Abstract: One of the foundational reasons for calculus is to understand macroscopic behavior of functions from its microscopic, or asymptotic behavior. First developed in Euclidean setting and then extended to the setting of smooth manifolds, calculus has been used in various settings in physics and in dynamics. However, when the underlying space in which the function measures certain quantities is not a smooth space, the notions of calculus do not make sense. In this talk we will discuss a few ways of doing calculus in settings that do not have smooth structures. 

Abstract: One of the ways of measuring the energy of a function is to consider the integral of the square of its derivative; this is what is done in order to measure the energy of a satellite in motion. When the motion is in an inviscid medium, the energy is the integral of a power of its "speed". These are examples of local energies, that is, two functions that agree outside of a small region have differing energies only in that region. In some phenomena such local behavior is not applicable. In this talk we will discuss some ways of measuring nonlocal energies by using local energies.