Archive: all, 2023, 2022, 2021, 2020
Results 11 to 20 of 159
Location: MALL
Title: Defining definable compactness
Can topological compactness be expressed as a first-order property within tame topology? Let's find out. In this talk I will present various attempts in the literature to capture this notion. We will go over the model theory behind them and present open questions.
Location: MALL
Title: Finality of forcing
Iterated forcing is a powerful tool for ouroboric arguments in set theory that rely on repeatedly creating or destroying some property until your construction eats its own tail and gives you your final result (in fact a similar argument may be applied to many ideas in set theory, especially when ordinals are involved. A simple example would be the \omega_1th stage of the Borel/projective hierarchies being no more than the union of their prior stages). To this end, it is often a helpful feature of an iterated notion of forcing that in the final model one has not introduced any new reals (subsets of \kappa, functions \lambda\to\lambda, etc) that are not already present in some intermediate stage. This behaviour is precisely captured by finality, which we shall define and give an exact characterisation of.
Location: MALL
Title: Definable groups in valued fields
I will discuss joint work with Gismatullin and Halupczok, giving the structure of definably (almost) simple groups definable in Henselian valued fields, possibly equipped with extra structure. I will also describe some other work on definable groups in valued fields.
Location: MALL
Title: Omega-categorical pseudofinite groups
I will discuss recent joint work with Katrin Tent on omega-categorical groups which are pseudofinite, i.e. satisfy every first order sentence which is true of all finite groups. It is fairly easy to show that they are nilpotent-by-finite, and we conjecture that they are finite-by-abelian-by-finite, and can reduce this to the nilpotent class 2 case. We show that certain class 2 groups constructed by amalgamation are NOT pseudofinite – in particular there is an example with supersimple theory which is not pseudofinite.
Location: MALL
Title: Eventually different, refining and dominating families at the uncountable
NOTE: the speaker will be joining us online.
We will discuss some recent results, including ZFC inequalities, concerning the higher Baire spaces analogues of some of the classical combinatorial cardinal characteristics of the continuum.
Of special interest for the talk will be the generalized bounding number, relatives of the generalized almost disjointness number, as well as the generalized refining and dominating numbers.
Location: MALL
Title: The Whitney embedding theorems and o-minimality
The Whitney embedding theorems (95%) and o-minimality (5%).
Location: MALL
Title: Weakly immediate types and T-convexity
For $T$ an o-minimal theory expanding RCF, a $T$-convex valuation ring on an o-minimal expansion of a RCF is a convex subring closed under continuous $T$-definable functions. This was first defined by Van Den Dries and Leweneberg who proved that the common theory $T_{\mathrm{convex}}$ of the expansions of models of $T$ by a non-trivial $T$-convex valuation ring is complete and weakly o-minimal. One of the key properties of the valuation theory of $T_{\mathrm{convex}}$ for power bounded $T$ is the so called residue-valuation property which can be restated as saying that every model of $T_{\mathrm{convex}}$ has a spherically complete maximal immediate extension. This is known to be false if $T$ defines an exponential. The goal of the talk will be to discuss potential analogues of the residue-valuation property in the exponential context.
Location: MALL
Title: Strong almost disjointness
A collection of unbounded subsets of $\omega_1$ is strongly almost disjoint if each pairwise intersection is finite. I will present Baumgartner's thinning out technique and use it to show that under Martin's Axiom + "failure of weak CH" every $\omega_1$-mad family has an $\omega_1$-mad strongly almost disjoint refinement. My presentation will be a mix between lecture and research talk. It has come to my ears that students of set theory in Leeds have not learned about $\Delta$-systems. For this reason I will present this powerful combinatorial result which is the essence of most ccc proofs. All subject to my bad management of time.
Location: MALL
Title: Monadic second-order definability of classes of matroids
Matroids can be seen as abstractions of geometrical configurations. Every (finite) matroid consists of a (finite) set called the ground set, and a collection of distinguished subsets called the independent sets. A classic example arises when the ground set is a finite set of vectors from a vector space, and the independent subsets are exactly the subsets that are linearly independent. Any such matroid is said to be representable. We can think of a representable matroid as being a geometrical configuration where the points have been given coordinates from a field. Another important class arises when the points are given coordinates from a group. Such a class is said to be gain-graphic.Monadic second-order logic is a natural language for matroid applications. In this language we are able to quantify only over subsets of the ground set. The importance of monadic second-order logic comes from its connections to the theory of computation, as exemplified by Courcelle's Theorem. This theorem provides polynomial-time algorithms for recognising properties defined in monadic second-order logic (as long as we impose a bound on the structural complexity of the input objects).It is natural to ask which classes of matroids can be defined by sentences in monadic second-order logic. When the class consists of the matroids that are coordinatized by a field we have a complete answer to this question. When the class is coordinatized by a group the problem becomes much harder.This talk will contain a brief introduction to matroids.Based on work with Sapir Ben-Shahar, Matt Conder, Daryl Funk, Mike Newman, and Gabriel Verret.
Location: MALL
Title: Weak Threads for Ladder Systems at Inaccessible $\kappa$
"Every club sequence has a weak thread" is a compactness property that implies simutaneously stationary reflection. In this talk, we will first explore weak threads for various ladder systems. Then we show it is consistent that every club sequence has a weak thread and there exists an almost disjoint ladder system given by vanishing branches of a $\kappa$-Suslin tree. This is joint work with Assaf Rinot and Zhixing You.