Manifolds with a-corners & moduli space of Morse flows

Manifolds with ordinary boundary/corners have found their presence in differential geometry and PDEs: they form Man^b or Man^c category; and for boundary value problems, they are nice objects to work on. Manifolds with analytical corners -- a-corners for short -- form a larger category Man^{ac} which contains Man^c, and they can in some sense be viewed as manifolds with boundary at infinity.
In this talk I'll walk you through the definition of manifolds with corners and a-corners, and give some examples to illustrate how the new definition will help.