Title: The modulus of curve families in R^n and its properties (Part 1 of 2)

Abstract: This talk will be an introduction to what is the modulus of a curve family, introduced in 1950 by Ahlfors and Beurling and generalized to its current form by Fuglede in 1957. We will define the modulus for curve families in R^n, but the definition makes sense in a general metric measure space. Although the definition is a bit cumbersome, the modulus has some very nice and useful properties, which lead naturally to the definition of quasiconformal mappings and also Newtonian spaces. These are, respectively, generalizations of conformal mappings and Sobelev spaces. We will prove these nice properties of the modulus and also calculate the modulus of certain families of curves.