Principal Investigator: Eduardo Gonzalez
The principal investigator will investigate, using symplectic geometry techniques, invariants associated to Hamiltonian group actions. The first project will study (joint with A. Ott, C. Woodward and F. Ziltener) the construction of gauged Gromov-Witten invariants on punctured surfaces with fixed holonomies. The PI will also investigate the particular case of toric actions on affine complex spaces. Our second project is on the study of the invariance of gauged Gromov-Witten potentials for toric actions on affine spaces. We will use established formulae to show that for Calabi-Yau wall-crossing the gauged Gromov-Witten potentials do not change. After this project is finished we will study the behaviour of the potential in the non-toric case. This will be done jointly with C. Woodward. Our third proposed activity, joint with H. Iritani, will explain the relation between Seidel elements and Mirror Transformations for nef toric non-singular projective varieties. We will then try to apply our methods to the non-nef case. Our last proposed project with A. Cotton-Clay will attempt to resolve a conjecture due to M. Thaddeus relating Floer cohomology for symplectomorphisms and orbifold (stacky) cohomology for global quotients by finite groups.
Our research will advance the interplay of several subjects in mathematics and physics where symmetries play a relevant role. The first project in this grant will advance the understanding and applications of symplectic geometry and gauge theory (the modern language of elementary particle physics) into physical theories called "gauged sigma models". Our other projects will provide a different approach to the Mirror Symmetry phenomena using symplectic geometry invariants.
Our NSF sponsored research advanced the interplay of several subjects in mathematics and physics where symmetries play a relevant role. The first project in this grant advanced the understanding and applications of symplectic geometry techniques, gauge theory (the modern language of elementary particle physics) and algebraic geometry into physical theories called sigma models or Gromov-Witten theory. Our research found applications into other theories in Physics, namely Mirror Symmetry. We were able to resolve the crepant conjecture, for a big class of spaces, some of which are of interest to mathematicians in other fields. This conjecture catapulted work in Gromov-Witten theory for several years. Our work used techniques common in symplectic geometry, and we applied it into algebraic geometry. We expect that our results will serve as a computational tool for future researchers in the fields of algebraic and symplectic geometry. The grant sponsored several projects which resulted in two published papers, three Internet available preprints and several other in-progress work. With the grant the PI was able to establish and continue international collaborations, as well as research visits to several countries, including Japan, Spain, Denmark, Mexico and Germany. This help us disseminate our results, sponsored by the NSF. More importantly, this grant helped the PI kick-start a new line of research on the Seidel representation which, as of now, has been the focus of research of many other researchers in the US and overseas. The PI established several contacts and research connections during the life of this award, not only domestically but from several international research centers. This highlights the impact NSF sponsored research has globally. Lastly, during the life of this award, the PI was active disseminating the results of his research in many institutions. The PI was active promoting the importance of mathematics to members of underrepresented groups, women and minorities, by delivering lectures and organizing conferences national and international on his field of research. The PI ran summer research programs for undergraduate students as well, where he encourage students to pursue a career in the mathematical sciences.
