Location: Roger Stevens LT23 (8.23)

Title: Algebraic minimality of automorphism groups of countable homogeneous structures

NOTE location change: Roger Stevens LT23 (8.23)

Permutation groups of a countable set are Hausdorff topological groups with the pointwise convergence topology. A Hausdorff topological group G is minimal if every bijective continuous homomorphism from G to another Hausdorff topological group is a homeomorphism. The Zariski topology is defined in a natural way for any group. However, a permutation group with the Zariski topology is not necessarily a topological group. When the Zariski topology is a topological group then it is minimal. In this talk we investigate the Zariski topology for the automorphism groups of some countable homogeneous structures. This is a joint work with Javier de la Nuez González.

Location: MALL 1

Title: Distality

Distality was introduced by Pierre Simon in 2012 as a property of NIP theories which captures "pure instability". It encompasses weakly o-minimal structures, the field of $p$-adics, and Presburger Arithmetic. We present the definitions of distality and distal cell decomposition. We describe recent applications of distal cell decomposition, and discuss an open problem.

Location: MALL 1

Title: M.e.c. limits of homogeneous structures

In this talk, I aim to give a glimpse into the study of m.e.c. limits, especially those of homogeneous structures. This is centred upon a conjecture of MacPherson, Steinhorn, Anscombe, and Wolf, that a homogeneous structure has a m.e.c. limit if and only if it is stable. I’ll give some background, talk through some examples and outline a novel method of tackling this question for certain unstable homogeneous structures.

Location: MALL 1

Title: Non-essential expansion of small theory and number of countable non-isomorphic models

NOTE date and time change: Thu 1.30pm

Authors: Bektur BAIZHANOV, Olzhas UMBETBAYEV, Tatyana ZAMBARNAYA

A. Let $T$ be a small theory, $p\in S(T)$ be a non-isolated type, $T_1:= T\cup p(\bar c)$ be a non-essential expansion. There are examples of small theories 1-2 such that:

$I(T, \omega)< \omega$, $I(T_1, \omega) = \omega$. (R. Woodrow, M.Peretyat'kin).

$I(T,\omega)= \omega$, $I(T_1, \omega)< \omega$ (B. Omarov)

Question. Is there a small theory such that $I(T, \omega) =2^{\omega}$, $I(T,\omega)\leq \omega$?

We discuss the usage of constant expansion on research of Vaught Conjecture.

B. Let $\mathfrak M$ be a model of a theory $T$. A finite diagram (S. Shelah) of $\mathfrak M$ is the collection of all $\emptyset$-definable complete types that are realized in $\mathfrak M$:

$$\mathcal D(\mathfrak M)=\{p\in S(T)\ |\ \mathfrak M\models p\}.$$

A dowry of a set $B\subseteq M$ is the following collection:

$$\mathcal D(B)=\{p\in S(T)\ | \text{ there exist }\mathfrak M\models T \text{ and } \bar b\in B \text{ such that } \bar b\in p(\mathfrak M)\}.$$

Conjecture. If $|\{ \mathcal D(\mathfrak M) \mid \mathfrak M\models T\}| >\omega$, then $|\{ \mathcal D(\mathfrak M) \mid \mathfrak M\models T\}| =2^{\omega}$, and, consequently, $I(T,\omega)= 2^{\omega}$.

Question. Let $P:=\{p_n\in S(T) \mid n<\omega\}$, $Q:=\{q_n\in S(T) \mid n <\omega\}$ be two families of non-isolated types such that for every $n<\omega$ there exists $\mathfrak M_n$ is realized first $n$ types from $P$ and is omitted first $n$ types from $Q$. Do there exist a countable model $\mathfrak M$ of $T$, such that it realizes each type from $P$ and omits each type from $Q$?

The criterion for the existence of such a theory will be shown.

Location: MALL 1

Title: Is o-minimality of the open-core an elementary property?

Let $M$ be a structure endowed with a dense linear order without endpoints. Its open-core is its reduced generated by the collection of all of its open definable sets (in any Cartesian power of $M$). We prove that the property of having an o-minimal open core isn't elementary. This is a joint work with Alexi Block-Gorman.

Location: Roger Stevens LT04

NOTE room & time change: Roger Stevens LT04, 2.15 PM