Subdiferential theory is conceived to deal with nonsmoothness, but omnipresence of pathologies imposes the use of ad-hoc assumptions or an ambient structure (paradigm). Two paradigms prevail the developments of the theory: convex analysis (convex paradigm) and semi-algebraic geometry (tame paradigm). In both paradigms, first-order information (knowledge of the subdifferential) determines the function up to a constant.
This 4-year project is inscribed in the broad area of Variational Analysis, focusing on the question of first-order characterization of functions and motivated by recent developments in asymmetric structures. The project proposes a novel approach to explore structure. The starting point is the remark that although convexity and tameness have different origins, they both rely on orientation: convexity is a unilateral theory (graphs and tangent spaces are replaced by epigraphs and tangent cones), while the model-theoretic definition of an o-minimal structure relies on the total order of the field (together with a finiteness property). This leads to the idea that exploring (some form of) asymmetry and controlling cardinality could provide ground for a consistent theory outside the realm of the aforementioned paradigms. The proposal contains two independent lines, which ultimately relate to this driving idea.
The first line of research is devoted to the question of determination in a broad sense. Based on the discovery that several functions are determined by their metric slope and critical values, an axiomatic de nition of a downhill-oriented operator is proposed: the idea is to assign a positive scalar (abstract descent) to each point of a given function, in a way that a comparison principle holds, and covering the instances of gradient moduli (for smooth functions), remoteness of subdiferentials (for convex functions), local or global strong slopes (for functions in a metric space) as well as average descents of stochastic processes. The working assumption is that abstract descent and values at points of 0-descent suffce to determine unambiguously a large class of functions.
The second line of the project proposes a study of spaces with an asymmetric distance. Semi-Lipschitz functions come into play (as natural morphisms). In this setting we aim to characterize absolutely minimal semi-Lipschitz functions via an adequate infinite-Laplace type operator and obtain insight on how differential calculus works in an asymmetric normed space. The ultimate objective is to deal with a canonical asymmetric subdifferential theory. This is based on the fact that convexity (as well as semialgebricity) are not distance-related notions, while differential calculus and Lipschitz functions are clearly affected by possible asymmetrizations of the space. Very little is known on this topic, which will be of increasing relevance in the future.
This proposal is mostly of theoretical nature. Nevertheless, although the project does not pretend to concrete applications by the end of its execution, several spin-off results are expected (already within a 2-year horizon) in connection with stochastic processes or metric geometry.
