Topology in Symplectic, Contact, Low Dimensions and Singularities

It is our natural desire to understand the structure of the space-time that we live in. This can be described in mathematical terms as a 4-dimensional space, or ""manifold"". Surprisingly, dimension 4 is the most mysterious and elusive of all dimensions. In dimensions both lower and higher than 4, we have a rather clear picture about questions related to continuity and differentiability properties. By contrast, dimension 4 is highly irregular in many significant aspects, and, despite several breakthrough results in the past fifty years, we still do not have a panoramic view of the landscape of 4-dimensional spaces.The doctoral network Top-SCoLDS will prepare a new generation of researchers to answer deep questions related to 4-dimensional manifolds. The subject can be approached from many perspectives, including algebraic and symplectic-contact geometry, geometric topology, knot theory, partial differential equations, and artificial intelligence. Top-SCoLDS will take a diversified approach to questions about 4-dimensional manifolds, leveraging the members' complementary expertise.Top-SCoLDS will bring together world-leading experts in these fields, who will join their efforts to provide a comprehensive educational path for 12 PhD students. The students will be exposed to diverse tools and techniques to attack cutting-edge research questions on 4-manifolds, giving them a wide perspective on the subject and the opportunity to thrive in this active research area.The PhD students will benefit from research, education, and complementary skills training. They will receive training from world leaders in the theory and application of machine learning, AI, and topological data analysis, and from a wide range of non-academic and industrial partners. This will enable these students to innovate, collaborate, and become leading professionals in mathematics, and will ensure that they can apply their skills beyond the academic context and excel in the industrial sector too.""