Gabriel Conant
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Gabriel Conant has a total of 9 co-authors. Here are the top co-authors by number of shared publications:
- Anand Pillay
Department of Mathematics, University of Notre Dame, Notre Dame, USA
3 shared publications - Isaac Goldbring
University of California, Irvine
2 shared publications - Uri Andrews
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1325
2 shared publications - Alex Kruckman
DEPARTMENT OF MATHEMATICS INDIANA UNIVERSITY BLOOMINGTON BLOOMINGTON, IN47405, USA
1 shared publication - Caroline Terry
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
1 shared publication Edgar A. Bering Iv
1 shared publication
Latest Publications
- Open Access
An axiomatic approach to free amalgamation
We use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures in relational languages, in which the class of algebraically closed substructures is closed under free amalgamation. We show that any free amalgamation theory has elimination of hyperimaginaries and weak elimination of imaginaries. With this result, we use several families of well-known homogeneous structures to give new examples of rosy theories. We then prove that, for free amalgamation theories, simplicity coincides with NTP2 and, assuming modularity, with NSOP3 as well. We also show that any simple free amalgamation theory is 1-based. Finally, we prove a combinatorial characterization of simplicity for Fraisse limits with free amalgamation, which provides new context for the fact that the generic $K_n$-free graphs are SOP3, while the higher arity generic $K^r_n$-free $r$-hypergraphs are simple.
Open AccessAn axiomatic approach to free amalgamation
We use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures in relational languages, in which the class of algebraically closed substructures is closed under free amalgamation. We show that any free amalgamation theory has elimination of hyperimaginaries and weak elimination of imaginaries. With this result, we use several families of well-known homogeneous structures to give new examples of rosy theories. We then prove that, for free amalgamation theories, simplicity coincides with NTP2 and, assuming modularity, with NSOP3 as well. We also show that any simple free amalgamation theory is 1-based. Finally, we prove a combinatorial characterization of simplicity for Fraisse limits with free amalgamation, which provides new context for the fact that the generic $K_n$-free graphs are SOP3, while the higher arity generic $K^r_n$-free $r$-hypergraphs are simple.
- Open Access
Pseudofinite groups and VC-dimension
We develop local NIP group theory in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure $G$ expanding a group, and left invariant NIP formula $\delta(x;\bar{y})$, we prove various aspects of "local fsg" for the right-stratified formula $\delta^r(x;\bar{y},u):=\delta(x\cdot u;\bar{y})$. This includes a $\delta^r$-type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on $\delta^r$-formulas, and generic compact domination for $\delta^r$-definable sets.Open AccessPseudofinite groups and VC-dimension
We develop local NIP group theory in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure $G$ expanding a group, and left invariant NIP formula $\delta(x;\bar{y})$, we prove various aspects of "local fsg" for the right-stratified formula $\delta^r(x;\bar{y},u):=\delta(x\cdot u;\bar{y})$. This includes a $\delta^r$-type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on $\delta^r$-formulas, and generic compact domination for $\delta^r$-definable sets. - Open Access
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 $\phi$-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.
Open AccessRemarks 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 $\phi$-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.
