I will present an abstract approach to finite Ramsey theory, which reveals the formal algebraic structure underlying results of that theory and which yields most of the results of the theory. I will illustrate how the general approach is applied by new concrete examples of Ramsey statements - the self-dual Ramsey statement, a Ramsey statement for finite trees, and a Ramsey statement for \(1\)-Lipschitz functions.
I will present an abstract approach to finite Ramsey theory, which reveals the formal algebraic structure underlying results of that theory and which yields most of the results of the theory. I will illustrate how the general approach is applied by new concrete examples of Ramsey statements - the self-dual Ramsey statement, a Ramsey statement for finite trees, and a Ramsey statement for \(1\)-Lipschitz functions.
I will provide a technology to extend Hjorth's ergodicity results for turbulent equivalence relations to a measure setting.
I will sketch the main ideas in the classification of the invariant random subgroups of the group of finitary permutations of the natural numbers.
I will describe the universal minimal flow of any topological group \(G\) as a space of filters on \(G\). I will show how notions from topological dynamics naturally translate into this language and give some applications to dynamics of groups of automorphisms of structures and the isometry group of the Urysohn space.
Algorithmic randomness provides a way to define a random outcome. Most research in algorithmic randomness has been conducted using computable measures. However, there are some interesting results that have come from considering noncomputable measures. In a surprising result, Levin established the existence of probability measures for which all infinite binary sequences are random. These measures are termed neutral measures. In this talk, I will introduce Levin's neutral measures and outline some their properties.
Given a Polish group \(G\) and a class \(\mathcal C\) of subgroups of \(G\), define a "universal \(\mathcal C\) subgroup of \(G\)" to be a subgroup \(K\) in \(\mathcal C\) such that every subgroup \(H\) in \(\mathcal C\) is a continuous homomorphic preimage of \(K\). I will prove that, for any Polish group \(G\), the countable power of \(G\) has a universal analytic subgroup. I will put this result in context and mention other similar results for the classes of compactly generated and \(K_\sigma\) subgroups.
We use determinacy to prove some results on Borel matchings and colorings of \(n\)-regular Borel graphs. We apply these results to prove that many-one equivalence on \(2^\omega\) can not be a uniformly universal countable Borel equivalence relation.
We use determinacy to prove some results on Borel matchings and colorings of \(n\)-regular Borel graphs. We apply these results to prove that many-one equivalence on \(2^\omega\) can not be a uniformly universal countable Borel equivalence relation.
We use determinacy to prove some results on Borel matchings and colorings of \(n\)-regular Borel graphs. We apply these results to prove that many-one equivalence on \(2^\omega\) can not be a uniformly universal countable Borel equivalence relation.
We use determinacy to prove some results on Borel matchings and colorings of \(n\)-regular Borel graphs. We apply these results to prove that many-one equivalence on \(2^\omega\) can not be a uniformly universal countable Borel equivalence relation.