Models and Sets Seminar

Location: MALL
Title: Strong Existential Closedness

NOTE time change: 3.00 PM

The strong existential closedness problem was introduced in Zilber's work on pseudoexponentiation. Since then, it has been naturally adapted to many situations in arithmetic geometry. In this talk I will introduce the problem, review some important Diophantine questions that are connected to it, and discuss some of the known results.

Speaker's homepage

Location: MALL
Title: Cardinal characteristics modulo nice ideals on $\omega$

Many of the standard cardinal characteristics of the continuum are defined in terms of a relation holding almost everywhere, where "almost everywhere" means on all but a finite set. A very natural generalisation is to take "almost everywhere" to mean on all but a member of a given ideal. I will talk about what happens when we do this, with the density 0 ideal on $\omega$ as a focal example.

Speaker's homepage

Location: Roger Stevens LT 08 (9.08)
Title: Interaction of determinacy and forcing

NOTE room change: Roger Stevens LT 08 (9.08)

Determinacy principles provide a unified theory of definable sets of reals beyond Borel and analytic sets, while forcing is an important technique to study the independence of properties of sets of reals. This suggests studying the interaction of the two: how robust are determinacy principles under well behaved forcings? I will talk about the history of this problem as well as recent joint results with Jonathan Schilhan and Johannes Schürz on iterations of proper forcings. A sample application of our results is the following: starting from a model of analytic determinacy, one can construct a model of analytic determinacy and the Borel conjecture.

Speaker's homepage

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.

Speaker's homepage

Location: MALL
Title: Shattering Domination

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 $\mathsf{AD}$.


This is joint work with Nam Trang.

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: Weak Essentially Undecidable Theories of Concatenation

Speaker's homepage



We sketch a proof of mutual interpretability of Robinson arithmetic and a weak finitely axiomatized theory of concatenation.

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.