- Open Access
Weakly minimal groups with a new predicate
Fix a weakly minimal (i.e., superstable $U$-rank $1$) structure $\mathcal{M}$. Let $\mathcal{M}^*$ be an expansion by constants for an elementary substructure, and let $A$ be an arbitrary subset of the universe $M$. We show that all formulas in the expansion $(\mathcal{M}^*,A)$ are equivalent to bounded formulas, and so $(\mathcal{M},A)$ is stable (or NIP) if and only if the $\mathcal{M}$-induced structure $A_{\mathcal{M}}$ on $A$ is stable (or NIP). We then restrict to the case that $\mathcal{M}$ is a pure abelian group with a weakly minimal theory, and $A_{\mathcal{M}}$ is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of $(\mathbb{Z},+)$. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form $(\mathcal{M},A)$. Most notably, we show that if $(G,+)$ is a weakly minimal additive subgroup of the algebraic numbers, $A\subseteq G$ is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of $A$ is a root of unity, then $(G,+,B)$ is superstable for any $B\subseteq A$.Open AccessWeakly minimal groups with a new predicate
Fix a weakly minimal (i.e., superstable $U$-rank $1$) structure $\mathcal{M}$. Let $\mathcal{M}^*$ be an expansion by constants for an elementary substructure, and let $A$ be an arbitrary subset of the universe $M$. We show that all formulas in the expansion $(\mathcal{M}^*,A)$ are equivalent to bounded formulas, and so $(\mathcal{M},A)$ is stable (or NIP) if and only if the $\mathcal{M}$-induced structure $A_{\mathcal{M}}$ on $A$ is stable (or NIP). We then restrict to the case that $\mathcal{M}$ is a pure abelian group with a weakly minimal theory, and $A_{\mathcal{M}}$ is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of $(\mathbb{Z},+)$. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form $(\mathcal{M},A)$. Most notably, we show that if $(G,+)$ is a weakly minimal additive subgroup of the algebraic numbers, $A\subseteq G$ is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of $A$ is a root of unity, then $(G,+,B)$ is superstable for any $B\subseteq A$. - 15 February 2019
- Published by Cambridge University Press (CUP) in The Journal of Symbolic Logic
- Vol. 84 (2) , 750-780
INDEPENDENCE IN GENERIC INCIDENCE STRUCTURES
We study the theory T m,n of existentially closed incidence structures omitting the complete incidence structure K m,n , which can also be viewed as existentially closed K m,n -free bipartite graphs. In the case m = n = 2, this is the theory of existentially closed projective planes. We give an $\forall \exists$ -axiomatization of T m,n , show that T m,n does not have a countable saturated model when m , n ≥ 2, and show that the existence of a prime model for T 2,2 is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model theoretic notions of complexity for T m,n . We show that T m,n is NSOP 1 , but not simple when m , n ≥ 2, and we show that T m,n has weak elimination of imaginaries but not full elimination of imaginaries. These results rely on combinatorial characterizations of various notions of independence, including algebraic independence, Kim independence, and forking independence.- 15 February 2019
- Published by Cambridge University Press (CUP) in The Journal of Symbolic Logic
- Vol. 84 (2) , 750-780
INDEPENDENCE IN GENERIC INCIDENCE STRUCTURES
We study the theory T m,n of existentially closed incidence structures omitting the complete incidence structure K m,n , which can also be viewed as existentially closed K m,n -free bipartite graphs. In the case m = n = 2, this is the theory of existentially closed projective planes. We give an $\forall \exists$ -axiomatization of T m,n , show that T m,n does not have a countable saturated model when m , n ≥ 2, and show that the existence of a prime model for T 2,2 is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model theoretic notions of complexity for T m,n . We show that T m,n is NSOP 1 , but not simple when m , n ≥ 2, and we show that T m,n has weak elimination of imaginaries but not full elimination of imaginaries. These results rely on combinatorial characterizations of various notions of independence, including algebraic independence, Kim independence, and forking independence.- 17 January 2019
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 230 (1) , 471-508
Stability and sparsity in sets of natural numbers
- 17 January 2019
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 230 (1) , 471-508
Stability and sparsity in sets of natural numbers
- 1 January 2019
- Published by Institute of Mathematics, Polish Academy of Sciences in Fundamenta Mathematicae
- Vol. 244 (1) , 1-16
Extending partial isometries of generalized metric spaces
We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal {K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any $A\in \mathcal {K}$ the- 1 January 2019
- Published by Institute of Mathematics, Polish Academy of Sciences in Fundamenta Mathematicae
- Vol. 244 (1) , 1-16
Extending partial isometries of generalized metric spaces
We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal {K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any $A\in \mathcal {K}$ the- 6 November 2018
- Published by Walter de Gruyter GmbH in Journal of Group Theory
- Vol. 22 (1) , 63-82
Definable sets containing productsets in expansions of groups
We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the “productset property”). We first show that the productset property holds for any definable subset A of an expansion of a discrete amenable group such that A has positive Banach density and the formula x ⋅ y ∈ A {x\cdot y\in A} is stable. For arbitrary expansions of groups, we consider a “1-sided” version of the productset property, which is characterized in various ways using coheir independence. For stable groups, the productset property is equivalent to this 1-sided version, and behaves as a notion of largeness for definable sets, which can be characterized by a natural weakening of model-theoretic genericity. Finally, we use recent work on regularity lemmas in distal theories to prove a definable version of the productset property for sets of positive Banach density definable in certain distal expansions of amenable groups.
