## ~ Abstracts ~

Character tables, the group conditions, and tractability
William Cocke, University of Wisconsin, USA

We examine the group condition on a finite matrix and show that together with invertability it determines a character table of a finite abelian group. We then show that such a condition is sufficient to establish a dichotomy theorem regarding the graph partition function.

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Computable numberings in hierarchies
Marina Dorzhieva, Novosibirsk State University, Russia

In mid-90s, Goncharov and Sorbi [1] described a new approach to computable numberings, extending the concept of computability to numberings of families of objects admitting a description in some effectively specified language.

In this approach we can define computable numberings in arithmetical, analytical and Ershov hierarchies.

We will discuss some classical and recent results concerning computable numberings in these hierarchies.

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Jun Le Goh, Cornell University, USA

I will present Lerman, Solomon, and Towsner (2012)'s proof that ADS does not imply SCAC. Time permitting, I will also outline proofs by Wang and Patey (independently) of nonimplications between conjunctions of various related statements.

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Tukey reducibility among Borel ideals on omega
Jialiang He, Sichuan University, China

Let P and Q be two directed posets, we call P is Tukey reduce to Q, denote as P \leq_T Q. If there exists a function from Q to P map cofinal subsets of Q into cofinal subsets of P. In this talk, we will discuss some results about Tukey relation among Borel ideals on omega.

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