A structure expanding a dense linear order is weakly o-minimal if any definable set in one variable is a finite union of points and convex sets. This is a more general condition compared to regular o-minimality, where we require these sets to be a finite union of points and intervals.
As a result, weak o-minimality doesn't have all the useful properties that o-minimality has, for example it's not preserved under elementary equivalence, nor can we show that any definable function is piecewise monotone and continuous.
In this talk I'll highlight some of the nice properties o-minimal structures have, give examples of weakly o-minimal structures which break those properties, and then show how we can salvage weaker versions of the properties in a weakly o-minimal setting.
Most of this talk will be based on a paper by Dugald Macpherson, David Marker and Charles Steinhorn called "Weakly O-minimal structures and real closed fields".
Location: MALL
Title: Cohesive powers of linear orders
A cohesive power of a computable structure is an effective analog of an ultrapower, where a cohesive set acts as an ultrafilter. We compare the properties of cohesive powers to those of classical ultrapowers. In particular, we investigate what structures arise as the cohesive power of $B$ over $C$, where $B$ varies over the computable copies of some fixed computably presentable structure $A$, and $C$ varies over the cohesive sets.
Let $\omega$, $\zeta$, and $\eta$ denote the respective order-types of $(\mathbb{N}, <)$, $(\mathbb{Z}, <)$, and $(\mathbb{Q}, <)$. We take omega as our computably presentable structure, and we consider the cohesive powers of its computable copies. If L is a computable copy of omega that is computably isomorphic to the standard presentation $(\mathbb{N}, <)$, then all of $L$'s cohesive powers have order-type $\omega + (\zeta \times \eta)$, which is familiar as the order-type of countable non-standard models of PA.
We show that it is possible to realize a variety of order-types other than $\omega + (\zeta \times \eta)$ as cohesive powers of computable copies of omega. For example, we show that there is a computable copy $L$ of omega whose power by any $\Delta_2$ cohesive set has order-type $\omega + \eta$. More generally, we show that it is possible to achieve order-types of the form $\omega +$ certain shuffle sums as cohesive powers of computable linear orders of type $\omega$.
Location: MALL
Title: The small Dowker space problem
It is well-known that the product of two normal topological spaces need not be normal, but what about the normality of the product of a normal space $X$ with the unit interval $[0,1]$? A counterexample space $X$ is called a "Dowker space". In 1972, Rudin proved that such a space exists, but it remains open whether there must exist a Dowker space of size $\aleph_1$. In this talk, we shall report on a joint work with Shalev and Todorcevic in which we present a weak sufficient condition for the existence of a small Dowker space.
The Erdős-Rado arrow relation is to the order property as the shattering domination relation is to the independence property. In attempting to create a faithful translation of the independence property in a first-order theory in the setting of abstract elementary classes, the problem of having no compactness becomes clear immediately. In a logical setting involving the compactness theorem, it is easy to find 'tree-indiscernible' sequences with the same EM-type as any arbitrary tree (as in the same way one can find order-indiscernible sequences with the same EM-type as any arbitrary sequence). However, without such tools, we are left with a much more blunt weapon: taking large, extant trees and finding within them structures that just so happen to be indiscernible. The shattering domination relation (and its numerous derivatives) is an attempt to measure how blunt that weapon is, that is to say how large a tree has to be before we can find an indiscernible sub-tree of a given size. In the setting of ZFC+GCH, this is solved, but it seems likely that in ZFC+¬GCH, it is independent.
