A lot of early work in descriptive set theory under the axiom of determinacy focused on generalizing the deep theory of the $\Sigma^1_1$ and $\Pi^1_1$ sets to higher levels of the projective hierarchy. Under determinacy hypotheses, the projective hierarchy exhibits periodic behavior and many facts about $\Sigma^1_1$ and $\Pi^1_1$ sets repeat at the higher odd levels of the hierarchy. We will review some of this periodic behavior and mention some new results about the next odd level, i.e. for $\Sigma^1_3$ and $\Pi^1_3$ sets. This is joint work with William Chan, Sandra Müller, and Farmer Schlutzenberg.

We will discuss the generalization of several properties of ultrafilter, such as being a P-point, a Q-point or a Selective ultrafilter, to finite additive measures. We will also examine the Rudin-Blass and the Rudin-Keisler orders and their generalizations to measures, highlighting the similarities and differences between the world of ultrafilters and the world of measures.

A cofinitary group is a subgroup of the permutations of $\omega$ such that every two elements are eventually different, i.e. their graphs have finite intersection. Maximal such cofinitary groups cannot be countable and using choice one may easily construct cofinitary groups which are maximal. Thus, the minimal size of a maximal cofinitary group defines a cardinal invariant $\mathfrak{a}_g$. In this talk I will discuss the history of definable/projective maximal cofinitary groups. In particular, I will present the construction of a co-analytic maximal cofinitary group, which is indestructible by various forcings. Thus, we obtain that (starting from L) many standard models for constellations of cardinal invariants in fact also have a witness for $\mathfrak{a}_g$ of optimal projective complexity. This is joint work with Vera Fischer and David Schrittesser

A Hagendorf order is a linear order X that this is strictly indecomposable to the right and such that any strict suborder of X (in the sense of bi-embeddability) embeds into an initial segment of X. The easiest examples of such orders are indecomposable ordinals. In the 70s, J. Hagendorf asked whether these are the only such orders and J. Larson could show (in ZFC) that this is at least the case among scattered orders. On the other hand, it can be shown that under Baumgartner's Axiom non-ordinal Hagendorf types exist. We further study the existence of such orders under other set theoretical assumptions, with a particular focus on real order types. The main open problem is whether non-ordinal Hagendorf orders can be constructed just from ZFC.

It is a well-known result that, after adding one Cohen real, the transcendence degree of the reals over the ground-model reals is continuum. We extend this result for a set X of finitely many mutually generic Cohen reals, by showing that, in the forcing extension, the transcendence degree of the reals over a combination of the reals in the extension given by each proper subset of X is also maximal. This is joint work with Ralf Schindler.

A space $X \subseteq 2^\omega$ is \emph{universally meager} if for any Polish space $Y$ and any continuous nowhere constant map $f:Y \rightarrow 2^\omega$ the preimage $f^{-1}[X]$ is meager in $Y$. We call a space \emph{totally imperfect} if it contains no copy of $2^\omega$. We present a forcing property $(\dagger)$, which is a strenthening of properness and implies that no dominating reals are added. It is known that many classical forcing posets like Cohen and Miller satisfy $(\dagger)$. We showed that property $(\dagger)$ is preserved by countable support iterations. We then used this preservation result to prove that if we have such an iteration of length $\omega_2$ over a model of CH, where the single forcings have size at most $\omega_1$, all universally meager sets $X \subseteq 2^\omega$ have size at most $\omega_1$ in the forcing extension. This has multiple applications: In the Miller model, we generalized our result of having no concentrated and $\gamma$-sets of size continuum to totally imperfect Hurewicz sets, which are universally meager by a result of Zakrzewski. Moreover, since Bartoszy\'{n}ski showed that all perfectly meager spaces are universally meager in the Miller model, we get that even all perfectly meager spaces have size stricly less than continuum in the Miller model. Miller proved that there exists a strong measure zero set of size $\omega_1$ iff there exists a Rothberger space of size $\omega_1$. Goldstern, Judah and Shelah constructed a forcing iteration for which there is a strong measure zero set of size $\omega_2$ in the extension. However, this iteration satisfies property $(\dagger)$ and Rothberger spaces are universally meager in this model. Hence our result implies that it is consistent with ZFC to have a strong measure zero set of size $\omega_2$, but no Rothberger space of size $\omega_2$. This is joint work with Piotr Szewczak and Lyubomyr Zdomskyy.