## Ben De Smet (Leeds)

Location: MALL

Title: The Whitney embedding theorems and o-minimality

The Whitney embedding theorems (95%) and o-minimality (5%).

Models (respectively Sets) is a weekly seminar of model theorists (respectively set theorists) in Leeds, that aims to foster collaboration and engagement in each other's research. Roughly twice a term, the two groups will meet together for a two-hour joint seminar. Please contact Mervyn Tong at mmhwmt (at) leeds.ac.uk if you have any questions.

Time and place: MALL 1, Wednesday 13.00 - 14.00 (Sets) and 14.00 - 15.00 (Models)

Current organisers: Hope Duncan (Sets) and Mervyn Tong (Models)

Results 1 to 10 of 94

Location: MALL

Title: The Whitney embedding theorems and o-minimality

The Whitney embedding theorems (95%) and o-minimality (5%).

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.

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: On stable and distal expansions of $(\mathbb{Z}, +)$ and $(\mathbb{Z}, <, +)$ by a unary predicate

Proper expansions of $(\mathbb{Z}, +)$ (respectively $(\mathbb{Z}, <, +)$) by a unary predicate have proven to be a rich source of interesting stable (respectively distal) structures. Precisely which predicates give a stable (respectively distal) expansion remains an open question that, not only is of independent interest, but also has the potential to provide counterexamples to open problems on model-theoretic dividing lines. In this workshop, we survey some results that provide a partial answer to this question and discuss strategies to tackle this and adjacent open problems. In particular, we will discuss the existence of non-distal NIP expansions of $(\mathbb{N}, +)$, and we will consider a candidate counterexample to the conjecture that a structure has a distal expansion if and only if every formula has the strong Erdős-Hajnal property.

Location: MALL

Title: Almost free abelian groups and anti-compactness principles

An abelian group is almost free if any subgroup of smaller cardinality is free. We review some theorems mainly by Magidor and Shelah centered around the question of when almost free abelian groups are free. This also serves as an introduction to some anti-compactness principles such as squares.

Location: MALL

Title: Transfer of NIP in finitely ramified henselian valued fields

By a technique of Jahnke and Simon, itself building on work of Chernikov and Hils, we investigate the transfer of NIP from the residue field to the valued field, in the case of a finitely ramified henselian valued field of mixed characteristic. This relies on understanding the complete theories of such fields, on a certain Ax--Kochen/Ershov principle, and the stable embeddedness of the residue field. This talk will touch on joint work with Franziska Jahnke, and another project additionally with Philip Dittmann.

Location: MALL

Title: The space of types with a spectral topology

NOTE time change: 4 pm GMT

Influenced by results in real algebraic geometry, Pillay pointed out in 1988 that the space of types of an o-minimal expansion of a real closed field admits a spectral topology. With this topology, this space is quasi-compact and $T_0$, yet not Hausdorff. Nonetheless, the subspace of all closed points turns out to be quasi-compact and Hausdorff. The purpose of this talk is to present some results on the spectral space of types for o-minimal and more generally NIP theories. This is joint work with Elías Baro and José Fernando.

Location: MALL

Title: Small ordered theories with a maximum spectrum of countable models

For theories of totally ordered structures, we consider questions concerning the number of non-isomorphic countable models. The approach is to consider realisations of 1-types, and formulas acting on the set of all realisations of a type.