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Wednesday, May 15, 2024
12:00 PM - 1:00 PM
Online Event

Logic Seminar

Invariant uniformization and reducibility
Michael Wolman, Department of Mathematics, Caltech,

Please note that the time is PST

Given sets X,Y and P⊆X×Y with projX(P)=X, a uniformization of P is a function f:X→Y such that (x,f(x))∈P for all x∈X. If now E is an equivalence relation on X, we say that P is E-invariant if x1Ex2⟹Px1=Px2, where Px={y:(x,y)∈P} is the x-section of P. In this case, an E-invariant uniformization is a uniformization f such that x1Ex2⟹f(x1)=f(x2).

Consider now the situation where X,Y are Polish spaces and P is a Borel subset of X×Y. In this case, standard results in descriptive set theory provide conditions which imply the existence of Borel uniformizations. These fall mainly into two categories: "small section" and "large section" uniformization results.

Suppose now that E is a Borel equivalence relation on X, P is E-invariant, and P has "small" or "large" sections. In this talk, we address the following question: When does there exist a Borel E-invariant uniformization of P?

We show that for a fixed E, every such P admits a Borel E-invariant uniformization iff E is smooth. Moreover, we compute the minimal definable complexity of counterexamples when E is not smooth. Our counterexamples use category, measure, and Ramsey-theoretic methods.

We also consider "local" dichotomies for such pairs (E,P). We give two new proofs of a dichotomy of Miller in the case where P has countable sections, the first using Miller's (G0,H0) dichotomy and Lecomte's ℵ0-dimensional G0 dichotomy, and the second using a new ℵ0-dimensional analogue of the (G0,H0) dichotomy. We also prove anti-dichotomy results for the "large section" case. We discuss the "Kσ section" case, which is still open.

This is joint work with Alexander Kechris.

For more information, please contact Math Dept. by phone at 626-395-4335 or by email at [email protected].