In this document we prove: Let $\mathbb K=(K,+,\cdot,v,\Gamma)$ be an algebraically closed valued field and let $(G,\oplus)$ be a $\mathbb K$-definable group that is either the multiplicative group or contains …
In this document we prove: Let $\mathbb K=(K,+,\cdot,v,\Gamma)$ be an algebraically closed valued field and let $(G,\oplus)$ be a $\mathbb K$-definable group that is either the multiplicative group or contains a finite index subgroup that is $\mathbb K$-definably isomorphic to a $\mathbb K$-definable subgroup of $(K,+)$. Then if $\mathcal G=(G,\oplus,\ldots)$ is a strongly minimal non locally modular structure definable in $\mathbb K$ and expanding $(G,\oplus)$, it interprets an infinite field. This document is the PhD thesis of the author and it was advised by professors Assaf Hasson and Alf Onshuus.
We prove that if a strongly minimal non-locally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.
We prove that if a strongly minimal non-locally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.
We prove that if a strongly minimal non-locally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.
We prove that if a strongly minimal non-locally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.
Let $\mathbb K=(K,+,\cdot,v,\Gamma)$ be a valued algebraically closed field of characteristic and $(G,\oplus)$ be a $\mathcal K$-interpretable group that is either locally isomorphic to $(K,+)$ or to $(K,\cdot)$. Then if …
Let $\mathbb K=(K,+,\cdot,v,\Gamma)$ be a valued algebraically closed field of characteristic and $(G,\oplus)$ be a $\mathcal K$-interpretable group that is either locally isomorphic to $(K,+)$ or to $(K,\cdot)$. Then if $\mathcal G=(G,\oplus,\ldots)$ is a strongly minimal non locally modular structure intepretable in $\mathbb K$, it interprets a field. We also present an strategy for proving the same without the assumption of having a definable group operation.
Abstract We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying …
Abstract We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.
We study $\mu$-stabilizers for groups definable in ACVF in the valued field sort. We prove that $\mathrm{Stab}^\mu(p)$ is a infinite definable subgroup of $G$ when $p$ is standard and unbounded. …
We study $\mu$-stabilizers for groups definable in ACVF in the valued field sort. We prove that $\mathrm{Stab}^\mu(p)$ is a infinite definable subgroup of $G$ when $p$ is standard and unbounded. In the particular case when $G$ is linear algebraic, we show that $\mathrm{Stab}^\mu(p)$ is a solvable algebraic subgroup of $G$, with $\mathrm{dim}(\mathrm{Stab}^\mu(p))=\mathrm{dim}(p)$ when $p$ is $\mu$-reduced and unbounded.
We study $\mu$-stabilizers for groups definable in ACVF in the valued field sort. We prove that $\mathrm{Stab}^\mu(p)$ is an infinite unbounded definable subgroup of $G$ when $p$ is standard and …
We study $\mu$-stabilizers for groups definable in ACVF in the valued field sort. We prove that $\mathrm{Stab}^\mu(p)$ is an infinite unbounded definable subgroup of $G$ when $p$ is standard and unbounded. In the particular case when $G$ is linear algebraic, we show that $\mathrm{Stab}^\mu(p)$ is a solvable algebraic subgroup of $G$, with $\mathrm{dim}(\mathrm{Stab}^\mu(p))=\mathrm{dim}(p)$ when $p$ is $\mu$-reduced and unbounded.
Let $\mathcal{K}:=(K;+, \cdot, D)$ be a differentially closed field with constant field $C$. Let also $E_j(x,y)$ be the differential equation of the the $j$-function. We prove a Zilber style classification …
Let $\mathcal{K}:=(K;+, \cdot, D)$ be a differentially closed field with constant field $C$. Let also $E_j(x,y)$ be the differential equation of the the $j$-function. We prove a Zilber style classification result for strongly minimal sets in the reduct $\mathcal{K}_{E_j}:=(K;+, \cdot, E_j)$ assuming an Existential Closedness (EC) conjecture for $E_j$. More precisely, assuming EC we show that in $\mathcal{K}_{E_j}$ all strongly minimal sets are geometrically trivial or non-orthogonal to $C$. The Ax-Schanuel inequality for the $j$-function and its adequacy play a crucial role in this classification.
We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation></mml:semantics></mml:math></inline-formula>is any non-locally modular …
We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation></mml:semantics></mml:math></inline-formula>is any non-locally modular strongly minimal structure interpreted in an algebraically closed field<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>of characteristic zero, then<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation></mml:semantics></mml:math></inline-formula>itself interprets<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>; in particular, any non-1-based structure interpreted in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>is mutually interpretable with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>. Notably, we treat both the ‘one-dimensional’ and ‘higher-dimensional’ cases of the conjecture, introducing new tools to resolve the higher-dimensional case and then using the same tools to recover the previously known one-dimensional case.