Graphons arising from graphs definable over finite fields

Definable regularity lemmas for NIP hypergraphs

We present a systematic study of the regularity phenomena for NIP hypergraphs and connections to the theory of (locally) generically stable measures, providing a model-theoretic hypergraph version of the results from [L. Lovász, B. Szegedy, "Regularity partitions and the topology of graphons", An irregular mind, Springer Berlin Heidelberg, 2010, 415-446]. Besides, we revise the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the results from [A. Chernikov, S. Starchenko, "Regularity lemma for distal structures", J. Eur. Math. Soc. 20 (2018), 2437-…

双亚纯几何的可定义伽罗瓦理论

Definable Galois theory for bimeromorphic geometry

通过研究紧复空间理论CCM中的模型论可定义绑定群，发展双亚纯几何的伽罗瓦理论，并应用于主亚纯丛的结构定理，同时给出绑定群为代数群的例子及其线性判别。

群的可定义等价关系与Zeta函数

Definable equivalence relations and zeta functions of groups

我们证明了理论的$p$ adics ${\mathbb Q}_p$承认消除的前提是我们添加一个排序${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$为每个$n$。我们还证明了在$p$中消除的矩阵是一致的。利用$p$ -adic和motivic积分，我们推导出由可定义的等价关系产生的某些形式zeta函数的一致合理性。对于正特征的局部域上可定义的等价关系，这也得到类似的结果。附录中包含了一个替代的证明，使用胞腔剖分，合理性（固定$p$），这些正式zeta函数，延伸到子分析的背景。作为一个应用，我们证明了合理性和一致性的zeta函数通过计数扭曲同构类的不可约表示的n-生成的幂零群，这些类似的结果Grunewald，西格尔和史密斯和杜Sautoy和Grunewald的子群zeta函数的n-生成的幂零群。

o-minimal GAGA and a conjecture of Griffiths

We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period ma…

多维渐近类

Multidimensional asymptotic classes

本文提出多维渐近类框架以统一刻画有限一阶结构中可定义集基数的强一致性条件，发展了其基础理论、给出大量实例，并引入广义可测结构以揭示超积的模型论与稳定性性质。

$p$-adic model theory, $p$-adic integrals, Euler products, and zeta functions of groups

We give a survey of Denef's rationality theorem on $p$-adic integrals, its uniform in $p$ versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes in congruence quotients of Chevalley groups over rings of integers of local fields. We then state results on analytic properties of Euler products of such $p$-adic integrals over all $p$, and an application to counting conjugacy classes in congruence quotients of certain algebraic groups over the rationals. We then briefly discuss zeta functions arising from …