A difference field is a field equipped with a given automorphism and a difference variety is the natural analogue of an algebraic varieties in this setting. Complex numbers with complex conjugation or finite fields with the Frobenius automorphism are natural examples of difference fields.
For finite fields and varieties over them, the celebrated Lang-Weil estimate gives a universal estimate of number of rational points of varieties over finite fields in terms of several notions of the complexities of the given variety. In this talk, we will discuss an analogue to Lang-Weil estimate for difference varieties in finite difference fields. The proof uses pseudofinite difference fields, where the automorphism is the nonstandard Frobenius. This is joint work with Martin Hils, Ehud Hrushovski and Tingxiang Zou.
An infinite series of real numbers is conditionally convergent if it converges, but the sums of the positive and of the negative terms are both divergent. How many infinite subsets of the naturals are necessary such that every conditionally convergent series has a subseries given by one of our infinite subsets that is divergent? The answer to this question is known as the subseries number ß, and was isolated as a cardinal characteristic of the continuum by Brendle, Brian and Hamkins.
I will present the Kinna-Wagner principles, which are weakenings of the Axiom of Choice and give a proof to what has so far been only a conjecture, which shows how these principles relate to the structure of intermediate models of ZF between a ground model and its forcing extension.
The intermediate model theorem states that whenever G is generic over V and V ⊆ M ⊆ V[G] are models of ZFC, then M is also a forcing extension of V . Unfortunately, this fails completely if we only assume ZF instead. Can more can be said? The goal of our talk is to present a generalization of the above theorem that works for ZF and talk about some of the recent progress made in the theory of symmetric extensions. This is joint with A. Karagila.
CW complexes are topological spaces built up dimension by dimension from Euclidean cells, with a subset declared to be open if its intersection with each of these cells is open. Unfortunately when you take the product of two CW complexes, the product topology does not in general satisfy this requirement. I will explain when exactly it does; it turns out that it depends on the cardinal $\mathfrak{b}$. For the old hands who have seen this talk multiple times before, there will also be something new, with details that I realised last week I ought to draw out more.
Lie algebras have an interesting relationship with model companions: Whether a given theory has a model companions depends nontrivially on the chosen language and restrictions on the Lie algebras. We will discuss several of these results, and how they can be applied to answer a question of Mennuni.
Martin's Conjecture is a proposed classification of Turing-invariant functions under the Axiom of Determinacy. Whether the classification holds for the ostensibly smaller class of order-preserving functions is open, but more tractable. In this talk, we’ll explain an approach to proving Martin’s Conjecture for order-preserving functions and discuss how far we can go. This is joint work with Patrick Lutz.
Henselian valued fields are a class of structures whose model theory has been much investigated. After an introduction to this area, I will present recent joint work with Anscombe and Jahnke in the finitely ramified setting. Prior knowledge of valued fields will not be assumed.
In this talk, we consider how much non-constructive principles are sufficient for Friedberg-Muchinik construction of degree $d$ such that $0<d<0'$. We will see that the only point we need a non-constructive principle is to show "if a recursive set $S$ of natural number has finite cardinality, then $S$ has an upper bound", which requires $\Sigma^0_1$ law of excluded middle.
Let $\mathsf{M}$ be the weak set theory (with powersets) axiomatised by: $\textsf{Extensionality}$, $\textsf{Pair}$, $\textsf{Union}$, $\textsf{Infinity}$, $\textsf{Powerset}$, transitive containment ($\textsf{TCo}$), $\Delta_0\textsf{-Separation}$ and $\textsf{Set-Foundation}$. In this talk I will discuss the relationship between two alternative versions of the set-theoretic collection scheme: $\textsf{Collection}$ and $\textsf{Strong Collection}$. Both of these schemes yield $\mathsf{ZF}$ when added to $\mathsf{M}$, but when restricted the $\Pi_n$-formulae (denoted $\Pi_n\textsf{-Collection}$ and $\textsf{Strong } \Pi_n\textsf{-Collection}$) these alternative versions of set-theoretic collection differ. In particular, over the theory $\mathsf{M}$, $\textsf{Strong }\Pi_n\textsf{-Collecton}$ is equivalent to $\Pi_n\textsf{-Collection}+\Sigma_{n+1}\textsf{-Separation}$. And, $\mathsf{M}+\textsf{Strong }\Pi_n\textsf{-Collection}$ proves the consistency of $\mathsf{M}+\Pi_n\textsf{-Collection}$. In this talk I will show that, despite this difference in consistency strength, every countable well-founded model of $\mathsf{M}+\Pi_n\textsf{-Collection}$ satisfies $\textsf{Strong } \Pi_n\textsf{-Collection}$. If time permits I will outline how this argument can be refined to show that $\mathsf{M}+\Pi_n\textsf{-Collection}+\Pi_{n+1}\textsf{-Foundation}$ proves $\Sigma_{n+1}\textsf{-Separation}$.
