Liang Yu Higher randomness theory and its applications
Higher randomness theory was initiated by Martin Lof and Sacks. Recently some significant progress was made. I shall give a survey on this area. Some applications of the theory will also be discussed.
Isaac Goldbring The Connes Embedding Problem, MIP*=RE, and the Completeness Theorem
The Connes Embedding Problem (CEP) is arguably one of the most famous open problems in operator algebras. Roughly, it asks if every tracial von Neumann algebra can be approximated by matrix algebras. Earlier this year, a group of computer scientists proved a landmark result in complexity theory called MIP*=RE, and, as a corollary, gave a negative solution to the CEP. However, the derivation of the negative solution of the CEP from MIP*=RE involves several very complicated detours through C*-algebra theory and quantum information theory. In this talk, I will present recent joint work with Bradd Hart where we show how some relatively simple model-theoretic arguments can yield a direct proof of the failure of the CEP from MIP*=RE while simultaneously yielding a stronger, Gödelian-style refutation of CEP. No prior background in any of these areas will be assumed.
Goldbring Chinese Logic 2020.pdf
Su Gao From Hrushovski’s Property to Vershik’s Conjecture
In 1992 Hrushovski proved that the class of finite graphs has the EPPA, which has sometimes been referred to as the Hrushovski property, namely that every finite graph can be extended to another finite graph so that all partial isomorphisms of the original graph can be extended to a full isomorphism of the extended graph. The Hrushovski property eventually guarantees some strong property of the automorphism group of the (countable) random graph. In 2008 Vershik made a conjecture that the Hall’s universal local finite group can be embedded into the automorphism group of the random graph as a dense subgroup. Vershik’s conjecture represents a much deeper understanding of the automorphism group of ultrahomogeneous structures such as the random graph.
In this talk I will talk about some recent results from joint work with Mahmood Etedadialiabadi, Francois La Ma\^itre, and Julien Melleray. We confirm Vershik’s conjecture not only for the random graph but also for a variety of structures known as ultraextensive structures. These include universal $K_n$-free graphs, universal metric spaces with a fixed countable distance set, and with a little stretch of the definition, the universal Urysohn metric space. Along the way we generalize and strengthen previous results by Bhattacharjee, Coulbois, Herwig, Lascar, McPherson, Pestov, Rosendal, and Solecki. Our study can be viewed as an attempt to understand the automorphism group of ultraextensive structures through the study of the isomorphism types of their dense locally finite subgroups. This study is far from complete. I will talk about two competing conjectures that are contradictory to each other.
Joel David Hamkins Set-theoretic and arithmetic potentialism: the state of current developments
Recent years have seen a flurry of mathematical activity in set-theoretic and arithmetic potentialism, in which we investigate a collection of models under various natural extension concepts. These potentialist systems enable a modal perspective—a statement is possible in a model, if it is true in some extension, and necessary, if it is true in all extensions. We consider the models of ZFC set theory, for example, with respect to submodel extensions, rank-extensions, forcing extensions and others, and these various extension concepts exhibit different modal validities. In this talk, I shall describe the state of current developments, including the most recent tools and results.
Dexue Zhang Mathematical proof and many-valued logic
A mathematical proof is a finite sequence of assertions of which each term is either an axiom or is deducted from the preceding ones. Independence of axioms is a fundamental problem in mathematics, with the Axiom of Parallels, the Axiom of Choice and the Continuum Hypothesis as most known examples. The dependence of a mathematical “theorem” on inference rules originated with Brouwer's intuitionism. Since the law of the excluded middle is no longer valid in intuitionistic logic, some inference rules, for instance, proof by contradiction, is not allowed in Brouwer's way of doing mathematics. While intuitionism focuses on what can be proved without the law of excluded middle, this talk is concerned with the question: Is there a “theorem” that cannot be proved when a specific inference rule, proof by contradiction for instance, is not allowed? We present two examples in order theory via a many-valued formulation of completely distributive lattices and continuous directed partially ordered sets.
