Abstract:

Abstract:

Abstract: In his seminal 1961 paper, Auslander gave a beautiful characterization of flatness of a finite module over a regular local ring $R$ in terms of torsion in tensor powers of the module. Almost 40 years later, Vasconcelos conjectured a generalization of this criterion to the category of finite type $R$-algebras. I will survey a recent development in this area, leading to the establishing of Vasconcelos' conjecture (and then some) in characteristic zero, by local analytic methods. These are joint works with E. Bierstone and P.D. Milman, and with Hadi Seyedinejad.

Abstract: For years, much effort were spent to connect geometry-physics to number theory. Recently, a deep relation between Gromov-Witten theory and number theory were discovered on the topic of quasi-modular forms. It has already generated a great deal of interest among Gromov-Witten theory community. Moreover, it appears to be the beginning of an exciting area, which could potentially benefit both subjects. In the talk, I will explain these interesting discoveries.