Final Report Summary - MODALAN (Model Theory and Algebraic Analysis)
The researcher Dr. Luca Prelli has worked on the interdisciplinary project MODALAN: Model Theory and Algebraic Analysis at CMAF, Centro de Matemática e Aplicações Fundamentais, of the University of Lisbon, Portugal. At the host institution there is scientific expertise that suits the researcher's needs excellently. It combines model theory and algebraic analysis both disciplines on which the project depends. This provides the best environment for the projected research. Within a period of 24 months, the aim of this project is to investigate new cohomological approaches to 1. o-minimal geometry and 2. algebraic study of D-modules. The nature of this proposal is strongly multidisciplinary since it involves different branches of mathematics as algebra, analysis, geometry and logic. Indeed we intend to apply cohomological and functorial methods coming from homological algebra and algebraic geometry to o-minimal geometry and to the study of D-modules (solution of PDE).
1. O-minimal geometry
Tasks: (1) prove Pillay's conjecture for orientable definably compact groups (in o-minimal structures with definable Skolem functions); (2) concerning o-minimal sheaf cohomology and geometry of semi-bounded sets we want to construct the Grothendieck's six operations for o-minimal sheaves and prove refined structure theorems for semi-bounded sets (including coverings by open cells with applications to (1)).
2. Algebraic study of D-modules
Tasks: (1) extension of the classical functorial constructions to the framework of subanalytic sheaves; (2) applications to the study of the solutions of D-modules.
This is the report of the scientific activity based the arguments related to MODALAN project (see Part B).
1 - O-minimal geometry.
- We established a fundametal invariance theorem for the cohomology with compact support in o-minimal structures admitting some suitable conditions (including compact definable groups involved in the Pillay's conjecture). The paper with the above result is under submission (available online: with Mario Edmundo, CMAF, Lisbon, Invariance of o-minimal cohomology with definably compact supports, arXiv:1205.6124).
- We studied definable proper maps, definitions and properties. We made a paper on this subject which is actually under submission (available online: with Mario Edmundo and Marcello Mamino, CMAF, Lisbon, On definably proper maps, arXiv:1404.6634).
- We proved the existence of the six Grothendieck operations in o-minimal structures admitting some suitable conditions (including compact definable groups involved in the Pillay's conjecture). A note with a report on this result has been published (with Mario Edmundo, CMAF, Lisbon, The six Grothendieck operations on o-minimal sheaves, C. R. Acad. Sci. Paris, Ser. I, N. 352, pp. 455-458).
The whole paper is available online (with Mario Edmundo, CMAF, Lisbon, The six Grothendieck operations on o-minimal sheaves, arXiv:1401.0846).
- Thanks to these results we have now all the cohomological ingredients required to prove the Pillay's conjecture. We are now writing a paper with the final proof of the conjecture.
- We generalized o-minimal sheaf theory to the case of T-topologies (a kind of Grothendieck topologies). We made a paper which has been accepted for pubblication (with Mario Edmundo, CMAF, Lisbon, Sheaves on T-topologies, to appear, Journal of the Math. Soc. of Japan).
- As an application of o-minimal sheaf cohomology to semi-algebraic geometry we found a sheaf theoretical proof of the de Rham Theorem for Schwartz functions on Nash manifolds. The paper with the above result has been accepted for pubblication (De Rham theorem for Schwartz functions on Nash manifolds, Israel Journal of Mathematics, Volume 197, Issue 1, pp. 131-137).
- We proved that in a semi-bounded o-minimal expansion of an ordered group every non-empty open denable set is a finite union of open cells. The paper with the above result has been accepted for pubblication (with Mario Edmundo and Pantelis Eleftheriou, CMAF, Lisbon, Coverings by open cells, Archive for Math. Logic, Volume 53, Issue 3, pp. 307-325).
- We studied locally definable manifolds and we proved: (i) the existence of universal locally definable covering maps; (ii) invariance results for locally definable covering maps, o-minimal fundamental groups and fundamental groupoids; (iii) monodromy equivalence for locally constant o-minimal sheaves; (iv) classification results for locally definable covering maps; (v) o-minimal Hurewicz and Seifert–van Kampen theorems. The paper with the above result has been accepted for pubblication (with Mario Edmundo and Pantelis Eleftheriou, CMAF, Lisbon, The universal covering map in o-minimal expansions of groups, Topology and its Applications, Volume 160, Issue 13, pp. 1530-1556).
2 - Algebraic study of D-modules.
- We ended and published our work on microlocalization of subanalytic sheaves (Microlocalization of subanalytic sheaves, Mémoires de la SMF 135).
- We ended and published our work on multi-specialization of subanalytic sheaves (with Naofumi Honda, Hokkaido University, Multi-specialization and multi-asymptotic expansions, Advances in Mathematics, Vol. 232, pp. 432-498). We are now investigating new generalizations.
- We performed the study of the multi-microlocalization, obtaining a fiber formula extending the classical one, vanishing results for the cohomology, including edge of the wedge theorems and an extimation of microsupport. We also made some applications to D-modules. The paper with the above results is under submission (available online: with Naofumi Honda, Hokkaido University and Susumu Yamazaki, Nihon University, Multi-microlocalization and microsupport; arXiv:1401.0746).