- 6 November 2018
- Published by Walter de Gruyter GmbH in Journal of Group Theory
- Vol. 22 (1) , 63-82
Definable sets containing productsets in expansions of groups
We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the “productset property”). We first show that the productset property holds for any definable subset A of an expansion of a discrete amenable group such that A has positive Banach density and the formula x ⋅ y ∈ A {x\cdot y\in A} is stable. For arbitrary expansions of groups, we consider a “1-sided” version of the productset property, which is characterized in various ways using coheir independence. For stable groups, the productset property is equivalent to this 1-sided version, and behaves as a notion of largeness for definable sets, which can be characterized by a natural weakening of model-theoretic genericity. Finally, we use recent work on regularity lemmas in distal theories to prove a definable version of the productset property for sets of positive Banach density definable in certain distal expansions of amenable groups.
- Open Access
A group version of stable regularity
We prove that, given $\epsilon>0$ and $k\geq 1$, there is an integer $n$ such that the following holds. Suppose $G$ is a finite group and $A\subseteq G$ is $k$-stable. Then there is a normal subgroup $H\leq G$ of index at most $n$, and a set $Y\subseteq G$, which is a union of cosets of $H$, such that $|A\vartriangle Y|\leq\epsilon|H|$. It follows that, for any coset $C$ of $H$, either $|C\cap A|\leq \epsilon|H|$ or $|C\setminus A|\leq \epsilon|H|$. This qualitatively generalizes recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$.Open AccessA group version of stable regularity
We prove that, given $\epsilon>0$ and $k\geq 1$, there is an integer $n$ such that the following holds. Suppose $G$ is a finite group and $A\subseteq G$ is $k$-stable. Then there is a normal subgroup $H\leq G$ of index at most $n$, and a set $Y\subseteq G$, which is a union of cosets of $H$, such that $|A\vartriangle Y|\leq\epsilon|H|$. It follows that, for any coset $C$ of $H$, either $|C\cap A|\leq \epsilon|H|$ or $|C\setminus A|\leq \epsilon|H|$. This qualitatively generalizes recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$. - Open Access
Multiplicative structure in stable expansions of the group of integers
We define two families of expansions of $(\mathbb{Z},+,0)$ by unary predicates, and prove that their theories are superstable of $U$-rank $\omega$. The first family consists of expansions $(\mathbb{Z},+,0,A)$, where $A$ is an infinite subset of a finitely generated multiplicative submonoid of $\mathbb{N}$. Using this result, we also prove stability for the expansion of $(\mathbb{Z},+,0)$ by all unary predicates of the form $\{q^n:n\in\mathbb{N}\}$ for some $q\in\mathbb{N}_{\geq 2}$. The second family consists of sets $A\subseteq\mathbb{N}$ which grow asymptotically close to a $\mathbb{Q}$-linearly independent increasing sequence $(\lambda_n)_{n=0}^\infty\subseteq\mathbb{R}^+$ such that $\{\frac{\lambda_n}{\lambda_m}:m\leq n\}$ is closed and discrete.Open AccessMultiplicative structure in stable expansions of the group of integers
We define two families of expansions of $(\mathbb{Z},+,0)$ by unary predicates, and prove that their theories are superstable of $U$-rank $\omega$. The first family consists of expansions $(\mathbb{Z},+,0,A)$, where $A$ is an infinite subset of a finitely generated multiplicative submonoid of $\mathbb{N}$. Using this result, we also prove stability for the expansion of $(\mathbb{Z},+,0)$ by all unary predicates of the form $\{q^n:n\in\mathbb{N}\}$ for some $q\in\mathbb{N}_{\geq 2}$. The second family consists of sets $A\subseteq\mathbb{N}$ which grow asymptotically close to a $\mathbb{Q}$-linearly independent increasing sequence $(\lambda_n)_{n=0}^\infty\subseteq\mathbb{R}^+$ such that $\{\frac{\lambda_n}{\lambda_m}:m\leq n\}$ is closed and discrete.