In this talk, I will describe a 2-functor, called "2-linearization", from spans of groupoids into 2-vector spaces (C-linear abelian categories). Using groupoids representing moduli stacks of flat connections, this gives rise, for every finite group G, to an "extended topological quantum field theory", a 2-functorial invariant for manifolds with corners. I will also discuss how to extend this to compact Lie groups, including measures on stacks, and a generalization of the category of distintegrations (a nice category of measure spaces) to stacks.

Manifolds are very nice objects in modern mathematics. However, the category of manifolds is not that pleasant. Many generalizations of manifolds are proposed around 1980's. Diffeological spaces are one of them, which were first defined by J. Souriau in 1980, and later on systematically developed by P. Iglesias-Zemmour, J. Baez, A. Hoffnung and others. In this talk, some known results on the basic properties of diffeological spaces and some of their differential geometric and topological aspects will be described. Some new results on the general topological aspects and categorical aspects will be presented at the end.

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I describe recent work with Greg Pearlstein proving that the zero locus of an admissible normal function is algebraic. I will explain why this is a generalization of a result of Cattani, Deligne and Kaplan showing that the Noether-Lefschetz locus is algebraic. I will also explain why the key step in the proof is a boundedness theorem for period maps. In addition, I will spend some time motivating the result and explaining how normal functions arose in Lefschetz's proof the Hodge conjecture for surfaces (the 1-1 theorem) and how they also are related to the Hodge conjecture for arbitrary varieities.

We will explain the symplectic cutting construction and some of its applications.