Mathematics

In this note, we provide a proof of a technical result of Erdős and Hajnal about the existence of disjoint type graphs with no odd cycles. We also prove that this result is sharp in a certain sense.

We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an AKE-style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new tool for analyzing valued fields we employ a relative quantifier elimination for pure short exact sequences of abelian groups.

We develop domain theory in constructive univalent foundations without Voevodsky's resizing axioms. In previous work in this direction, we constructed the Scott model of PCF and proved its computational adequacy, based on directed complete posets (dcpos). Here we further consider algebraic and continuous dcpos, and construct Scott's D ∞ model of the untyped λ -calculus. A common approach to deal with size issues in a predicative foundation is to work with information systems or abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. Here we instead accept that dcpos may be large and work with type universes to account for this. For instance, in the Scott model of PCF, the dcpos have carriers in the second universe U 1 and suprema of directed families with indexing type in the first universe U 0 . Seeing a poset as a category in the usual way, we can say that these dcpos are large, but locally small, and have small filtered colimits. In the case of algebraic dcpos, in order to deal with size issues, we proceed mimicking the definition of accessible category. With such a definition, our construction of Scott's D ∞ again gives a large, locally small, algebraic dcpo with small directed suprema.

We investigate fields of characteristic 0 and dp-rank 2. While we do not obtain a classification, we prove that any unstable field of characteristic 0 and dp-rank 2 admits a unique definable V-topology. If this statement could be generalized to higher ranks, we would obtain the expected classification of fields of finite dp-rank. We obtain the unique definable V-topology by investigating the "canonical topology" defined in earlier work. Contrary to earlier expectations, the canonical topology need not be a V-topology. However, we are able to characterize the canonical topology (on fields of dp-rank 2 and characteristic 0) in terms of differential valued fields. This differential valued structure is obtained through a partial classification of "2-inflators," a sort of generalized valuation that arises naturally in fields of finite rank. Additionally, we give an example of a dp-rank 2 expansion of ACVF with a definable set of full rank and empty interior. This example interferes with certain strategies for proving the henselianity conjecture.

We prove that unstable dp-finite fields admit definable V-topologies. As a consequence, the henselianity conjecture for dp-finite fields implies the Shelah conjecture for dp-finite fields. This gives a conceptually simpler proof of the classification of dp-finite fields of positive characteristic. For n≥1 , we define a local class of " W n -topological fields", generalizing V-topological fields. A W 1 -topology is the same thing as a V-topology, and a W n -topology is some higher-rank analogue. If K is an unstable dp-finite field, then the canonical topology is a definable W n -topology for n=dp-rk(K) . Every W n -topology has between 1 and n coarsenings that are V-topologies. If the given W n -topology is definable in some structure, then so are the V-topological coarsenings.

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 "dense pairs" to construct uncountably many dp-minimal expansions of (Z,+,S) . These constructions are applications of the Mordell-Lang conjecture and are the first examples of "non-modular" dp-minimal expansions of (Z,+) . We canonically associate an o-minimal expansion R of (R,+,×) , an R -definable circle group H , and a character Z→H to a "non-modular" dp-minimal expansion of (Z,+,S) . We also construct a "non-modular" dp-minimal expansion of (Z,+, Val p ) from the character Z→ Z × p , k↦exp(pk) .

It is shown that every dp-minimal integral domain R is a local ring and for every non-maximal prime ideal p of R , the localization R p is a valuation ring and p R p =p . Furthermore, a dp-minimal integral domain is a valuation ring if and only if its residue field is infinite or its residue field is finite and its maximal ideal is principal.

We study dp-minimal infinite profinite groups that are equipped with a uniformly definable fundamental system of open subgroups. We show that these groups have an open subgroup A such that either A is a direct product of countably many copies of F p for some prime p , or A is of the form A≅ ∏ p Z α p p × A p where α p <ω and A p is a finite abelian p -group for each prime p . Moreover, we show that if A is of this form, then there is a fundamental system of open subgroups such that the expansion of A by this family of subgroups is dp-minimal. Our main ingredient is a quantifier elimination result for a class of valued abelian groups. We also apply it to (Z,+) and we show that if we expand (Z,+) by any chain of subgroups ( B i ) i<ω , we obtain a dp-minimal structure. This structure is distal if and only if the size of the quotients B i / B i+1 is bounded.

In the spirit of Hjorth's turbulence theory, we introduce "unbalancedness": a new dynamical obstruction to classifying orbit equivalence relations by actions of Polish groups which admit a two side invariant metric (TSI). Since abelian groups are TSI, unbalancedness can be used for identifying which classification problems cannot be solved by classical homology and cohomology theories. In terms of applications, we show that Morita equivalence of continuous-trace C ∗ -algebras, as well as isomorphism of Hermitian line bundles, are not classifiable by actions of TSI groups. In the process, we show that the Wreath product of any two non-compact subgroups of S ∞ admits an action whose orbit equivalence relation is generically ergodic against any action of a TSI group and we deduce that there is an orbit equivalence relation of a CLI group which is not classifiable by actions of TSI groups.

The o-minimal structure generated by the restricted Pfaffian functions, known as restricted sub-Pfaffian sets, admits a natural measure of complexity in terms of a format F , recording information like the number of variables and quantifiers involved in the definition of the set, and a degree D recording the degrees of the equations involved. Khovanskii and later Gabrielov and Vorobjov have established many effective estimates for the geometric complexity of sub-Pfaffian sets in terms of these parameters. It is often important in applications that these estimates are polynomial in D . Despite much research done in this area, it is still not known whether cell decomposition, the foundational operation of o-minimal geometry, preserves polynomial dependence on D . We slightly modify the usual notions of format and degree and prove that with these revised notions this does in fact hold. As one consequence we also obtain the first polynomial (in D ) upper bounds for the sum of Betti numbers of sets defined using quantified formulas in the restricted sub-Pfaffian structure.