Erik Walsberg

activity

20132026

most citedTame Topology over Dp-minimal Structures

12 citations · 63 across the 29 of their papers we have counts for

collaborators

33 papers

math.LO2026

Building trees in large fields

Elliot Kaplan, Angus Matthews, Erik Walsberg

We show that large rosy fields are bounded, and substantially simplify the proofs that large stable fields are separably closed and that large simple fields are bounded. Our proofs…

math.LO2022

When is the étale open topology a field topology?

Philip Dittmann, Erik Walsberg, Jinhe Ye

We investigate the following question: Given a field $K$ , when is the étale open topology $E_{K}$ induced by a field topology? On the positive side, when $K$ is the fractio…

math.LO2022★ 1 cited

Interpolative fusions II: Preservation results

Alex Kruckman, Minh Chieu Tran, Erik Walsberg

We study interpolative fusion, a method of combining theories $T_{1}$ and $T_{2}$ in distinct languages in a "generic" way over a common reduct $T_{\cap}$ , to obtain a theory $T_{\cup}^{*}$ .…

math.AG2021★ 3 cited

The étale open topology over the fraction field of a henselian local domain

Will Johnson, Erik Walsberg, Jinhe Ye

Suppose that $R$ is a local domain with fraction field $K$ . If $R$ is Henselian then the $R$ -adic topology over $K$ refines the étale open topology. If $R$ is regular then the étal…

math.AG2021

Topological proofs of results on large fields

Erik Walsberg

We use the recently introduced étale open topology to prove several facts about large fields. We show that these facts lift to a very general topological setting.

math.LO2021★ 5 cited

Éz fields

Erik Walsberg, Jinhe Ye

Let $K$ be a field. The étale open topology on the $K$ -points $V (K)$ of a $K$ -variety $V$ was introduced in our previous work. The étale open topology is non-discrete if and only i…

math.LO2021

Notes on trace equivalence

Erik Walsberg

We introduce trace definability, a weak notion of interpretability, and trace equivalence, a weak notion of equivalence for first order structures and theories. In particular we ge…

math.LO2020★ 2 cited

Galois groups of large fields with simple theory (with an appendix by Philip Dittmann)

Anand Pillay, Erik Walsberg

Suppose that $K$ is an infinite field which is large (in the sense of Pop) and whose first order theory is simple. We show that $K$ is {\em bounded}, namely has only finitely many…

math.LO2020★ 7 cited

The étale-open topology and the stable fields conjecture

Will Johnson, Chieu-Minh Tran, Erik Walsberg +1

For an arbitrary field $K$ and $K$ -variety $V$ , we introduce the étale-open topology on the set $V (K)$ of $K$ -points of $V$ . This topology agrees with the Zariski topology, Euclide…

math.LO2020

A P-adic structure which does not interpret an infinite field but whose Shelah completion does

Erik Walsberg

We give a $p$ -adic example of a structure whose Shelah completion interprets $Q_{p}$ but which does not (provided an extremely plausible conjecture holds) interpret an infin…

math.LO2020

Nippy proofs of p-adic results of Delon and Yao

Erik Walsberg

Let $K$ be an elementary extension of $Q_{p}$ , $V$ be the set of finite $a \in K$ , $st$ be the standard part map $K^{m} \to Q_{p}^{m}$ , and $X \subseteq K^{m}$ be…

math.LO2020★ 1 cited

Dp-minimal expansions of $(Z, +)$ via dense pairs via Mordell-Lang

Erik Walsberg

This is a contribution to the classification problem for dp-minimal expansions of $(Z, +)$ . Let $S$ be a dense cyclic group order on $(Z, +)$ . We use results on "de…

math.LO2020★ 2 cited

Generalizing a theorem of Bès and Choffrut

Erik Walsberg

Bès and Choffrut recently showed that there are no intermediate structures between $(R, <, +)$ and $(R, <, +, Z)$ . We prove a generalization: if $R$ …

math.LO2019★ 5 cited

Directed sets and topological spaces definable in o-minimal structures

Pablo Andujar Guerrero, Margaret E. M. Thomas, Erik Walsberg

We study directed sets definable in o-minimal structures, showing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expan…

math.LO2019★ 3 cited

An NIP structure which does not interpret an infinite group but whose Shelah expansion interprets an infinite field

