Will Johnson
Let $K$ be a type-definable infinite field in an NIP theory. If $K$ has characteristic $p > 0$, then $K$ is Artin-Schreier closed (it has no Artin-Schreier extensions). As a consequence, $p$ does not divide the degree of any finite separable extension of $K$. This generalizes a theorem of Kaplan, Scanlon, and Wagner.
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Artin-Schreier extensions in NIP and simple fields
我们证明了NIP域没有Artin-Schreier扩张，而简单域只有有限个。
Artin-Schreier extensions and combinatorial complexity in henselian valued fields
本文通过显式公式控制混合特征henselian值域的p-扩张，将NIP分类推广到NIPn语境，并证明NIPn henselian值域在NIP剩余域下是NIP的，同时讨论NTP2的驯良性。
For G a group definable in an NIP theory we prove that there is a smallest type-definable subgroup H of G such that the quotient G/H is stable. This generalizes the existence of G^00, the smallest type-definable subgroup of G of bounded index.
We prove a definable version of Matoušek's $(p,q)$-theorem in NIP theories. This answers a question of Chernikov and Simon. We also prove a uniform version. The proof builds on a proof of Boxall and Kestner who proved this theorem in the distal case, utilizing the notion of locally compressible types which appeared in the work of the author with Bays and Simon.
We investigate what henselian valuations on ordered fields are definable in the language of ordered rings. This leads towards a systematic study of the class of ordered fields which are dense in their real closure. Some results have connections to recent conjectures on definability of henselian valuations in strongly NIP fields. Moreover, we obtain a complete characterisation of strongly NIP almost real closed fields.
On large externally definable sets in NIP
本文研究了$ω_1$有限子集的共尾系统，证明其不能在NIP结构中定义，并肯定回答了Chernikov和Simon 2013年的问题：NIP理论中任何不可数外部可定义集均含无限可定义子集。
Let T be an NIP L-theory and T' be an enrichment. We give a sufficient condition on T' for the underlying L-type of any definable (respectively invariant) type over a model of T' to be definable (respectively invariant) as an L-type. Besides, we generalise work of Simon and Starchenko on the density of definable types among non forking types to this relative setting. These results are then applied to Scanlon's model completion of valued differential fields.
Let $\mathcal{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R},<,+)$ by closed subsets of $\mathbb{R}^n$ and continuous functions $f : \mathbb{R}^m \to \mathbb{R}^n$. Then $\mathcal{R}$ is generically locally o-minimal. It follows that if $X \subseteq \mathbb{R}^n$ is definable in $\mathcal{R}$ then the $C^k$-points of $X$ are dense in $X$ for any $k \geq 0$. This follows from a more general theorem on $\mathrm{NIP}$ expansions of locally compact groups, which itself follows from a result on quotients of definable sets by equivalence relations which are externally definable and $\bigwedge$-d…
Externally definable sets and dependent pairs II
我们继续研究NIP理论中的外部可定义集合的结构以及通过新谓词扩展后的NIP的保持。最重要的是：有限集上的类型是一致可定义的;在模型上，公式的非分叉实例族（参数范围在一个类型可定义的集合上）可以被许多不变类型所覆盖;我们给出了在远端理论中由一个新谓词展开的有界性的一些准则;命名一个任意小的不可辨别序列保留NIP，而命名一个大的序列则不保留;有一些NIP理论模型，在这些模型上，所有的1型都是可定义的，但不是所有的n型都是可定义的。
We study the definable topological dynamics $(G(M), S_G(M))$ of a definable group acting on its type space, where $M$ is either an $o$-minimal structure or a $p$-adically closed field, and $G$ a definable amenable group. We focus on the problem raised by Neweslki of whether weakly generic types coincide with almost periodic types, showing that the answer is positive when $G$ has boundedly many global weakly generic types. We also give two "minimal counterexamples" where $G$ has unboundedly many global weakly generic types, extending the main results of "On minimal flows, definably amenable gro…
We study definable topological dynamics of some algebraic group actions over an arbitrary NIP field $K$. We show that the Ellis group of the universal definable flow of $\mathrm{SL}_2(K)$ is non-trivial if the multiplicative group of $K$ is not type-definably connected, providing a way to find multiple counterexamples to the Ellis group conjecture, particularly in the case of dp-minimal fields. We also study some structure theory of algebraic groups over $K$ with definable f-generics.
We study the algebraic implications of the non-independence property (NIP) and variants thereof (dp-minimality) on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a definable henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson's preprint "dp-minimal fields", arXiv: 1507.02745v1, July 2015.
