In the fall of 2000, the Notre Dame logic community was lucky enough to get Greg Hjorth, Rodney G. Downey, Zoé Chatzidakis, and Paola D'Aquino to come to Notre Dame. Each of them came for about a month and taught a month long graduate class on a topic of their choice. The following (refereed) articles are refinements of these lectures.

These classes were attended by Professors Steve Buechler, Peter Cholak, Julia Knight and Sergei Starchenko. The graduate students attending were Bijan Afshordel (visiting from Germany), Andrew Arana, Alexander Berenstein, Tom Doherty, Jacob Heidenreich, Evgueni Vassiliev, and Rebecca Weber.

We would like to thank the Notre Dame Department of Mathematics for allowing the logicians at Notre Dame to split one of the advanced graduate logic classes into four separate pieces and for a portion of the required funding. We would like to thank the Graduate School at Notre Dame for another portion of the funding. Many thanks to Julia Knight, the Charles L.Huisking Professor of Mathematics, Notre Dame, who used her chair account for the remaining funding.

Most of all wewant to thank Greg Hjorth, Rodney G. Downey, Zoé Chatzidakis, and Paola D'Aquino for coming to Notre Dame and providing us with a very interesting series of lectures, and for the time and energy needed to turn them into the following articles. Our thanks.