the U.S.A. I also used the grant to visit and invite experts in convex geometry. I want to thank FWF since this grant experience helps me to find a tenure-track position at Shanghai University (from 1.10.2021). I hope to continue working closely with the group at TU Wien in the future.