As our world, with time included, is four dimensional, exploring similarities and differences of 4-dimensional spaces through geometric and topological methods leads to better understanding of the universe we live in. Smooth and symplectic structures on 4-dimensional spaces are broadly featured in theoretical physics; e.g. in classical mechanics, string and quantum field theories. The research projects invoke new ideas and techniques, creating leverage to address several important problems regarding the topology of smooth and symplectic 4-manifolds, and that of contact 3-manifolds and their fillings. A key aspect of this program is the reduction of many intricate problems to fairly simple algebraic relations between curves on surfaces, which sets an excellent ground to present problems accessible to graduate and advanced undergraduate students. The PI will engage and mentor students in related research topics.
This is a project in low dimensional geometry and topology, focusing on a variety of profound questions involving symplectic 4-manifolds and contact 3-manifolds. Some of the projects on 4-manifolds are pertinent to classification of symplectic Calabi-Yau surfaces, diversity of Lefschetz pencils and fibrations, novel constructions of small exotic and symplectic 4-manifolds, and complexity in various stable equivalences. As for 3-manifolds, the PI will probe the complexity in Giroux correspondence in terms of open book support genus, the characterization of newly discovered contact 3-manifolds with arbitrarily large Stein fillings, diversity of complex curves bounding links in the 3-sphere, and generalizations of quasi-positive links. Gauge theory, new mapping class group techniques and symplectic surgeries will play a vital role in this program.
