Elenco seminari del ciclo di seminari
“INTRODUCTION TO INVOLUTIVE STRUCTURES AND GLOBAL SOLVABILITY AND COHOMOLOGY OF TUBE STRUCTURES ON COMPACT MANIFOLDS”

This short course consists of two lectures, each two hours long. The first lecture introduces involutive structures, the associated differential complex, and key questions of solvability and regularity. We explore important examples and techniques for working in this framework. The focus is on practical, hands-on knowledge, enabling participants to begin research quickly and providing essential literature for those seeking deeper understanding. In the second lecture, I discuss some new techniques to study the differential complexes in the case of tube structures on M x T of corank m, in which M is a compact manifold and T is the m-torus. By systematically employing partial Fourier series, for complex tube structures, we completely characterize global solvability, in a given degree, in terms of a weak form of hypoellipticity, thus generalizing existing results and providing a broad answer to an open problem proposed by Hounie and Zugliani (Math Ann 369(3–4):1177–1209, 2017). We also obtain new results on the finiteness of the cohomology spaces in intermediate degrees. In the case of real tube structures, we extend an isomorphism for the cohomology spaces originally obtained by Dattori da Silva and Meziani (Math Nachr 289(17–18):2147–2158, 2016) in the case in which M is a n-torus . Moreover, we establish necessary and sufficient conditions for the differential operator to have closed range in the first degree. We conclude the lecture with some open problems.

This short course consists of two lectures, each two hours long. The first lecture introduces involutive structures, the associated differential complex, and key questions of solvability and regularity. We explore important examples and techniques for working in this framework. The focus is on practical, hands-on knowledge, enabling participants to begin research quickly and providing essential literature for those seeking deeper understanding. In the second lecture, I discuss some new techniques to study the differential complexes in the case of tube structures on M x T of corank m, in which M is a compact manifold and T is the m-torus. By systematically employing partial Fourier series, for complex tube structures, we completely characterize global solvability, in a given degree, in terms of a weak form of hypoellipticity, thus generalizing existing results and providing a broad answer to an open problem proposed by Hounie and Zugliani (Math Ann 369(3–4):1177–1209, 2017). We also obtain new results on the finiteness of the cohomology spaces in intermediate degrees. In the case of real tube structures, we extend an isomorphism for the cohomology spaces originally obtained by Dattori da Silva and Meziani (Math Nachr 289(17–18):2147–2158, 2016) in the case in which M is a n-torus . Moreover, we establish necessary and sufficient conditions for the differential operator to have closed range in the first degree. We conclude the lecture with some open problems.