A bstract The Distance Conjecture states that an infinite tower of modes becomes exponentially light when approaching distance point in field space. We argue the inherent path-dependence this statement can be addressed combining with recent Tameness Conjecture. latter asserts effective theories are described by tame geometry and implements strong finiteness constraints on coupling functions spa...

In this paper we study a specific subclass of abstract elementary classes. We construct a notion of independence for these AEC’s and show that under simplicity the notion has all the usual properties of first order non-forking over complete types. Our approach generalizes the context of א0 -stable homogeneous classes and excellent classes. Our set of assumptions follow from disjoint amalgamatio...

In this survey we present the criterion for tameness of strongly simply connected algebras due to Brustle, de la Pena and Skowronski. We recall relevant concepts representation theory discuss some applications connections other problems.

We show that any finitely generated metabelian group can be embedded in a metabelian group of type F3. More generally, we prove that if n is a positive integer and Q is a finitely generated abelian group, then any finitely generated ZQ-module can be embedded in a module that is n-tame. Combining with standard facts, the F3 embedding theorem follows from this and a recent theorem of R. Bieri and...

We introduce a generalized notion of finiteness that provides structural principle for the set effective theories can be consistently coupled to quantum gravity. More concretely, we propose Tameness Conjecture states all valid are labelled by definable parameter space and must have scalar field spaces coupling functions using tame geometry built from an o-minimal structure. give brief introduct...

We show, assuming a mild set-theoretic hypothesis, that if an abstract elementary class (AEC) has a superstable-like forking notion for models of cardinality λ and a superstable-like forking notion for models of cardinality λ+, then orbital types over models of cardinality λ+ are determined by their restrictions to submodels of cardinality λ. By a superstable-like forking notion, we mean here a...

Fixing a complete Riemannian metric g on Rn, we show that a local diffeomorphism f : Rn → Rn is bijective if the height function f · v (standard inner product) satisfies the Palais-Smale condition relative to g for each for each nonzero v ∈ Rn. Our method substantially improves a global inverse function theorem of Hadamard. In the context of polynomial maps, we obtain new criteria for invertibi...

This paper discusses applications of LERF to tame covers of certain hyperbolic 3-manifolds. For example, if M = H/Γ is an orientable noncompact finite volume hyperbolic 3-manifold and Γ is LERF, then the cover of M corresponding to any subgroup of Γ generated by two parabolic elements is tame. 2000 Mathematics Subject Classification 57M50, 20H10