- As an application of sheaf theory on subanalytic sites we constructed functorially the sheaf of relative tempered distributions (in generally one can construct sheaves of functions with relative growth conditions). The paper with the above result has been published (with Teresa Monteiro Fernandes, CMAF, Lisbon, Relative subanalytic sheaves, Fundamenta Mathematicae, Vol. 226, n. 1, pp. 79-100).
1. O-minimal geometry
Tasks: (1) prove Pillay's conjecture for orientable definably compact groups (in o-minimal structures with definable Skolem functions); (2) concerning o-minimal sheaf cohomology and geometry of semi-bounded sets we want to construct the Grothendieck's six operations for o-minimal sheaves and prove refined structure theorems for semi-bounded sets (including coverings by open cells with applications to (1)).
2. Algebraic study of D-modules
Tasks: (1) extension of the classical functorial constructions to the framework of subanalytic sheaves; (2) applications to the study of the solutions of D-modules.
This is the report of the scientific activity based the arguments related to MODALAN project (see Part B).
1 - O-minimal geometry.
- We established a fundametal invariance theorem for the cohomology with compact support in o-minimal structures admitting some suitable conditions (including compact definable groups involved in the Pillay's conjecture). The paper with the above result is under submission (available online: with Mario Edmundo, CMAF, Lisbon, Invariance of o-minimal cohomology with definably compact supports, arXiv:1205.6124).
- We studied definable proper maps, definitions and properties. We made a paper on this subject which is actually under submission (available online: with Mario Edmundo and Marcello Mamino, CMAF, Lisbon, On definably proper maps, arXiv:1404.6634).
- We proved the existence of the six Grothendieck operations in o-minimal structures admitting some suitable conditions (including compact definable groups involved in the Pillay's conjecture). A note with a report on this result has been published (with Mario Edmundo, CMAF, Lisbon, The six Grothendieck operations on o-minimal sheaves, C. R. Acad. Sci. Paris, Ser. I, N. 352, pp. 455-458).
The whole paper is available online (with Mario Edmundo, CMAF, Lisbon, The six Grothendieck operations on o-minimal sheaves, arXiv:1401.0846).
- Thanks to these results we have now all the cohomological ingredients required to prove the Pillay's conjecture. We are now writing a paper with the final proof of the conjecture.
- We generalized o-minimal sheaf theory to the case of T-topologies (a kind of Grothendieck topologies). We made a paper which has been accepted for pubblication (with Mario Edmundo, CMAF, Lisbon, Sheaves on T-topologies, to appear, Journal of the Math. Soc. of Japan).
- As an application of o-minimal sheaf cohomology to semi-algebraic geometry we found a sheaf theoretical proof of the de Rham Theorem for Schwartz functions on Nash manifolds. The paper with the above result has been accepted for pubblication (De Rham theorem for Schwartz functions on Nash manifolds, Israel Journal of Mathematics, Volume 197, Issue 1, pp. 131-137).
- We proved that in a semi-bounded o-minimal expansion of an ordered group every non-empty open denable set is a finite union of open cells. The paper with the above result has been accepted for pubblication (with Mario Edmundo and Pantelis Eleftheriou, CMAF, Lisbon, Coverings by open cells, Archive for Math. Logic, Volume 53, Issue 3, pp. 307-325).
- We studied locally definable manifolds and we proved: (i) the existence of universal locally definable covering maps; (ii) invariance results for locally definable covering maps, o-minimal fundamental groups and fundamental groupoids; (iii) monodromy equivalence for locally constant o-minimal sheaves; (iv) classification results for locally definable covering maps; (v) o-minimal Hurewicz and Seifert–van Kampen theorems. The paper with the above result has been accepted for pubblication (with Mario Edmundo and Pantelis Eleftheriou, CMAF, Lisbon, The universal covering map in o-minimal expansions of groups, Topology and its Applications, Volume 160, Issue 13, pp. 1530-1556).
2 - Algebraic study of D-modules.
- We ended and published our work on microlocalization of subanalytic sheaves (Microlocalization of subanalytic sheaves, Mémoires de la SMF 135).
- We ended and published our work on multi-specialization of subanalytic sheaves (with Naofumi Honda, Hokkaido University, Multi-specialization and multi-asymptotic expansions, Advances in Mathematics, Vol. 232, pp. 432-498). We are now investigating new generalizations.
- We performed the study of the multi-microlocalization, obtaining a fiber formula extending the classical one, vanishing results for the cohomology, including edge of the wedge theorems and an extimation of microsupport. We also made some applications to D-modules. The paper with the above results is under submission (available online: with Naofumi Honda, Hokkaido University and Susumu Yamazaki, Nihon University, Multi-microlocalization and microsupport; arXiv:1401.0746).
- As an application of sheaf theory on subanalytic sites we constructed functorially the sheaf of relative tempered distributions (in generally one can construct sheaves of functions with relative growth conditions). The paper with the above result has been published (with Teresa Monteiro Fernandes, CMAF, Lisbon, Relative subanalytic sheaves, Fundamenta Mathematicae, Vol. 226, n. 1, pp. 79-100).