Location: MALL
Title: On preserving AD via forcings
Speaker's homepage
It is well-known that forcings preserve $\mathsf{ZFC}$, i.e., any set generic extension of any model of $\mathsf{ZFC}$ is again a model of $\mathsf{ZFC}$. How about the Axiom of Determinacy ($\mathsf{AD}$) under $\mathsf{ZF}$? It is not difficult to see that Cohen forcing always destroys $\mathsf{AD}$, i.e., any set generic extension of a model of $\mathsf{ZF}+ \mathsf{AD}$ via Cohen forcing is not a model of $\mathsf{AD}$. Actually it is open whether there is a forcing which adds a new real while preserving $\mathsf{AD}$. In this talk, we present some results on preservation & non-preservation of $\mathsf{AD}$ via forcings, whose details are as follows:
Starting with a model of $\mathsf{ZF} + \mathsf{AD}^{+} + V = L(P(R))$, any forcing increasing $\Theta$ destroys $\mathsf{AD}$.
It is consistent relative to $\mathsf{ZF} + \mathsf{AD}_R$ that $\mathsf{ZF} + \mathsf{AD}^{+} +$ There is a forcing which increases $\Theta$ while preserving $\mathsf{AD}$.
In $\mathsf{ZF}$, no forcings on the reals preserve $\mathsf{AD}$. (This is an improvement of the result of Chan and Jackson where they additionally assumed $\Theta$ is regular.)
In $\mathsf{ZF} + \mathsf{AD}^{+} + V = L(P(R)) + \Theta$ is regular, there is a forcing on $\Theta$ which adds a new subset of $\Theta$ while preserving $\mathsf{AD}$.
Location: MALL
Title: Decidability of Theories of Modules of Prüfer domains
Speaker's homepage
An integral domain is Prüfer if its localisation at each maximal ideal is a valuation domain. Many classically important rings are Prüfer domains. For instance, they include Dedekind domains and hence rings of integers of number fields; Bézout domains and hence the ring of complex entire functions and the ring of algebraic integers; the ring of integer valued polynomials with rational coefficients and the real holomorphy rings of formally real fields.
Over the last 15 years, efforts have been made to characterise when the theory of modules of (particular types of) Prüfer domains are decidable. I will give an overview of such decidability results culminating in recently obtained elementary conditions completely characterising when the theory of modules of an arbitrary Prüfer domain is decidable.
Location: MALL
Title: Diamonds, Compactness, and Global Scales
Speaker's homepage
In pursuit of an understanding of the relations between compactness and approximation principles we address the question: To what extent do compactness principles assert the existence of a diamond sequence? It is well known that a cardinal $\kappa$ that satisfies a sufficiently strong compactness assumption must also carry a diamond sequence. However, other results have shown that certain weak large cardinal assumptions are consistent with the failure of the full diamond principle. We will discuss this gap and describe recent results with Jing Zhang which connect this problem to the existence of a certain global notion of cardinal arithmetic scales.
Location: MALL
Title: Freeness and typical behavior for algebraic structures
Speaker's homepage
The talk is on joint work with Johanna Franklin and Turbo Ho. Gromov asked “What is a typical group?” He was thinking of finitely presented groups. He proposed an approach involving limiting density. In 2013, I conjectured that for elementary first order sentences $\varphi$, and for group presentations with $n$ generators ($n\geq 2$) and a single relator, the limiting density for groups satisfying $\varphi$ always exists, with value $0$ or $1$, and the value is $1$ iff $\varphi$ is true in the non-Abelian free groups. The conjecture is still open, but there are positive partial results by Kharlampovich and Sklinos, and by Coulon, Ho, and Logan. We ask Gromov's question about structures in other equational classes, or algebraic varieties in the sense of universal algebra. We give examples illustrating different possible behaviors. Focusing on languages with just finitely many unary function symbols, we prove a result with conditions sufficient to guarantee that the analogue of the conjecture holds. The proof uses a version of Gaifman's Locality Theorem, plus ideas from random group theory and probability.
Location: MALL
Title: Higher independence at regular cardinals
Speaker's homepage
In the first part of this talk I will introduce the classical concept of a maximal independent family and its main properties. The second part of the talk will be devoted to deal with the generalisation of independence for regular uncountable cardinals. I will show the differences and similarities with the classical setting, as well as new lines of research that appear when dealing with this generalisation.
Finally, I will mention some recent results of Vera Fischer and myself regarding independence.