Erik Walsberg

We describe one.

math.LO2019

Externally definable quotients and NIP expansions of the real ordered additive group

Erik Walsberg

Let $R$ be an $NIP$ expansion of $(R, <, +)$ by closed subsets of $R^{n}$ and continuous functions $f : R^{m} \to R^{n}$ . Then $\math…

math.LO2019★ 4 cited

Dp and other minimalities

Pierre Simon, Erik Walsberg

A first order expansion of $(R, +, <)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, $p$ -adic fields, ord…

math.LO2019★ 1 cited

Dp-minimal expansions of discrete ordered abelian groups

Erik Walsberg

If $Z$ is a dp-minimal expansion of a discrete ordered abelian group $(Z, <, +)$ and $Z$ does not admit a nontrivial definable convex subgroup then $Z$ …

math.LO2019

Coarse dimension and definable sets in expansions of the ordered real vector space

Erik Walsberg

Suppose $E \subseteq R$ is nowhere dense. If $(R, <, +, (x \mapsto λ x)_{λ \in R}, E)$ does not define every bounded Borel subset of every $R^{n}$ the…

cs.LO2019★ 2 cited

Continuous Regular Functions

Alexi Block Gorman, Philipp Hieronymi, Elliot Kaplan +5

Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function $f : [0, 1] \to [0, 1]$ is $r$ -regular if there is a Büchi automaton that accepts precisely the set of base $r…

math.LO2018★ 9 cited

Interpolative Fusions I

Alex Kruckman, Minh Chieu Tran, Erik Walsberg

We define the interpolative fusion $T_{\cup}^{*}$ of a family $(T_{i})_{i \in I}$ of first-order theories over a common reduct $T_{\cap}$ , a notion that generalizes many examples of random…

math.LO2018

Expansions of the real field by discrete subgroups of Gl $_{n} (C)$

Philipp Hieronymi, Erik Walsberg, Samantha Xu

Let $Γ$ be an infinite discrete subgroup of Gl $_{n} (C)$ . Then either $(R, <, +, \cdot, Γ)$ is interdefinable with $(R, <, +, \cdot, λ^{Z})$ for some…

math.LO2017★ 2 cited

A family of dp-minimal expansions of $(Z; +)$

Minh Chieu Tran, Erik Walsberg

We show that the cyclically ordered-abelian groups expanding $(Z; +)$ contain a continuum-size family of dp-minimal structures such that no two members define the same subs…

math.LO2017

A tetrachotomy for expansions of the real ordered additive group

Philipp Hieronymi, Erik Walsberg

Let $R$ be an expansion of the ordered real additive group. When $R$ is o-minimal, it is known that either $R$ defines an ordered field isomorphic to…

Building trees in large fields

When is the étale open topology a field topology?

Interpolative fusions II: Preservation results

The étale open topology over the fraction field of a henselian local domain

Topological proofs of results on large fields

Éz fields

Notes on trace equivalence

Galois groups of large fields with simple theory (with an appendix by Philip Dittmann)

The étale-open topology and the stable fields conjecture

A P-adic structure which does not interpret an infinite field but whose Shelah completion does

Nippy proofs of p-adic results of Delon and Yao

Dp-minimal expansions of (Z,+) via dense pairs via Mordell-Lang

Generalizing a theorem of Bès and Choffrut

Directed sets and topological spaces definable in o-minimal structures

An NIP structure which does not interpret an infinite group but whose Shelah expansion interprets an infinite field

Externally definable quotients and NIP expansions of the real ordered additive group

Dp and other minimalities

Dp-minimal expansions of discrete ordered abelian groups

Coarse dimension and definable sets in expansions of the ordered real vector space

Continuous Regular Functions

Interpolative Fusions I

Expansions of the real field by discrete subgroups of Gln​(C)

A family of dp-minimal expansions of (Z;+)

A tetrachotomy for expansions of the real ordered additive group

Dp-minimal expansions of $(Z, +)$ via dense pairs via Mordell-Lang

Expansions of the real field by discrete subgroups of Gl $_{n} (C)$

A family of dp-minimal expansions of $(Z; +)$