Submissions (2)
Accepted (2):
Embeddings of tree-like continua in the plane — Logan Hoehn <loganh@nipissingu.ca>
There are a number of interesting open problems in continuum theory that hinge on determining which tree-like continua can be embedded in the plane. Up to now, there are very few techniques available to show that a given tree-like continuum cannot be embedded in the plane. However, for a special class of tree-like continua, those which are inverse limits of simplicial inverse systems of trees, there is some hope that an algorithm may exist for checking planarity. I will describe this state of affairs, and pose some questions and computational challenges. At the same time, recent results are revealing that more tree-like continua can be embedded in the plane than perhaps were expected. I will discuss two such results: 1) Suppose $Y$ is any continuum of the form $Y = X \cup R$, where $X$ is an arc-like continuum, $R$ is a ray, $X \cap R = \emptyset$, and $\overline{R} \setminus R \subseteq X$. Then $Y$ can be embedded in the plane. 2) Suppose $Y$ is any continuum of the form $Y = K \cup \bigcup_{n=1}^\infty A_n$, where $K$ is a Knaster continuum and $\{A_n: n = 1,2,\ldots\}$ is a family of pairwise disjoint arcs, each intersecting $K$ in a single point, with $\mathrm{diam} A_n \to 0$. Then $Y$ can be embedded in the plane. This is joint work with Andrea Ammerlaan and Ana Anušić.
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Non-existence of common models for certain classes of continua — Eiichi Matsuhashi <matsuhashi@riko.shimane-u.ac.jp>
A continuum $X$ is called a \emph{common model} for a class $\mathcal{C}$ of continua if every member of $\mathcal{C}$ is a continuous image of $X$. One of the natural questions in continuum theory is whether a given class of continua admits a common model, and if not, how the non-existence of common models can be established. In this talk, we will discuss several recent results concerning the non-existence of common models for classes of continua arising in hyperspace theory and the theory of indecomposable continua. The main tool is a recent theorem on meandering continua, which provides a general method for establishing non-existence results. As applications, we will present new classes of continua associated with Whitney properties and Whitney reversible properties, together with several classes related to indecomposable continua, and show that these classes do not admit common models.
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