Large cardinals, substructures, and Chang's Conjecture

One of the basic results in model theory is Lowenheim-Skolem. It states that every infinite model has infinite sub-models of any size. Elementary substructures basically catch all intrinsic properties of the large structure (first order properties). Obtaining substructure with more similarity to the original structure (second order properties) is more subtle, and often independent of the standard axioms of set theory. 

In this talk I will discuss a special case of a second-order version of the Lowenheim-Skolem theorem - Chang's Conjecture. This principle is deeply connected to large cardinal axioms, in a ways that are not fully understood yet. I will present some of the definitions in this area and discuss some cases in which the Chang's Conjecture holds.