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Kyle Gannon

Publications and source records attributed to Kyle Gannon.

At least 19 recordsLinked to original sources
arXiv2026

Generically stable Keisler measures

Given a first-order theory $T$ (in discrete or continuous logic) and a Borel-definable global Keisler measure $\mu$ in $T$, we show that the following conditions are equivalent: $(i)$ $\mu$ is a frequency interpretation measure; $(ii)$ $\mu$ is definable and its canonical "random extension" $r_\mu$ is generically stable in the randomization theory $T^R$; $(iii)$ $\mu$ is "self-averaging". This result establishes a robust notion of generic stability for Keisler measures, which resolves a long-term research objective from previous work. The implications $(i)\Rightarrow(ii)\Rightarrow (iii)$ were previously established by the authors (for $T$ discrete). The primary focus of this paper is the reverse implications $(iii)\Rightarrow (ii)\Rightarrow(i)$, which we obtain through the use of AI models.

math.LO↗
arXiv2026

Model theory of convolution algebras

This paper deals with the model theory of convolution algebras $(L^1(G),*)$ for locally compact groups $G$, seen as Banach lattices equipped with the convolution product $*$. We first prove transfer principles for elementary equivalence and elementary embeddings when the underlying group $G$ is discrete, namely, $(\ell^1(G),*) \equiv (\ell^1(H),*)$ implies $G \equiv H$, while the converse holds when $G$ and $H$ are $\omega$-saturated (likewise for elementary substructures). Without $\omega$-saturation, the converse fails. Although pure Banach lattices are model-theoretically tame, our results imply that adding convolution yields wild behavior. For example, we prove that if $G$ is any locally compact, non-discrete group, then the formula $d(x,x*y)$ is unstable with respect to $\mathrm{Th}(L^1(G),*)$. Moreover, we show that if $G$ is discrete and contains a particular configuration of amenable subgroups, then the formula $d(x*y,z)\mathbin{\dot{-}}\frac{1}{2}$ witnesses $\mathrm{TP}_2$ with respect to $\mathrm{Th}(\ell^1(G),*)$. As a consequence, if $G$ contains an infinite abelian subgroup, then $\mathrm{Th}(\ell^{1}(G),*)$ has $\mathrm{TP}_{2}$. We prove similar results in the locally compact non-discrete setting using the notion of an approximate identity. Finally, we prove a `continuous-by-discrete' approximation theorem. Namely, convolution algebras of connected abelian Lie groups admit metric embeddings into ultraproducts of convolution algebras over finite abelian groups.

math.LO↗
arXiv2026

Measures and stability in a model, revisited

This article is written in celebration of the 8th Kazakh-French Logical Colloquium. We expand on an unpublished research note of the second author. We record some results concerning local Keisler measures with respect to a formula which is stable in a model. We prove that in this context, every local Keisler measure on the associated local type space is a weighted sum of (at most countably many) types. Using this observation, we give an elementary proof of the commutativity of the Morley product in this context. We then give a functional analytic proof that the double limit property lifts to the appropriate evaluation map on pairs of local measures. We end with some comments on the NOP and local measures in the (properly) stable context.

math.LO↗
arXiv2025

Upside down and backwards

We investigate the semigroup of invariant types through the lens of Ellis theory; primarily focusing on definably amenable NIP groups. In this context, we observe that the collection of strong right $f$-generic types forms the unique minimal left ideal and thus, the Ellis subgroups are isomorphic to $G/G^{00}$ via the canonical quotient map. As consequence of the Newelski-Pillay conjecture, the Ellis subgroups of the semigroup of invariant types are abstractly isomorphic to the Ellis subgroups of the semigroup of finitely satisfiable types in the definable amenable NIP setting. We are interested in the existence of natural isomorphisms from invariant Ellis subgroups to finitely satisfiable Ellis subgroups and we determine when these isomorphisms can be witnessed by variants of the canonical NIP retraction map. Several limiting examples are provided. Outside of the NIP context, we provide an abelian group (and thus definably amenable) with an $\emptyset$-definable (dfg) type in which the invariant Ellis subgroups and finitely satisfiable Ellis subgroups not isomorphic.

