We discuss the equivalence of perfect obstruction theories (extensively used in formation of Gromov-Witten invariants) and quasi-smooth derived schemes. The latter can be thought of as a derived zero locus. When the zero locus is locally given by a complete intersection, one gets a classical scheme.

Symplectomorphism groups are one of the classical infinite-dimensional Lie groups that have been studied. Arnold's paper in 1966, where he used methods of infinite-dimensional Lie theory to study the hydrodynamics of a perfect incompressible fluid, has motivated intensive research in infinite-dimensional Lie theory. He showed that the geodesics on the group of volume preserving diffeomorphisms are essentially solutions of the Euler's equations. In a fundamental paper in 1970 Marsden and Ebin studied some infinite-dimensional groups in more details which included symplectomorphism groups. In this talk we study a special class of symplectomorphism groups that resemble compact Lie groups in a particular way. We see there is a similar notion of the so-called maximal tori in the symplectomorphism groups of toric manifolds. As a consequence we see there is an analogue of the Schur-Horn-Kostant convexity theorem in this infinite-dimensional setting. It also should be mentioned that these results are a generalization of results that were obtained by Bao-Ratiu 1997, Bloch-Flaschka-Ratiu 1993 and El-hadrami 1996 for special cases of toric manifolds.

The goal of this talk is to introduce some of the most basic notions in the field of topology. We focus on the concept of a surface or 2-dimensional manifold. A surface is a mathematical abstraction of the familiar concept of a surface made of paper - like the surface of a sphere, the Mobius strip, and so on. We will spend time constructing these surfaces and then I will demonstrate the tools an algebraic topologist would use to classify all possible surfaces.

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.

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.