In the first half of the talk, I will review the theory of shifted symplectic structures in derived algebraic geometry and its applications to Donaldson-Thomas theory of Calabi-Yau 3-folds and 4-folds.

In the second half of the talk, I will first introduce symplectic pushforwards as a general operation in shifted symplectic geometry. Basic examples, including cotangent bundles, critical loci, and Hamiltonian reduction, can be understood as special cases of this operation. Moreover, this unification enables us to provide an etale local structure theorem for shifted symplectic Artin stacks.

I will then introduce Lagrangian classes as a new type of virtual classes in enumerative geometry, whose existence was conjectured by Joyce. Lagrangian classes not only unify most of the modern enumerative structures, but also provide the following new enumerative structures: (1) cohomological field theories for gauged linear sigma models; (2) cohomological Hall algebras for 3-Calabi-Yau categories; (3) relative Donaldson-Thomas invariants for 4-dimensional log Calabi-Yau pairs; (4) algebraic Fukaya categories for hyper-kahler varieties; (5) refined surface counting invariants for Calabi-Yau 4-folds. This is joint work in progress with Adeel Khan, Tasuki Kinjo, and Pavel Safronov.