math.LO↗
arXiv2025

Generic sampling and invariant measures on the space of $k$-uniform hypergraphs

We prove a model-theoretic representation theorem for the distribution of an ergodic exchangeable $k$-uniform hypergraph: every such measure arises as the pushforward of the countably-iterated Morley product of a global Borel-definable Keisler measure over the countable universal homogeneous $k$-uniform hypergraph. We show this by starting with a Borel $k$-hypergraphon $W$ and constructing a Keisler measure $\mu_{W}$ such that generic sampling with respect to $\mu_{W}$ yields the same invariant measure as does the standard hypergraphon sampling procedure with respect to $W$. When $k = 2$, our results give a new representation theorem for ergodic exchangeable graphs via Keisler measures over a monster model of the Rado graph.

math.CO↗
arXiv2025

Convolution semigroups for automorphism dynamics

Initially motivated by Hrushovski's paper on definability patterns, we obtain homeomorphisms between Ellis semigroups related to natural actions of the automorphism groups of first order structures and certain collections of types and Keisler measures. Thus, we can transfer the semigroup operation from these Ellis semigroups to the corresponding collections of types and Keisler measures. By generalizing this transferred product, we obtain a new convolution operation for invariant types and measures in arbitrary first-order theories. We develop its general theory and prove several correspondence theorems between idempotent measures and closed subgroups of the automorphism group of a sufficiently large (so-called monster) model with respect to the relatively definable topology. Via the affine sort construction, we demonstrate that this new notion of convolution encodes the standard definable convolution operation over definable groups.

math.LO↗
arXiv2024

Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture

We study idempotent measures and the structure of the convolution semigroups of measures over definable groups. We isolate the property of generic transitivity and demonstrate that it is sufficient (and necessary) to develop stable group theory localizing on a generically stable type, including invariant stratified ranks and connected components. We establish generic transitivity of generically stable idempotent types in important new cases, including abelian groups in arbitrary theories and arbitrary groups in rosy theories, and characterize them as generics of connected type-definable subgroups. Using tools from Keisler's randomization theory, we generalize some of these results from types to generically stable Keisler measures, and classify idempotent generically stable measures in abelian groups as (unique) translation-invariant measures on type-definable fsg subgroups. This provides a partial definable counterpart to the classical work of Rudin, Cohen and Pym for locally compact topological groups. Finally, we provide an explicit construction of a minimal left ideal in the convolution semigroup of measures for an arbitrary countable NIP group, from a minimal left ideal in the corresponding semigroup on types and a canonical measure constructed on its ideal subgroup. In order to achieve it, we in particular prove the revised Ellis group conjecture of Newelski for countable NIP groups.

math.LO↗
arXiv2024

Model theoretic events

We develop a notion of sampling, called \emph{generic sampling}, for the context of global Keisler measures where the standard product is replaced by the Morley product. Choosing a point randomly in this space with respect to our distribution yields a \emph{random generic type} in infinitely many variables. We investigate several natural model-theoretic events and provide conditions under which they occur for almost all random generic types.

math.LO↗
arXiv2024

A note on transfer maps and the Morley product in NIP theories

In an important (yet unpublished) research note, Ben Yaacov describes how to turn a global Keisler measures into a type over a monster model of the randomization. This transfer methods allow one to turn questions involving measures into those involving types (in continuous logic). Assuming that T is NIP, we show that the Morley product commutes with the transfer map for finitely satisfiable measures. We characterize when the Morley product commutes with the restriction map for pairs of global finitely satisfiable types in the randomization. We end by making some brief observations about the Ellis semigroup in this context.

math.LO↗
arXiv2023

Generic stability, randomizations, and NIP formulas

We prove a number of results relating the concepts of Keisler measures, generic stability, randomizations, and NIP formulas. Among other things, we do the following: (1) We introduce the notion of a Keisler-Morley measure, which plays the role of a Morley sequence for a Keisler measure. We prove that if $\mu$ is fim over $M$, then for any Keisler-Morley measure $\lambda$ in $\mu$ over $M$ and any formula $\varphi(x,b)$, $\lim_{i \to \infty} \lambda(\varphi(x_i,b)) = \mu(\varphi(x,b))$. We also show that any measure satisfying this conclusion must be fam. (2) We study the map, defined by Ben Yaacov, taking a definable measure $\mu$ to a type $r_\mu$ in the randomization. We prove that this map commutes with Morley products, and that if $\mu$ is fim then $r_\mu$ is generically stable. (3) We characterize when generically stable types are closed under Morley products by means of a variation of ict-patterns. Moreover, we show that NTP$_2$ theories satisfy this property. (4) We prove that if a local measure admits a suitably tame global extension, then it has finite packing numbers with respect to any definable family. We also characterize NIP formulas via the existence of tame extensions for local measures.

