Speaker: Paul Brodhead Title: Effective Capacity and Random Closed Sets Abstract: I will discuss recent joint work, with Douglas Cenzer, investigating the connection between measure and capacity in the space of closed subsets of Cantor space. For any computable measure, a computable capacity may be defined by letting T(Q) be the measure of the family of closed sets which have nonempty intersection with Q. We prove an effective version of the Choquet's theorem by showing that every computable capacity may be obtained from a computable measure in this way. Furthermore, we establish conditions which characterize when the capacity of a random closed set is nonnegative. Finally, we construct, for certain measures, an effectively closed set with positive capacity and with Lebesgue measure zero. Speaker: Alan Dow Title: Normality and Cohen/Random reals Abstract: The consistency of the Normal Moore Space Conjecture relied heavily on the fact that adding either Cohen or Random reals preserved when sets could not be separated by open sets in a topological space. In response to a question from the Open Problems in Topology II by Ohta and Yamazaki, we investigate preservation questions about separating sets by continuous real-valued functions. Speaker: Damir Dzafarov Title: Stable Ramsey's Theorem and Measure Abstract: The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are non-null in a certain effective measure-theoretic sense. We show that uniformity results for the two principles agree below $\bf 0'$ but are not the same in general. We also answer the analogs of two well known questions about the stable Ramsey's theorem by showing that our weaker principle does not imply $COH$ or $WKL_0$ in the context of reverse mathematics. Speaker: Johanna Franklin Title: Randomness and the Ershov hierarchy Abstract: To determine whether a real is Martin-Loef random, we use Martin-Loef tests: uniform sequences of r.e. subsets of the Cantor space whose measures converge uniformly to 0. In this talk, I will define new notions of randomness in which the subsets of the Cantor space are $n$-r.e. for some $n$ greater than 1. I will present various ways in which a test can be considered to be $n$-r.e. and describe the classes of $n$-r.e. random reals. In one case, the resulting class of random reals is a proper subclass of the Martin-Loef random reals but properly contains the Demuth random reals and the weakly 2-random reals. Speaker: Denis Hirschfeldt Title: Reverse Mathematics of Model Theoretic Principles Abstract: I will discuss how tools from reverse mathematics and computability theory can be used to establish close connections between basic model theoretic results, combinatorial principles, and notions of genericity. In particular, I will discuss joint work with Shore and Slaman on the Atomic Model Theorem (the fact that every complete atomic theory in a countable language has a countable atomic model), and with Lange and Shore on the Homogeneous Model Theorem (a characterization of the sets of types of countable theories that are type spectra of homogeneous models). Speaker: Bart Kastermans Title: Formalizing Set Theory; on a simple example. Abstract: We all know the vast difference between the notion of proof as described in an introductory logic class, and the one we use on a daily basis when doing mathematics. The simple notion of proof has a large benefit that it is easy to check. Ideally all our proofs would be given with this much detail (next to the intuitive explanation) so that a higher degree of certainty can be obtained. In practice this is undoable, or at least highly unpleasant, to do by hand. To facilitate this certain computer systems have been developed to help with this. In this talk I'll give a short introduction to such a system, and show how I have used it to formalize a simple example of a cofinitary group. Speaker: Julia Knight Title: Describing free groups Abstract: Sela showed that the free groups on more than one generator are all elementarily equivalent. There is related work by Karlampovich and Myasnikov. Jacob Carson, Valentina Harizanov, Karen Lange, Christina Maher, Charles McCoy, Andrei Morozov, Sara Quinn, John Wallbaum and I looked for the simplest possible infinitary descriptions of the different free groups. To show that our descriptions are optimal, we used index sets. We obtained sharp results for the free groups of finite rank. For the group of infinite rank, we showed that there is a computable $\Pi_4$ description, but we could not show that this is optimal. Wallbaum and McCoy continued working, and succeeded in doing this. We also considered the problem of finding a basis for a given copy $G$ of the free group of infinite rank. The large