September 20
Sam Coskey,
Boise State University
Jumps of equivalence relations and scattered linear orders
This is joint work with John Clemens. We will begin this talk by discussing the problem of classifying the countable scattered linear orders. Here a linear order is called scattered if the rational order doesn’t embed into it. The class of such orders admits a ranking function valued in the ordinals; we will study the corresponding classification problem for each fixed rank. We will show that each increase in rank results in a “jump” in the complexity of the classification problem. In the second part of the talk we will define a family of jump operators on equivalence relations, each associated with a fixed countable group. The jump in the case of scattered linear orders is that associated with the group Z of integers. We will discuss the basic theory of these jump operators. Finally, we will discuss the question of when such a jump operator is proper, in the sense that the jump of E is strictly above E in the Borel reducibility order.