Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.

Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work.

]]>We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian _{ p }-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated.

Each fixed integer nnhas associated with it ⌊n2⌋ rhombs: ρ1,ρ2,…,ρ⌊n2⌋, where, for each 1≤h≤⌊n2⌋, rhomb ρ_{h}ρh is a parallelogram with all sides of unit length and with smaller face angle equal toh×πn radians.

An Oval is a centro-symmetric convex polygon all of whose sides are of unit length, and each of whose turning angles equalsℓ×πn for some positive integer ℓℓ. A (n,k)(n,k)-Oval is an Oval with 2k2k sides tiled with rhombsρ1,ρ2,…,ρ⌊n2⌋; it is defined by its Turning Angle Index Sequence, a kk-composition of nn. For any fixed pair (n,k)(n,k) we count and generate all (n,k)(n,k)-Ovals up to translations and rotations, and, using multipliers, we count and generate all (n,k)(n,k)-Ovals up to congruency. For odd nn if a (n,k)(n,k)-Oval contains a fixed number λλ of each type of rhombρ1,ρ2,…,ρ⌊n2⌋ then it is called a magic (n,k,λ)(n,k,λ)-Oval. We prove that a magic (n,k,λ)(n,k,λ)-Oval is equivalent to a (n,k,λ)(n,k,λ)-Cyclic Difference Set. For even nn we prove a similar result. Using tables of Cyclic Difference Sets we find all magic (n,k,λ)(n,k,λ)-Ovals up to congruency for n≤40n≤40.

Many related topics including lists of (n,k)(n,k)-Ovals, partitions of the regular 2n2n-gon into Ovals, Cyclic Difference Families, partitions of triangle numbers, uu-equivalence of (n,k)(n,k)-Ovals, etc., are also considered.

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