group of authors succeeded in showing that there is a basis that is $\Pi^0_2(G)$. Wallbaum and McCoy showed that this is best possible. Speaker: Geoff Laforte Title: Equivalence structures and isomorphisms in the difference hierarchy Abstract: Effective categoricity can be examined via isomorphisms available in Ershov's difference hierarchy and in related hierarchies. There are various possibilities here for the notion of isomorphism and some interesting results involving even structures as simple as equivalence relations. (This is joint work with Doug Cenzer and jeff Remmel.) Speaker: Tamas Matrai Title : On F_{\sigma} p-ideals and Tukey reducibility} Abstract: The structure of all subsets of natural numbers with almost inclusion embeds into the family of F_{\sigma} p-ideals partially ordered by Tukey reducibility. I relate this result to previously known cofinal diversity results and Banach space constructions. Speaker: Jan Reimann, UC Berkeley Title: Randomness and Definability Abstract: The duality between measures and the sets they "charge" is a central theme in modern analysis. An effective analogue of this question is: Given a real X, does there exist a (probability) measure relative to which X is effectively random in the sense of Martin-Loef (so that X is not an atom of the measure)? It turns out the view from logic opens up a new perspective stemming from the interplay between randomness and definability strength. I will try to present various aspects of this relation. These will include determinacy, the use of higher infinities, rank functions for coanalytic sets, Hausdorff measures and potential theory. Speaker: Jeff Remmel, UCSD Title: $\Sigma^0_1$ and $\Pi^0_1$ structures We study computability theoretic properties of $\Sigma _{1}^{0}$ and $\Pi_{1}^{0}$ equivalence structures and how they differ from computable equivalence structures or equivalence structures that belong to the Ershov difference hierarchy. Our investigation includes the complexity of isomorphisms between $\Sigma _{1}^{0}$ equivalence structures and between $\Pi _{1}^{0}$ equivalence structures. Speaker: Mariya Soskova, Sofia, Bulgaria Title: Characterizing the strength of the local theory of the enumeration degrees Abstract: In this joint project with Hristo Ganchev we prove that the the first order theory of Peano arithmetic can be interpreted in the local theory of the enumeration degrees. We use the coding of standard models of arithmetic method introduced by Nies, Shore and Slaman. Our main coding tool is the notion of a K-pair, introduced and used by Kalimullin to show that the enumeration jump is definable. Speaker: Simon Thomas Title: Borel Determinacy and the Word Problem for Finitely Generated Groups Abstract: It is well-known that for each $A \in 2^{\mathbb{N}}$, there exists a finitely generated group $G_{A}$ such that the word problem for $G_{A}$ is Turing equivalent to $A$. In this talk, we show that there does not exist a Borel choice of $G_{A}$ such that if $A \equiv_{T} B$, then $G_{A} \cong G_{B}$. Speaker: Ferit Toska, UF Title: Computability of Countable Subshifts Abstract: The computability of countable subshifts and their members is examined. Results include the following. Subshifts of Cantor-Bendixson rank one contain only eventually periodic elements. Any rank one subshift, in which every limit point is periodic, is decidable. Subshifts of rank two may contain members of arbitrary Turing degree. In contrast, effectively closed (\pz) subshifts of rank two contain only computable elements, but \pz subshifts of rank three may contain members of arbitrary c.~e. degree. There is no subshift of rank $\omega$.Speaker: Guohua Wu Title: Separating cuppable degrees from low_n-cuppables Abstract: In 2000, Li, Wu and Zhang gave a hierarchy of cuppable degrees: a c.e. degree a is low_n-cuppable (n>0), if there is a low_n c.e. degree cupping a to 0', and proved that there is a degree low_2 cuppable, but not low-cuppable. In this talk, I will present a recent result (joint work with Greenberg and Ng) about the existence of a cuppable degree, but not low_n cuppable for any n. Speaker: Beatriz Zamora-Aviles Title: Structure of order ideals Abstract: Let BH1 be the set of bounded positive operators of norm at most one on an infinite dimensional Hilbert space. We give a definition of analytic P-ideals on this space and give a characterization of them. This characterization is a non-commutative version of the well-known characterization of analytic P-ideals on omega due to S. Solecki. We also obtain a characterization of certain meager subsets of this space. This is a non-commutative analog to Jalali-Naini and Talagrand characterization of meager subsets of the cantor space.