The proposed research lies on the interaction between Gromov--Witten theory and other subjects in mathematics and physics, including birational geometry, moduli of curves, and mirror symmetry. The main themes of the proposal are the functoriality of Gromov--Witten theory under crepant transformations and the mirror symmetry in the orbifold category.
Gromov--Witten theory lies in the intersection of many exciting research areas in mathematics and physics. On the one hand, the theory itself has remarkable conjectural structures. Investigating these structures requires some new insights into the theory and input from other areas. This provides a lot of interesting problems for classical subjects in mathematics. On the other hand, it also helps to discover deep relations and connections between existing mathematics.
Gromov--Witten theory is a rapidly developing field with many deep and interesting connections to areas of current research in both mathematics and physics. Often ideas from one area can be applied to another area. Sometimes, different ideas from different areas converge to form a new project. These kinds of interactions have inspired many new developments and produced many beautiful results. This project has its origin from ideas in at least three different areas. Firstly, from physics, string theory predicts that some different-looking spaces will share equivalent physical phenomena. In other words, these spaces will be indistinguishable from string-theoretic points of view. This prediction has far-reaching consequences in Gromov--Witten theory which appear to be quite important but also quite difficult to rigorously demonstrate from a purely mathematical point of view. (Technically, these are called Crepant Resolution Conjecture or more generally Crepant Transformation Conjeccture.) Secondly, from the classical study of geometry of spaces mathematicians realized that some different-looking spaces have the same "geometric invariants". One class of examples is called "K-equivalent" spaces. It is a deep mathematical result, proved in the late 1990s, that K-equivalent spaces share identical geometric numbers called Betti numbers. However, upon closer examination, some more refined structures can be different. One natural question follows: Is there a way to "modify" or "deform" the refined structures so that they become the same for these different spaces? It turns out that the so-called quantum deformation from Gromov--Witten theory is the key to this modification. (Technically, the refined structures are the ring structures of the cohomology rings and the quantum deformations of the cohomology rings are the quantum cohomology rings defined by the Gromov--Witten theory. They are equivalent only after analytic continuation.) Thirdly, from a purely Gromov--Witten point of view, one wants to understand what kind of operations are "natural" in this theory. This is a difficult question, as it turns out that a lot of things which are "natural" in the classical theory no longer hold after the "quantum deformation" in the Gromov--Witten theory. Therefore, one has to look for new kinds of natural operations. These three points of view converge in this project and form the central question of the study. This work supported by the NSF makes significant contributions to understanding this important direction. Together with many collaborators, the PI was able to formulate precise mathematical statements of this phenomenon and prove them in many key cases with newly devised tools. This project has produced a number of publications in journals and conference proceedings, all of them peer-reviewed. In particular, together with Prof. Lin and Prof. Wang from National Taiwan University, three papers are published. These papers clarify and generalize the previous results from three complex dimensional spaces to higher dimensions. Furthermore, the PI often collaborated with graduate students and postdocs, in addition to two international collaborators. Thus, this project also provides a fertile ground for training students and new PhDs. In particular, this funding has supported two graduate students, one of whom has graduated and is now a postdoc in Purdue University. Two of the published papers are joint works with my two different PhD students. In addition to this, the education component of this project includes two series of lectures in a few summer schools in Grenoble and in Taipei.
