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Functorial and algebraic methods in the study of systems of linear partial differential equations

Final Report Summary - FUNALGPDE (Functorial and algebraic methods in the study of systems of linear partial differential equations.)

The research objectives, the methodology and the interdisciplinary setting of the project "Functorial and algebraic methods in the study of systems of linear partial differential equations" are located in a rapidly developing area of forefront research. They link different areas of mathematics as the analysis of systems of partial differential equations, the algebraic study of geometric objects and the modern algebraic approach through category and sheaf theory.
The project FUNALGPDE was supposed to last two years. During the first, Dr. Morando was at RIMS in Kyoto in contact with Prof. M. Kashiwara and Prof. T. Mochizuki, learning from them the classical and new results on D-module theory and algebraic analysis. The second period was spent by Dr. Morando at the University of Padova, Department of Mathematics, within the local team of Algebraic Analysis.

The original scientific objectives were mainly focused in three directions. Let us briefly recall them here.

1 Riemann-Hilbert correspondence and subanalytic site

One of the most beautiful results of D-module theory is the regular Riemann–Hilbert correspondence. Such a result generalizes the 21st Hilbert problem and in its most general statement reads as an equivalence between the bounded derived category of D-modules with regular holonomic cohomology groups (objects of analytic nature) and the bounded derived category of sheaves of C-vector spaces with C-constructible cohomology groups (objects of topological and combinatorial nature). In his proof of this correspondence, Kashiwara
used techniques and tools coming from analysis (tempered distributions) and geometry (subanalytic sets). These were further developed in the '90 by Kashiwara–Schapira and they finally found in Astérisque 271 a natural framework with the subanalytic site and the complex of sheaves of tempered holomorphic functions. In 2003, Kashiwara-Schapira suggested interesting applications of these new tools to the study of irregular holonomic D-modules.

A first objective of Dr. Morando was to prove a conjecture of Kashiwara-Schapira on the constructibility on the subanalytic site of the complexes tempered solutions of holonomic D-modules. Such an objective has been fully achieved in two steps. First he obtained a preliminary and partial result which appeared as: “Preconstructibility of Tempered Solutions of Holonomic D-modules”, Int Math Res Notices (2012) doi: 10.1093/imrn/rns247. Then, recently, he proved the conjecture in full generality and presented it in a preprint available to the attention of the scientific community in: “Constructibility of tempered solutions of holonomic D-modules” arXiv:1311.6621.

2 Fourier transform for tempered solutions of D-modules

The Fourier–Laplace transform is an essential tool in the study of linear PDE and their solutions. B. Malgrange, C. Sabbah and T. Mochizuki studied the Fourier–Laplace transform in relation to the irregular Riemann–Hilbert correspondence on the Riemann sphere. Recently their results found interesting application to mirror symmetry and other fields.
An objective of the project FUNALGPDE was to follow the results proved by the above authors in order to obtain their counterpart in the subanalytic setting for tempered solutions.
Since the beginning Dr. Morando was aware of some technical constraints which would have restricted the study of the Fourier transform for D-modules and their tempered solutions to certain particular subcategories of holonomic D-modules, in the same spirit of the work of L. Daia. Recently, a paper by D'Agnolo-Kashiwara has appeared in which the category of enhanced ind-sheaves is introduced and studied along with the associated complex of enhanced tempered solutions of holonomic D-modules. Dr. Morando understood that this was the natural language to study a topological Fourier transform in full generality. He spent the second period of the project deepening his knowledge on such a new setting. He is now involved in matching the abstract and natural functorial formulae obtained via D'Agnolo-Kashiwara framework with the classical explicit results of C. Hien, T. Mochizuki and C. Sabbah.

3 Sheaves on the subanalytic site:

An objective of the project FUNALGPDE was to widen the application of subanalytic sheaves to the context of D-module theory and classical analysis. In particular we wanted to develop the work of Honda–Morando on tempered ultradistributions and to give a geometric description of the constructible property for sheaves on the subanalytic site by means of classical spaces as real spectra studied by Coste–Roy and Berkovich or Huber spaces.
As concern this topic, Dr. Morando intended to establish a bridge between some classical local constructions in which real blow-up spaces and functions with sectorial growth conditions appeared and the global and more natural setting of sheaves on the subanalytic site. In particular, specialists in the classical theory of local irregular Riemann-Hilbert correspondence on curves were searching for results of commutation between the direct and inverse image functors and the moderate De Rham functor. The importance of this problem and the results obtained until now can be understood from the recent work of C. Sabbah and T. Mochizuki. On the other hand, classical results of Kashiwara-Schapira give the commutation between the tempered De Rham functor on holonomic D-modules and the direct and inverse image functors. Furthermore, the spaces of functions with sectorial moderate growth can be obtained from tempered functions on subanalytic sets by means of the specialization on the subanalytic site defined by L. Prelli and Honda-Prelli. At the moment being, in collaboration with Prof. Honda and Dr. Prelli, he is trying to establish a clear and functorial connection between the two languages.

The other objectives of project FUNALGPDE concerning dissemination, interdisciplinary collaboration and development of Dr. Morando career are witnessed in the following list of achievements and activities.

In last years Dr. Morando had some interviews for permanent positions for which he obtained very good scores though not successful. At the moment he has a two years position at Augsburg University where he collaborates with Prof. Hien both on all the themes of project FUNALGPDE and on new joint projects.
Dr. Morando organized or took part actively in many international events as conferences and workshops, mainly at the forefront of mathematical research but also at the level of popularization to a wide public.
Dr. Morando was invited to many institution for exposing his results and collaborate with local researchers from other domains of research. Both establishing new connections and renewing old ones, greatly strengthening his scientific position as a European researcher.
Dr. Morando fully accomplished the mission of spreading the knowledge acquired during this project to enrich the European scientific community in many forms. In particular, in Padua, he taught a course on D-modules and differential complexes for the Doctoral School in Mathematical Sciences.