Yijia Chen Understanding Some Graph Parameters by Infinite Model Theory
Graphs parameters, e.g., tree-width, clique-width, tree-depth, capture various structural information of graphs. They have found numerous applications in graph theory and computer science. There is also a long history of studying those parameters from a logic perspective. Among others, Courcelle's theorem states that every property definable in monadic second-order logic (e.g., 3-colorability, Hamiltonian path) can be decided in linear time on graphs of bounded tree-width. Recently, it has been discovered that logic, in particular infinite model theory, provides another novel perspective on some important graph parameters. I will use tree-depth as an example to illustrate the connection between：
the Los-Tarski theorem and the forbidden subgraph characterization of bounded tree-depth,
Craig's interpolation theorem and the collapse of monadic second-order logic to first-order logic on graphs of bounded tree-depth.
I will also discuss their extension to graphs of bounded shrub-depth, a graph parameter generalizing tree-depth to dense graphs, which yields a version of Courcelle's theorem with respect to fast parallel algorithms.
Zhikun She 基于可达集上下近似的混成系统安全性验证研究
信息-物理系统（CPS）是一种将计算与物理过程相结合的系统，是多学科交叉融合的产物。混成系统作为一类将离散事件系统和微分方程相结合的动力系统，被认为是研究信息-物理系统的一个非常有效的数学模型。由于混成系统安全性验证是一不可判定问题，本报告将主要围绕动力系统可达集的上下近似展开：首先，引入演化函数并借助李导数给出它的泰勒级数展开；接着，基于部分和公式，提出两种计算可达集上下近似的方法并给予实现；最后，与最近两种可达集上下近似方法的计算结果比较展示了我们方法的优越性。
Jizhan Hong Some model-theoretic aspects of certain valued Frobenius fields
In this talk, I will present the quantifier elimination result I obtained for valued Frobenius fields in a first-order language and illustrate one its applications to the study of definable sets over valued omega-free PAC fields.
Jinhe Ye The étale open topology and the stable fields conjecture
For any field $K$, we introduce natural topologies on $K$-points of varieties over $K$, which is defined to be the weakest topology such that étale morphisms are open. This topology turns out to be natural in a lot of settings. For example, when $K$ is algebraically closed, it is easy to see that we have the Zariski topology, and it picks up the valuation topology in many henselian valued fields. Moreover, many topological properties correspond to the algebraic properties of the field. As an application, we will show that large stable fields are separably closed. Joint work with Will Johnson, Chieu-Minh Tran, and Erik Walsberg.
Yong Liu A priority argument in 0^(4)
Priority method is quite a foundamental tool to use for people who are interested in Turing degrees, especially the degrees below the Halting Problem. Introduced by Lachlan and Harrington, the tree method (a form of priority method) allows a classification of the complexity of an argument in terms of the complexity of the computation of the final outcomes of the requirements in its construction. In this sense of classification, a 0^(n)-priority argument requires 0^(n) to locate the final outcome of each requirement for the sets. All constructions that we know so far requires 0^(2) to compute a true path. As a requirement can be assigned to finitely many nodes on this true path, the outcome of the last node is considered as the final outcome, and p^(1) is needed to locate it. Overall, it is a 0^(3)-priority argument.
In this talk, we will discuss some basics about the complexity of the priority argument and give an example that uses 0^(4)-priority argument where the true path is computed only in 0^(3). More precisely, we show that there is no strong minimal pair in r.e. degrees. I.e., given any incomparable r.e. sets A, B, there is a nonrecursive set X below A such that the join of B and X does not compute A.
This is a joint work with Cai Mingzhong, Liu Yiqun, Peng Cheng, and Yang Yue.
Guozhen Shen A Choice-Free Cardinal Equality
For a cardinal $\mathfrak{a}$, let $\mathrm{fin}(\mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $\mathfrak{a}$. It is proved without the aid of the axiom of choice that, for all infinite cardinals $\mathfrak{a}$ and all natural numbers $n$,
\[
2^{\mathrm{fin}(\mathfrak{a})^n}=2^{[\mathrm{fin}(\mathfrak{a})]^n}.