Groups, measures, and the NIP
我们讨论的措施，不变测度可定义的群体，和一般性，往往在一个NIP（故障的独立性）的环境。我们完成了证明第三作者的结构相关的可定义紧群$G$在饱和的$o$ -最小结构紧李群。我们还证明了关于这类$G$的其他一些结构性结果，例如在$G$的可定义子集上存在一个左不变的可加概率测度.最后，我们引入了紧控制的概念（紧空间对可定义集的控制），并在$o$ -极小情形下给出了一些新的结果。
Translating between NIP integral domains and topological fields
该研究证明NIP域上的可定义环拓扑与NIP整环紧密关联，初等等价下NIP拓扑域源自NIP整环，据此得出NIP域可定义环拓扑的局部有界性等性质，并为NIP局部环henselian猜想提供等价表述。
On f-generic types in NIP groups
该研究刻画了NIP理论中可定义群的可定义可和性，证明其等价于存在全局f-泛型类型，还推广了相关定理并应用于dp-极小群，同时给出类型可定义群对应结果的正确证明。
Externally definable fsg groups in NIP theories
本文证明NIP结构中外部可定义的fsg群与可解释群可定义同构，利用诚实定义与群块重构方法，并推广了外部可定义集合与群的相关结果，包括证明Eleftheriou猜想。
Let $K$ be an NIP field and let $v$ be a henselian valuation on $K$. We ask whether $(K,v)$ is NIP as a valued field. By a result of Shelah, we know that if $v$ is externally definable, then $(K,v)$ is NIP. Using the definability of the canonical $p$-henselian valuation, we show that whenever the residue field of $v$ is not separably closed, then $v$ is externally definable. In the case of separably closed residue field, we show that $(K,v)$ is NIP as a pure valued field.
Maximal stable quotients of invariant types in NIP theories
本文证明了NIP理论中不变类型的实现集上存在最精细的相对类型可定义等价关系且具有稳定商，该结论是已有关于类型可定义群极大稳定商结果的对应物。
Trace definability IV: higher arity notions
本文引入了k-迹可定义性概念，证明了k-迹可定义于NIP理论的理论是k-NIP，且2-迹可定义于稳定理论的理论是2-NFOP。所有已知的k-NIP理论都可以k-迹可定义于NIP理论中，并构造了通用理论D_k(T)。
Combining a characterisation by Bélair, Kaplan, Scanlon and Wagner of certain $\rm NIP$ valued fields of characteristic $p$ with Dickson's construction of cyclic algebras, we provide examples of noncommutative $\rm NIP$ division ring of characteristic $p$ and show that an $\rm NIP$ division ring of characteristic $p$ has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite $\rm NIP$ fields have no Artin-Schreier extension. The result extends to ${\rm NTP}_2$ division rings of characteristic $p$, using results of Chernikov, Kaplan and Simon. We also highli…
We initiate a systematic study of the convolution operation on Keisler measures, generalizing the work of Newelski in the case of types. Adapting results of Glicksberg, we show that the supports of generically stable (or just definable, assuming NIP) measures are nice semigroups, and classify idempotent measures in stable groups as invariant measures on type-definable subgroups. We establish left-continuity of the convolution map in NIP theories, and use it to show that the convolution semigroup on finitely satisfiable measures is isomorphic to a particular Ellis semigroup in this context.
We initiate the study of definable V-topolgies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if $(K,v_1,v_2)$ is a bi-valued NIP field with $v_1$ henselian (resp. t-henselian) then $v_1$ and $v_2$ are comparable (resp. dependent). As a consequence Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah's conjecture is equivalent to the statement th…
We solve two problems from the paper "On maximal stable quotients of definable groups in NIP theories" by M. Haskel and A. Pillay, which concern maximal stable quotients of groups type-definable in NIP theories. The first result says that if $G$ is a type-definable group in a distal theory, then $G^{st}=G^{00}$ (where $G^{st}$ is the smallest type-definable subgroup with $G/G^{st}$ stable, and $G^{00}$ is the smallest type-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from $T$ to the hyperimaginary expansion $T^{heq}$. The second r…
Let $K$ be an elementary extension of $\mathbb{Q}_p$, $V$ be the set of finite $a \in K$, $\mathrm{st}$ be the standard part map $K^m \to \mathbb{Q}^m_p$, and $X \subseteq K^m$ be $K$-definable. Delon has shown that $\mathbb{Q}^m_p \cap X$ is $\mathbb{Q}_p$-definable. Yao has shown that $\dim \mathbb{Q}^m_p \cap X \leq \dim X$ and $\dim \mathrm{st}(V^n \cap X) \leq \dim X$. We give new $\mathrm{NIP}$-theoretic proofs of these results and show that both inequalities hold in much more general settings. We also prove the analogous results for the expansion $\mathbb{Q}^{\mathrm{an}}_p$ of $\mathbb…
We present a systematic study of the regularity phenomena for NIP hypergraphs and connections to the theory of (locally) generically stable measures, providing a model-theoretic hypergraph version of the results from [L. Lovász, B. Szegedy, "Regularity partitions and the topology of graphons", An irregular mind, Springer Berlin Heidelberg, 2010, 415-446]. Besides, we revise the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the results from [A. Chernikov, S. Starchenko, "Regularity lemma for distal structures", J. Eur. Math. Soc. 20 (2018), 2437-…
Stable embeddedness and NIP
本文给出了完备理论T中谓词P稳定嵌入的充分条件：P的诱导0-可定义结构具有“有限秩”，P在T中具有NIP，P是1-稳定嵌入的。这推广了Hasson和Onshuus最近在P在T中是o-极小的情况下的工作。
We show that every unstable NIP theory admits a V-definable linear quasi-order, over a finite set of parameters. In particular, if the theory is omega-categorical, then it interprets an infinite linear order. This partially answers a longstanding open question.
