Abstract: In early 2010, a mathematical theory of GLSM was proposed by Fan-Jarvis-Ruan to generalize both Gromov-Witten theory and FJRW-theory. The mathematical GLSM theory produced an open moduli space, in contrast to the traditional moduli theory where the compactness is required. Then, a cosection (constructed out of superpotential) localized the theory to the critical locus. The above theory is theoretically beautiful, but useless in computation. Recently, a delicate compactification of GLSM (logarithmic GLSM) was constructed to remedy the above defect. Its localization formula is proved to be extremely effective to solve many outstanding problems in the subject of Gromov-Witten theory, including BCOV axioms of higher genus Gromov-Witten theory of quintic 3-fold, r-spin conjecture relating r-spin virtual cycle and locus of holomorphic differential, modularity of Gromov-Witten theory of elliptic fibration and so on. In the talk, we will survey the above developments. These are joint works with Shuai Guo, Felix Janda and Qile Chen.

About the speaker

Prof. Yongbin Ruan studied from 1978 at Sichuan University with Benke Certificate of graduation followed by a master's degree in 1985. In 1985/86 he was a teaching assistant at the University of Wisconsin–Madison. In 1991 he received his Ph.D. from the University of California, Berkeley with thesis Gauge theory and its applications to Riemannian Geometry under the supervision of Robion Kirby (and Tomasz Mrowka). As a postdoc he was at Michigan State University. In 1993 he became an assistant professor at the University of Utah. At the University of Wisconsin–Madison he became in 1995 an associate professor and in 1999 a full professor. He was a professor at University of Michigan from 2006 to 2019 and Bill Fulton Collegiate Chair of Mathematics from 2013 to 2019. Since 2019, he is a professor at Institute for Advanced Study in Mathematics at Zhejiang University.

He has been a visiting professor at the ETH Zurich, in Hong Kong, and at MIT. He was in 1993 and again in 2004 at the Institut des hautes études scientifiques, in 1993 at the Max Planck Institute for Mathematics, in 1994 at Cambridge's Isaac Newton Institute, and in 1994 at the Mathematical Sciences Research Institute.

In 1998 he was an Invited Speaker with talk Quantum Cohomology and its Applications at the International Congress of Mathematicians in Berlin. From 1995 to 1997 he was a Sloan Research Fellow. He became a Fellow of the American Mathematical Society in the class of 2015.