math.LO↗
arXiv2023

Concerning Keisler Measures over ultraproducts

As consequence of the VC theorem, any pseudo-finite measure over an NIP ultraproduct is generically stable. We demonstrate a converse of this theorem and prove that any finitely approximable measure over an ultraproduct is itself pseudo-finite (even without the NIP assumption). We also analyze the connection between the Morley product and the pseudo-finite product. In particular, we show that if $\mu$ is definable and both $\mu$ and $\nu$ are pseudo-finite, then the Morley product of $\mu$ and $\nu$ agrees with the pseudo-finite product of $\mu$ and $\nu$. Using this observation, we construct generically stable idempotent measures on pseudo-finite NIP groups.

math.LO↗
arXiv2022

An invitation to extension domination

Motivated by the theory of domination for types, we introduce a notion of domination for Keisler measures called extension domination. We argue that this variant of domination behaves similarly to its type setting counterpart. We prove that extension domination extends domination for types and that it forms a preorder on the space of global Keisler measures. We then explore some basic properties related to this notion (e.g. approximations by formulas, closure under localizations, convex combinations). We also prove a few preservation theorems and provide some explicit examples.

math.LO↗
arXiv2022

Definable convolution and idempotent Keisler measures II

We study convolution semigroups of invariant/finitely satisfiable Keisler measures in NIP groups. We show that the ideal (Ellis) subgroups are always trivial and describe minimal left ideals in the definably amenable case, demonstrating that they always form a Bauer simplex. Under some assumptions, we give an explicit construction of a minimal left ideal in the semigroup of measures from a minimal left ideal in the corresponding semigroup of types (this includes the case of SL$_{2}(\mathbb{R})$, which is not definably amenable). We also show that the canonical push-forward map is a homomorphism from definable convolution on $\mathcal{G}$ to classical convolution on the compact group $\mathcal{G}/\mathcal{G}^{00}$, and use it to classify $\mathcal{G}^{00}$-invariant idempotent measures.

math.LO↗
arXiv2021

Sequential approximations for types and Keisler measures

This paper is a modified chapter of the author's Ph.D. thesis. We introduce the notions of sequentially approximated types and sequentially approximated Keisler measures. As the names imply, these are types which can be approximated by a sequence of realized types and measures which can be approximated by a sequence of `averaging measures' on tuples of realized types. We show that both generically stable types (in arbitrary theories) and Keisler measures which are finitely satisfiable over a countable model (in NIP theories) are sequentially approximated. We also introduce the notion of a smooth sequence in a measure over a model and give an equivalent characterization of generically stable measures (in NIP theories) via this definition. In the last section, we take the opportunity to generalize the main result of [8].

math.LO↗
arXiv2021

Keisler measures in the wild

We investigate Keisler measures in arbitrary theories. Our initial focus is on Borel definability. We show that when working over countable parameter sets in countable theories, Borel definable measures are closed under Morley products and satisfy associativity. However, we also demonstrate failures of both properties over uncountable parameter sets. In particular, we show that the Morley product of Borel definable types need not be Borel definable (correcting an erroneous result from the literature). We then study various notions of generic stability for Keisler measures and generalize several results from the NIP setting to arbitrary theories. We also prove some positive results for the class of frequency interpretation measures in arbitrary theories, namely, that such measures are closed under convex combinations and commute with all Borel definable measures. Finally, we construct the first example of a complete type which is definable and finitely satisfiable in a small model, but not finitely approximated over any small model.

math.LO↗
arXiv2021

Definable convolution and idempotent Keisler measures

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.

math.LO↗
arXiv2019

Remarks on generic stability in independent theories

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of "generic stability" in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as $ϕ$-types in simple theories that are definable and finitely satisfiable in a small model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs.

math.LO↗