On NIP and invariant measures
我们研究分叉，Lascar强类型，Keisler措施和可定义的群体，假设下的$NIP$（不是独立属性），继续方面的数学。主要成果包括：（i）如果$p = tp(b/A)$不在$A$上分叉，则$b$在$A$上的Lascar强类型与$b$在$A$上的紧致强类型一致，并且$p$的任何全局非分叉扩张在$bdd(A)$上都是Borel可定义的（ii）Keisler测度和可定义群的类似陈述，包括$G^{000} = G^{00}$对于$G$可定义服从的事实，（iii）定义，“一般稳定”类型和群的特征和性质（iv）具有可满足一般的群上平移不变Keisler测度的唯一性（vi）实闭域的$o$ -极小展开中可定义紧交换群的紧控制猜想的证明。
Forking and invariant types in monadic NIP theories
本文在一元NIP理论中证明不变全局类型的强分解定理，并由此得出分叉的逐坐标判据及可定义类型的稠密性。
On $ω$-categorical groups and rings with NIP
证明了具有NIP的$ω$ -范畴环是有限幂零的.证明了具有NIP和fsg的$ω$ -范畴群是有限幂零群.我们还注意到至少有一个强正则类型的$ω$ -范畴群是阿贝尔群。进一步，我们得到了每个$ω$ -范畴的特征单$p$ -群都有一个无穷的可定义阿贝尔子群.另外假设存在一个非代数的，在$\emptyset$上一般稳定的类型，这样的群是阿贝尔群。
External definability and groups in NIP theories
证明了NIP理论中可定义群的许多性质和不变量，如可定义顺从性、G/G^{00}等，当通过模型M的外部可定义集合M^{ext}传递到谢拉展开理论时，根据这些结果，我们继续研究的“可定义的拓扑动力学”的团体在NIP理论。特别地，我们证明了埃利斯群猜想的埃利斯群G/G^{00}在一些新的情况下，包括可定义的顺从群在o-极小结构。
We study invariant types in NIP theories. Amongst other things: we prove a definable version of the (p,q)-theorem in theories of small or medium directionality; we construct a canonical retraction from the space of M-invariant types to that of M-finitely satisfiable types; we show some amalgamation results for invariant types and list a number of open questions.
We apply the work of Bourgain, Fremlin and Talagrand on compact subsets of the first Baire class to show new results about phi-types for phi NIP. In particular, we show that if M is a countable model, then an M-invariant phi-type is Borel definable. Also the space of M-invariant phi-types is a Rosenthal compactum, which implies a number of topological tameness properties.
Local definable $(p,q)$-theorem and a refinement
该研究证明了任意一阶理论中NIP公式的可定义$(p,q)$猜想，给出其精细化并应用于稠密对。
Glivenko-Cantelli classes and NIP formulas
本文给出NIP公式的几个新等价形式，利用已知结果的新证明，强调Keisler测度比类型更复杂，并证明了一阶理论中公式具有NIP与DBSC可定义性、Baire-1/2可定义性及Rosenthal紧统等条件的等价性。
Groups definable in o-minimal structures: structure theorem, G^000, definable amenability and bounded orbits
解决了O-极小结构群的特殊情形下的一些公开问题，如G^00与G^000的等价性，可定义顺从性与一类有界轨道的存在性的等价性。我们证明了G到G^00函子的几乎正合性。我们问进一步的问题，在NIP理论与有界轨道的类型。
The following conjecture is due to Shelah-Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non-trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.