\]
On the other hand, it is proved consistent with $\mathsf{ZF}$ that there exists an infinite cardinal $\mathfrak{a}$ such that
\[
2^{\mathrm{fin}(\mathfrak{a})}<2^{\mathrm{fin}(\mathfrak{a})^2}<2^{\mathrm{fin}(\mathfrak{a})^3}<\dots<2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}.
\]
Mou Bo Plural Set Theory Based on The Idea of Limitation of Size
Oystein Linnebo once pointed out that there exist two plural set theory based on the idea of limitation of size. The first is Stephen Pollard’s plural set theory. It is comprised of John Mayberry’s idea of limitation of size and David Lewis’s mereology. Pollard used his plural set theory to reformulate George Boolos’ Frege-von Neumann set theory, he also recapitulated Tyler Burge’s aggregate theory. The second is John Burgess’s plural set theory. It is composed of Boolos’ logical aspects and Paul Bernays’ non-logical aspects. Concretely speaking, Boolos supplied the idea of limitation of size and plural logic, and Bernays offered the reflection axiom. Linnebo attacked Burgess’s work. He thought the work of Burgess is a regress with regard to Bernays’ work. Stewart Shapiro gived up on the plural set theory. On the basis of Burgess’ another work, he reconstructed the Bernays-Boolos set theory and proposed the Frege-Zermelo-Bernays-Boolos set theory. The advantages of the latter set theory consists in adding Frege’s extension operator and Zermelo’s categority theorem to the former set theory.
Kun Gao 数学直觉——从康德到当代认知科学
数学直觉是数学知识论的一个核心概念，也是任何一种令人满意的数学哲学不可回避的问题。但关于数学直觉的性质和功能，哲学家们持有十分不同的意见。其中，比较有影响的基本可以概括为三种，分别是康德-布劳威尔式数学直觉概念、弗雷格-哥德尔式数学直觉概念和希尔伯特式数学直觉概念。另一方面，当代认知科学围绕数学直觉和一般的人类数学认知能力展开了一些实验研究，这些研究对关于数学直觉的传统哲学思辨构成严峻挑战。它们引领我们走向一种更加自然化的数学直觉观，同时也为解决数学哲学中实在论与反实在论之间的持久争论提供了关键线索。
Jialiang He Square of Menger groups
This work is cooperated with Yinhe Peng and Liuzhen Wu.
I will present a few constructions for square of Menger group problem in metrizable sense and generally sense in this talk.
Under cov(M)=c, for any n\geq 1, there is a subgroup G of Z^N such that G^n is Menger but G^{n+1} is not Menger.
Under cov(M)=d=cf(d)$, for any n\geq 1, there is a subgroup G of R such that G^n is Menger but G^{n+1} is not Menger.
For any n\geq 1, there is a Menger subgroup G of R^{\omega_1} such that G^n is Menger but G^{n+1} is not Lindelof in ZFC.
According to the known result, product of Menger subspace of Z^N is Menger in Miller model, this shows Menger group square problem is independent with ZFC in the metrizable sense. But for nonmetrizable topological group, the answer is no.
Weiguang Peng On eigen-distributions for multi-branching Boolean trees
Game tree is an important model of computation in the area of theoretical computer science. Our work contributes to the series of studies on finding the equilibrium point of AND-OR game trees with respect to different kinds of distributions and classes of algorithms. In this talk, we will mainly discuss the optimal depth-first algorithms and equilibria of independent distributions on multi-branching trees.
Chieu-Minh Tran Small measure expansion in locally compact groups and the linear-nonlinear dividing line
I will present a number of results from recent joint works with Yifan Jing and Ruixiang Zhang and discuss connections to the linear-nonlinear dividing line in model theory. These include a classification of connected unimodular groups and compact subsets allowing minimal measure expansion; this answers a question by Kemperman in 1964 and confirms conjectures by Griesmer and Tao. Another result is a generalization of the Brunn-Minkowski inequality to all locally compact groups, resolving a problem studied by MacBeath in 1960 and answering questions by Hrushovski and Tao.