Invited speakers:
Jun Le Goh (UW-Madison)
Title: Two research directions in the enumeration degrees
Abstract: Enumeration reducibility is a notion of computing with positive information. It is more appropriate than Turing reducibility for measuring the complexity of objects such as continuous real-valued functions (Miller 2004). The equivalence classes under enumeration reducibility form a degree structure known as the e-degrees, partially ordered by enumeration reducibility. We shall discuss two lines of research concerning this structure. The first is the study of subclasses of the e-degrees and their first-order definability. The second concerns properties of the e-degrees as a partial order. The new results in this talk are joint work with Kalimullin, Miller, Soskova and (separately) Lempp, Ng, Soskova.
Xi He (Sichuan University)
Title: 理想极限点集的相关研究
Abstract: 理想极限点集，是将普通极限点集定义中的有限集换为理想后所得到的点集。一般来说，理想极限点集并不一定是闭集，并且其复杂度不易计算。我们将介绍理想极限点集的相关研究，着重于其是否是博雷尔集，以及低复杂度（闭集，F_\sigma）时的一些性质，并且指出这些研究与某些组合性质之间的联系。
Meiyan Liu (Sichuan University)
Title: The finer selection principle
Abstract: Tsaban introduced the following selection hypothesis.
\begin{itemize}
\item[$\bullet$] \textbf{$\bigcup_{k}(\Gamma, \Gamma)$}: For all $\mathcal{U}_{1}, \mathcal{U}_{2}, \cdots \in \Gamma$, none containing a finite subcover, there are finite $\mathcal{F}_{1} \subseteq \mathcal{U}_{1}, \mathcal{F}_{2} \subseteq \mathcal{U}_{2}, \cdots$ such that $|\mathcal{F}_{n}| \leq k$ for all $n$, and $\{\bigcup \mathcal{F}_{n}: n \in \omega\} \in \Gamma$.
\end{itemize}
We prove the following results.
\begin{itemize}
\item[(1)] Assume the Continuum Hypothesis, for each positive constant $k \in \omega$, there is a $\mathfrak{b}$-\emph{k}-scale \emph{X} such that $X \cup \mathrm{Fin}(\omega)$ satisfies $\bigcup_{k+1}(\Gamma, \Gamma)$ but not $\bigcup_{k}(\Gamma, \Gamma)$.
\end{itemize}
The above result means that for two distinct positive constants \emph{c}, \emph{d}, $\bigcup_{c}(\Gamma, \Gamma)$ and $\bigcup_{d}(\Gamma, \Gamma)$ can be separated.
\begin{itemize}
\item[(2)] Assume $\mathfrak{b}=\mathfrak{c}$, there is a set $X \subseteq 2^{\omega \times \omega}$ such that $X \cup \mathrm{Fin}(\omega \times \omega)$ satisfies $\bigcup_{\mathrm{id}}(\Gamma, \Gamma)$ but not $\bigcup_{k}(\Gamma, \Gamma)$ for all positive constants $k \in \omega$.
\item[(3)] Assume the Continuum Hypothesis, there is a set \emph{X} without perfect subsets, such that \emph{X} satisfies $\bigcup_{\mathrm{fin}}(\Gamma, \Gamma)$ but not $\bigcup_{\mathrm{id}}(\Gamma, \Gamma)$.
\end{itemize}
The above two results answer Tasban's conjecture and Zdomskyy's problem.
Guangyin Ma (Beijing Jiaotong University)
Title: 可数斯坦纳三元系同构关系的完备性
Abstract: 在数学的许多分支中，对所研究对象进行分类非常重要。基于分类问题本质上是等价关系的事实，自20世纪80年代起，描述集合论工作者发展出一套关于等价关系复杂程度的数学理论——不变量描述集合论。波莱尔完备作为等价关系复杂程度的众多基准之一，等同于说其所对应的分类问题恰好可以用某类可数结构作为完全不变量进行分类。我们将从图是忠实波莱尔完备的这一经典结果出发，先后构造图到3一致超图的忠实归约以及3一致超图到部分斯坦纳三元系的忠实归约。最后，通过改进以上3一致超图到部分斯坦纳三元系的归约使之成为到斯坦纳三元系的忠实归约，从而证明斯坦纳三元系是忠实波莱尔完备的。
David Schrittesser (University of Toronto)
Title: Mad families, regularity, and determinacy
Abstract: In 2019, together with Asger Törnquist, we answered a longstanding question asked by Mathias and showed that Universal Ramsey Regularity implies that there are no mad families. Our proof also applies to generalized mad families, where the finite ideal is replaced by iterated Fubini products of the Frechet ideal.This theorem has a counterpart in our older theorem, that (generalized) mad families cannot exist in point classes where determinacy holds, which in turn generalizes Mathias proof that no analytic mad families exist. Both questions are related to interesting open questions in set theory about the connections between the Ramsey property, Determinacy, and the existence of mad families.
In this talk I will review some of these questions as well as sketch our proof(s).
Minchao Wu (Australian National University)
Title: Machine learning for theorem proving
Abstract: Automated / interactive theorem proving has been a great tool to establish correctness of programs and mathematics. Despite the well-known undecidability of any fairly expressive systems, it is possible to leverage machine learning to help find proofs in a humanlike way. In this talk, I will be focusing on interactive theorem proving (ITP) and its automation. I will be describing one of my recent projects, TacticZero, which is a framework that learns ITP in an end-to-end manner without using human proofs. I'll also talk about the recent discoveries, challenges, and future work in this field.
BoKai Yao (Notre Dame University)
Title: Reflection Principles in ZF(C) with Urelements
Abstract: I will first identify a hierarchy of natural principles, including reflection principles, in ZFC set theory with urelements. As a result, the most natural way of axiomatizing ZFC with urelements turns out to be too weak. I will then turn to reflection principles in the context of ZF with urelements. Many questions remain open. The second part of my presentation is joint work with Joel David Hamkins.
Yang Zheng (Nankai University)
Title: On equivalence relations induced by Polish groups
Abstract: In this talk, We introduce a kind of orbit equivalence relations which can well describe the structure and properties of Polish groups. Given a Polish group G, let E(G) be the right coset equivalence relation G^{\omega}/cG, cG where is the group of all convergent sequences in G. We investigate the complexity of E(G) for various Polish groups G. This is joint work with Longyun Ding.