Externally definable sets and dependent pairs
证明了一阶NIP理论中的外可定义集具有诚实定义，给出了Shelah展开定理的一个新证明。我们还讨论了一个弱的概念，稳定嵌入真在这种情况下。这些结果，然后用来证明一个一般定理依赖对，特别是回答了一个问题的鲍德温和Benedikt命名一个不可辨别的序列。
We investigate bounds in Ramsey's theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek [B. Bukh, J. Matoušek. "Erdős-Szekeres-type statements: Ramsey function and decidability in dimension $1$", Duke Mathematical Journal 163.12 (2014): 2243-2270] from the semialgebraic case to arbitrary polynomially bounded $o$-minimal expansions of $\mathbb{R}$, and show that it doesn't hold in $\mathbb{R}_{\exp}$. This provides a new combinatorial characterization of polynomial boundedness for $o$-minimal s…
First, an example of a 2-dependent group without a minimal subgroup of bounded index is given. Second, all infinite n-dependent fields are shown to be Artin-Schreier closed. Furthermore, the theory of any non separably closed PAC field has the IPn property for all natural numbers n and certain properties of dependent (NIP) valued fields extend to the n-dependent context.
On some NIP Fragments of Fields
本文研究了某些字段理论中的NIP公式集，证明了分离闭赋值域的NIP性质，并探讨了Henselian赋值域与残余域、赋值群的NIP性质之间的关系。
We prove the dp-finite case of the Shelah conjecture on NIP fields. If K is a dp-finite field, then K admits a non-trivial definable henselian valuation ring, unless K is finite, real closed, or algebraically closed. As a consequence, the conjectural classification of dp-finite fields holds. Additionally, dp-finite valued fields are henselian. Lastly, if K is an unstable dp-finite expansion of a field, then K admits a unique definable V-topology.
Upside down and backwards
研究NIP背景下不变类型与有限可满足类型的Ellis子群同构性及自然同构条件
We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_σ$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over $\emptyset$. As an easy conclusion of our main theorem, we get the main result from our recent paper joint with Andand Pillay, which says that for any strong type defined on a single complete type over $\emptyset$, smoothness is equivalent to type-definability. We also explai…
On groups and fields interpretable in $\mathrm{NTP_2}$ fields
本文在NIP和NTP_2背景下发展模型论工具，证明从泛型数据构造可定义群同态的抽象定理，并由此得到代数有界完美NTP_2域及NIP微分域中可定义顺从群的结构定理与可解释域的分类结果。
Some results on NIP groups and their Ellis groups
本文在NIP群中发展分段f-generic理论，证明非分段f-generic可定义集构成理想，还得到NIP群Ellis群大小上界、有限阿基米德秩等相关结果，为解决同构类型问题提供进展。
Canonical quotients in model theory
本文研究模型论中的典范商，将modeling property扩展至连续理论以研究超可定义集的$n$-依赖性，并证明极大WAP商流和极大tame商流的Ellis群不依赖于monster model的选择。
On certain definable coarsenings of valuation rings and their applications
本文研究了估值环的可定义粗化及其在分枝理想和 Kähler 微分中的应用，探讨了 Galois 扩展与凸子群的关系。
Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture
本文研究可定义群上的幂等Keisler测度，提出一般传递性并证明其足以发展局部稳定群论，在阿贝尔群及rosy群等新情形确立一般稳定幂等型的一般传递性，分类幂等测度，并构造可数NIP群测度卷积半群的极小左理想，证明修正的Newelski猜想。
Ordered henselian valued fields: definability and Borel sets
研究有序赫恩塞利赋值域的可定义性与博雷尔集性质，通过量化消除和维度缩减定理，揭示其与NIP域分类猜想的联系。
Approximate equivalence relations
本文在可定义测度下研究近似等价关系，基于连续逻辑提出概率逻辑框架，利用Kim-Pillay空间完全分类二元相关性，推广范畴性定理与近似子群的李模型定理，并证明NIP理论中可定义测度自同构群的阿基米德有限维性。
Groups and fields with NTP2
NTP 2是由Shelah定义的一阶理论的一大类，推广了简单理论和NIP理论。NTP 2结构的代数例子是由p-自进和某些值差域的超积给出的（例如生活在特征为0的代数闭值域上的非标准弗罗贝尼乌斯自同构）。在这篇笔记中，我们提出了一些结果的组和域定义的NTP 2结构。最重要的是，我们隔离的链条件可定义的正规子群，并使用它来显示，任何NTP 2字段只有100多个Artin-Schreier扩展。我们还讨论了一个更强的链条件来自强加的理论负担的界限（一个适当的模拟重量），并表明，每一个强相关的价值领域是Kaplansky。
Newelski's Conjecture for $o$-Minimal and $p$-Adic Groups
本文研究了 $o$-最小和 $p$-adic 组的 Ellis 组，并证明 Newelski 猜想在特定条件下成立。
We study a class of tame $\mathcal{L}$-theories $T$ of topological fields and their $\mathcal{L}_δ$-extension $T_δ^*$ by a generic derivation $δ$. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. We show that the associated expansion by a generic derivation has $\mathcal{L}$-open core (i.e., every $\mathcal{L}_δ$-definable open set is $\mathcal{L}$-definable) and derive both a cell decomposition theorem and a transfer result of elimination of imaginaries. Other tame properties of $T$ such as relative elimination of field sor…
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