Title: Pseudocompact Paratopological and Quasitopological Groups

by Mikhail Tkachenko (Metropolitan Autonomous University) as part of Topological Groups

Lecture held in Elysium.

Abstract

Pseudocompactness is an interesting topological property which acquires very specific

features when applied to different algebrotopological objects. A celebrated theorem

of Comfort and Ross published in 1966 states that the Cartesian product of an arbitrary

family of pseudocompact topological groups is pseudocompact. We present a survey

of results related to the validity or failure of the Comfort-Ross’ theorem in the realm of

semitopological and paratopological groups and give some examples showing that

pseudocompactness fails to be stable when taking products of quasitopological groups.

Title: Luzin’s (N) and randomness reflection

by Linda Brown Westrick (Penn State) as part of Computability theory and applications

Abstract

We show that a computable real-valued function f has Luzin’s property (N) if and only if it reflects Pi^1_1-randomness, if and only if it reflects Delta^1_1-randomness relative to Kleene’s O, and if and only if it reflects Kurtz randomness relative to Kleene’s O. Here a function f is said to reflect a randomness notion R if whenever f(x) is R-random, then x is R-random as well. If additionally f is known to have bounded variation, then we show f has Luzin’s (N) if and only if it reflects weak-2-randomness, and if and only if it reflects Kurtz randomness relative to 0′. This links classical real analysis with algorithmic randomness. Joint with Arno Pauly and Liang Yu.

Title: Automorphism argument and reverse mathematics

by Keita Yokoyama (Japan Advanced Institute of Science and Technology) as part of Computability theory and applications

Abstract

In the study of models of Peano (or first-order) arithmetic, there are

many results on recursively saturated models and their automorphisms.

Here, we apply such an argument to models of second-order arithmetic

and see that any countable recursively saturated model (M,S) of WKL_0*

is isomorphic to its countable coded omega-submodel if

Sigma_1-induction fails in (M,S). From this result, we see some

interesting but weird properties of WKL_0* with the absence of

Sigma_1-induction such as the collapse of analytic hierarchy. This

argument can also be applied to the reverse mathematical study of

Ramsey’s theorem for pairs (RT22), and we see some new relations

between the computability-theoretic characterizations of RT22 and the

famous open question on the first-order part of RT22+RCA_0.

This work is a part of a larger project joint with Marta Fiori

Carones, Leszek Kolodziejczyk, Katarzyna Kowalik and Tin Lok Wong.