Classification from a computable viewpoint

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory. 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

Isomorphism relations on computable structures

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Algebraic Methods in Computational Complexity

Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. In some of the most exciting recent progress in Computational Complexity the algebraic theme still plays a central role. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Also the areas of derandomization and coding theory have experimented important advances. The seminar aimed to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and the goal of the seminar was to play an important role in educating a diverse community about the latest new techniques

Systematics 2008 - Programme and Abstracts

The Göttingen conference „Systematics 2008“ is the first joint meeting of the Gesellschaft für Biologische Systematik (GfBS) and the German Botanical Society, section Biodiversity and Evolutionary Biology (DBG), being the 10th Annual Meeting of the GfBS and the 18th International Symposium „Biodiversity and Evolutionary Biology“ of the DBG. The conference programme covers biological systematics in the widest sense and provides ample opportunities for oral and poster presentations on new advances in plant, animal and microbial systematics. This volume brings together the abstracts of invited speaches from the plenary sessions on „Progress in Deep Phylogeny“, „Speciation and Phylogeography“, and „New Trends in Biological Systematics“ as well as those of submitted talks and poster sessions.The Göttingen conference „Systematics 2008“ is the first joint meeting of the Gesellschaft für Biologische Systematik (GfBS) and the German Botanical Society, section Biodiversity and Evolutionary Biology (DBG), being the 10th Annual Meeting of the GfBS and the 18th International Symposium „Biodiversity and Evolutionary Biology“ of the DBG. The conference programme covers biological systematics in the widest sense and provides ample opportunities for oral and poster presentations on new advances in plant, animal and microbial systematics. This volume brings together the abstracts of invited speaches from the plenary sessions on „Progress in Deep Phylogeny“, „Speciation and Phylogeography“, and „New Trends in Biological Systematics“ as well as those of submitted talks and poster sessions

Programme and abstracts / Systematics 2008 : Göttingen 7-11 April 2008 / 10th Annual Meeting of the Gesellschaft für Biologische Systematik

The Göttingen conference Systematics 2008 is the first joint meeting of the Gesellschaft für Biologische Systematik (GfBS) and the German Botanical Society, section Biodiversity and Evolutionary Biology (DBG), being the 10th Annual Meeting of the GfBS and the 18th International Symposium Biodiversity and Evolutionary Biology of the DBG. The conference programme covers biological systematics in the widest sense and provides ample opportunities for oral and poster presentations on new advances in plant, animal and microbial systematics. This volume brings together the abstracts of invited speaches from the plenary sessions on Progress in Deep Phylogeny, Speciation and Phylogeography, and New Trends in Biological Systematics as well as those of submitted talks and poster sessions.The Göttingen conference Systematics 2008 is the first joint meeting of the Gesellschaft für Biologische Systematik (GfBp. and the German Botanical Society, section Biodiversity and Evolutionary Biology (DBG), being the 10th Annual Meeting of the GfBS and the 18th International Symposium Biodiversity and Evolutionary Biology of the DBG. The conference programme covers biological systematics in the widest sense and provides ample opportunities for oral and poster presentations on new advances in plant, animal and microbial systematics. This volume brings together the abstracts of invited speaches from the plenary sessions on Progress in Deep Phylogeny, Speciation and Phylogeography, and New Trends in Biological Systematics as well as those of submitted talks and poster sessions.Kongress: International Symposium Biodiversity and Evolutionary Biology ; 18 (Göttingen) : 2008.04.07-11 Annual meeting of the Gesellschaft für Biologische Systematik ; 10 (Göttingen) : 2008.04.07-11 Joint meeting of the Gesellschaft für Biologische Systematik and the German Botanical Society, Section Biodiversity and Evolutionary Biology ; 1 (Göttingen): 2008.04.07-1