Archives of: Boise Set Theory Seminar

John Clemens: Relative primeness of equivalence relations

Wednesday, November 11 from 3 to 4pm
Room: MP 207
Speaker: John Clemens (BSU)

Title: Relative primeness of equivalence relations

Abstract: Let $E$ and $F$ be equivalence relations on the spaces $X$ and $Y$. We say that $E$ is prime to $F$ if: whenever $\varphi: X \rightarrow Y$ is a homomorphism from $E$ to $F$, there is a continuous embedding $\rho$ from $E$ to itself so that the range of $\varphi \circ \rho$ is contained in a single $F$ class. That is to say, $\varphi$ is constant (up to $F$-equivalence) on a set on which $E$ maintains its full complexity with respect to Borel reducibility. When $E$ is prime to $F$, $E$ fails to be Borel-reducible to $F$ in a very strong way. I will discuss this notion and show that many non-reducibility results in the theory of Borel equivalence relations can be strengthened to produce primeness results. I will also discuss the possibility of new types of dichotomies involving the notion of primeness.

Marion Scheepers: Infinite games with limited memory

Wednesday, October 14 from 3 to 4pm
Room: MP 207
Speaker: Marion Scheepers (BSU)
Title: Playing an infinitely long game when you have limited memory

Abstract: In some infinite games one of the two players has a clear advantage, provided that the player has perfect memory. In this talk we briefly survey some results when the winning perfect information player has a limited memory.

Liljana Babinkostova: The selective strong screenability game

Wednesday, September 30 from 3 to 4pm
Room: MP 207
Speaker: Liljana Babinkostova (BSU)
Title: The selective strong screenability game

Abstract: Selective versions of screenability and of strong screenability coincide in a large class of spaces. We show that the corresponding games are not equivalent in even such standard metric spaces as the closed unit interval. We identify sufficient conditions for ONE to have a winning strategy, and necessary conditions for TWO to have a winning strategy in the selective strong screenability game.

John Clemens: Isomorphism of homogeneous structures

Wednesday, September 16 from 3 to 4pm
Room: MP 207
Speaker: John Clemens (BSU)
Title: Isomorphism of homogeneous structures

Abstract: The theory of Borel reducibility of equivalence relations can be used to gauge the complexity of the isomorphism problem for a collection of countable structures. Certain classes, such as that of graphs and trees, are known to have an isomorphism problem of maximal complexity. We may also consider only the homogeneous structures, those whose automorphism group acts transitively on the structure. I will discuss the question of when the isomorphism problem for a collection of homogeneous structures is as complicated as that for all such structures. This may be viewed as asking when information may be coded into a structure without using “local” coding. In particular, we can show that this is true for graphs, but not for trees.

Samuel Coskey: Amalgamation properties and conjugacy

Wednesday, September 2 from 3 to 4pm
Room: MP 207
Speaker: Samuel Coskey (BSU)

Title: Amalgamation properties and conjugacy

Abstract: In this talk we aim to classify automorphisms of ultrahomogeneous structures up to conjugacy. (Ultrahomogeneous structures are highly self-symmetric and can be viewed as a kind of generic limit of their finite pieces.) In past talks I gave several examples of homogeneous structures and settled the classification problem for their automorphisms. After reviewing the classical combinatorial framework, we will generalize the methods that worked for specific examples to broader classes of ultrahomogeneous structures.

Stuart Nygard: The density topology

Wednesday, April 29 from 3 to 4pm
Room: Math 124
Speaker: Stuart Nygard (BSU)
Title: The density topology

Abstract: In the Euclidean topology, open sets are defined by unions of open intervals. Can we remove some “small” set from the intervals and still have a meaningful topology? Yes. We define a topology using sets that have locally full measure. That is, a set will have an open neighborhood around a point if “almost every” point nearby belongs to the set. We will show how the topology naturally arises on R and other spaces. No knowledge of Lebesgue measure or topology is assumed.

Samuel Coskey: López-Escobar’s theorem for metric structures

Wednesday, April 8 from 3 to 4pm
Room: Math 124
Speaker: Samuel Coskey (BSU)
Title: López-Escobar’s theorem for metric structures

Abstract: The classical López-Escobar theorem states that any Borel class of countable structures may be axiomatized using an appropriate infinitary logic. One application of this theorem is to relate topological and model-theoretic versions of Vaught’s conjecture. In this talk we present a variant of López-Escobar’s theorem for metric structures, which implies that Borel classes of separable metric structures may be axiomatized in the appropriate infinitary continuous logic. As a consequence we obtain a new implication between the topological Vaught conjecture and a version for metric structures. This is joint work with Martino Lupini.

Samuel Coskey: Introduction to continuous logic

Wednesday, March 4 from 3 to 4pm
Room: Math 124
Speaker: Samuel Coskey (BSU)
Title: An introduction to continuous logic

Abstract: Continous logic is a proper generalization of first order logic where the usual binary truth values are replaced by the unit interval $[0,1]$. The models for this logic are metric structures, which are metric spaces together with continuous functions and $[0,1]$-valued relations. Just as ordinary logic has typical applications in discrete math, continuous logic has applications in analysis. In this talk we will introduce just the basic concepts and theory of continuous logic.

Randall Holmes: Preliminaries for Proving the Consistency of NF

Wednesday, February 11 from 3 to 4pm
Room: Math 124
Speaker: Randall Holmes (BSU)
Title: Preliminaries for Proving the Consistency of NF

Abstract: We will discuss some preliminary machinery intended for use in a consistency proof of Quine’s set theory New Foundations.
No particular familiarity with New Foundations is presupposed.

Samuel Coskey: Conjugacy and homogeneous graphs, II

Wednesday, February 4 from 3 to 4pm
Room: Math 124
Speaker: Samuel Coskey (BSU)
Title: Conjugacy and homogeneous graphs, II

Abstract: This talk is continued from last semester. We study the conjugacy relation on the automorphism group of a homogeneous structure from the point of view of Borel complexity theory. In particular, we present three examples of homogeneous graphs whose corresponding conjugacy problems have three different complexities.