北京雁栖湖应用数学研究院

In the study of reverse mathematics, numerous conservation theorems are established using low basis theorems and their variations. Specifically, the proof-theoretic/first-order strength of Ramsey's theorem for pairs and its variations are calibrated in this way. In this talk, we introduce a method for converting model-theoretic $\Pi^1_1$-conservation theorems by means of low-like basis theorems into proof interpretations. We will then overview the study of the first-order strength of Ramsey's theorem for pairs and reproving several conservation theorems together with polynomial-size proof transformations.

The CFI-graphs, named after Cai, Fuerer, and Immerman, are central to the study of the expressive power of first-order logic and fixed-point logic with counting. They are often colored graphs, and the coloring plays a key role in many proofs. As usual, it is not hard to remove the coloring by some extra graph gadgets, but at the cost of blowing up the size of the graphs and changing some key parameters of the graphs as well. This might lead to suboptimal bounds for their applications. In this talk, I will give a detailed account of the CFI-graphs, both colored and uncolored, and show they serve the same purposes for most applications. This is joint work with Joerg Flum and Mingjun Liu.

In this talk we review a classic structuring of hyperfinite Borel equivalence relations, namely ``Borel $\mathbb{Z}$-orders", define the extendibility of $\mathbb{Z}$-orders over $E_0$ to larger equivalence relations, relate it to a special case of the Union Problem, and we explore a forcing notion which forces new Borel $\mathbb{Z}$-orders as its generic objects.

I will give a survey of recent developments in Solovay reducibility for computably approximable reals, focusing on connections to analytical concepts and their structural properties.

We study of the complexity of several notions in ring theory by methods of computable structure theory. First, we discuss the problem of being typical subclasses of semiperfect rings for computable rings. For instance, we obtain that the index set of computable local rings is $\Pi^{0}_{2}$-complete within the index set of computable rings. We define a computably enumerable (c.e. for short) ring as the quotient ring of a computable ring modulo a c.e. congruence relation and view such rings as structures in the language of rings together with a binary relation. We further discuss the problem of being typical subclasses of semiperfect rings for computably enumerable rings and prove that the index set of c.e. semisimple rings is $\Sigma^{0}_{3}$-complete. As applications of the complexity results on local rings and semisimple rings, we also obtain the optimal complexity results on other closely connected classes of rings, such as the small class of finite direct products of fields and the more general class of semiperfect rings.

Many problems of Borel combinatorics can be formulated as the existence of Borel equivariant maps from the Bernoulli shifts to some subshifts of finite type. We consider the case in which the acting group is a finitely generated free abelian group. We show that in the one-dimensional case the problem is computable, and in the two-dimensional case, the continuous problem is c.e.-complete.

A recent discovery by the speaker is that any subtopos of the effective topos is presented by a bilayer of an oracle and a majority notion. As tools for controlling an oracle, there is Weihrauch reducibility, and as tools for controlling a majority notion, there are the Rudin-Keisler order and the Katetov order. Based on the former, the speakers have obtained several applications to constructive reverse mathematics, such as constructing a topos that separates several semiconstructive principles (we have used a priority argument to construct some of such toposes!). Regarding the latter, the speakers' recent developments show that the geometric inclusion order on subtoposes (Lawvere-Tierney topologies) obtained from subobjects of double negation sheaves in the effective topos corresponds to a game-theoretic variant (iterated Fubini product) of the Katetov order on lower sets. In this talk, I will provide an overview of these new discoveries regarding subtoposes of effective topos made by the speaker and collaborators over the past few years.

Abstract: The class of $[0,1]$-valued random variables on atomless probability spaces is an elementary class in continuous logic. The theory of this class is denoted by ARV. ARV is complete, separably categorical, omega-stable, and admits quantifier elimination. During this talk, we will characterize saturated models of ARV, and give explicit formulas between types. Finally, we will discuss type spaces of ARV and Wasserstein spaces.

This study investigates the asymptotic behavior of f-vector components in random Vietoris-Rips complexes constructed from a nonhomogeneous Poisson process on $\mathbb{R}^d$. To systematically characterize these asymptotic behaviors, we analyze these asymptotic properties across three regimes: the sparse, critical, and dense phases of point cloud density. By parameterizing the distance threshold in terms of time $t$, we reframe the f-vector components as a stochastic process denoted by $F_k(t)$. This talk proved the normalization, asymptotic limits, and expected values of $F_k(t)$, culminating in the establishment of the functional strong law of large numbers for these processes.

In this talk, I will first review two fundamental connections linking abelian lattice-ordered groups, Bezout/GCD domains, and MV-algebras: the Jaffard-Ohm one-to-many correspondence and Mundici's functorial equivalence. Then, I will demonstrate how these bridges highlight the significant cross-fertilization and mutual development among these three research areas.

This talk is inspired by Kazuyuki Tanaka’s book, Turing and Meta-Puzzles. A meta-puzzle, in his terminology, is not a single puzzle treated in isolation but rather a collection of puzzles considered together as if forming a unified whole. As an example, we will present a puzzle we devised in a finite setting and then discuss its extension to an infinite version. The aim is to illustrate how the shift from finite to infinite can raise new conceptual and logical issues.

In this talk, I talk about weak first order theories of concatenation and its essential undecidability. I also discuss the interpretability between these theories and weak first order arithmetic.