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# Topology - Continuum Theory

DMS Continuum Theory/Set Theoretic Topology Seminar
Apr 10, 2019 02:00 PM
Parker Hall 246

Speaker: Alexander Shibakov (flying his plane from Tennessee)

Title: Precompact groups and convergence.

Abstract: We will consider the question of the existence of precompact sequential group topologies on countable groups and show that such topologies fail to exist in many cases, answering some questions of D. Dikranjan. We will also give a (consistent) example of a "barely Fréchet" countably compact space disproving a conjecture of D. Shakhmatov from 1990.

DMS Continuum Theory/Set Theoretic Topology Seminar
Apr 03, 2019 02:00 PM
Parker Hall 246

Speaker: David Lipham

Abstract: We construct a connected space with a dispersion point which has a basis of open sets with discrete boundaries.  By a result of Tymchatyn, it embeds into a rational continuum.  A continuum is rational if it has a basis of open sets with countable boundaries.  This should answer a question of Grispolakis & Tymchatyn (c. 1975/1977). The example generates an indecomposable connected set which also embeds into a rational continuum.  On the other hand, we prove that no indecomposable semicontinuum can be embedded into a rational continuum.”

DMS Set Theoretic Topology/Continuum Theory Seminar
Feb 20, 2019 02:00 PM
Parker Hall 246

Speaker: Michel Smith will continue

Abstract: We prove that if $$X$$ is an inverse limit of Hausdorff arcs and $$M \subset X$$  is a hereditarily indecomposable continuum, then $$M$$ is a metric continuum.  We also provide an example to argue that, unlike in the metric situation, there exist continua which are inverse limits of Hausdorff arcs which cannot be embedded in the product of two Hausdorff arcs.  This implies that the inverse limit situation needs to be considered separately from products of two Hausdorff arcs.

DMS Set Theoretic Topology/Continuum TheorySeminar
Feb 13, 2019 02:00 PM
Parker Hall 246

Speaker: Michel Smith

Abstract: We prove that if $$X$$ is an inverse limit of Hausdorff arcs and $$M \subset X$$  is a hereditarily indecomposable continuum, then $$M$$ is a metric continuum.  We also provide an example to argue that, unlike in the metric situation, there exist continua which are inverse limits of Hausdorff arcs which cannot be embedded in the product of two Hausdorff arcs.  This implies that the inverse limit situation needs to be considered separately from products of two Hausdorff arcs.

DMS Continuum Theory Seminar
Oct 03, 2018 02:00 PM
Parker Hall 328

Speaker: David Lipham will continue speaking on last week's topic.

Abstract.  I plan to talk about "Singularities of meager composants and filament composants" in metric continua.  Given a continuum $$Y$$ and a point $$x$$ in $$Y$$,

• ​the meager composant of $$x$$ in $$Y$$ is the union of all nowhere dense subcontinua $$Y$$ containing $$x$$;
• the filament composant of $$x$$ in $$Y$$ is the union of all filament subcontinua of $$Y$$ containing $$x$$ (a subcontinuum $$L$$ is filament if there is a neighborhood of $$L$$ in which the component of $$L$$ is nowhere dense); and
• a meager/filament composant $$P$$ is said to be singular if there exists $$y$$ in $$Y-P$$ such that every connected subset of $$P$$ limiting to $$y$$ has closure equal to $$P$$ ($$y$$ is called a singularity of $$P$$).

To avoid trivial singularities, I will usually ​assume $$P$$ is dense in $$Y$$.

I will prove that each singular dense meager composant of a continuum $$Y$$ is homeomorphic to a traditional composant of an indecomposable continuum, even though $$Y$$ may be decomposable.  If $$Y$$ is homogeneous and has singular dense meager or filament composants, then I conjecture $$Y$$ must be indecomposable (based on some partial results in this direction).

DMS Continuum Theory Seminar
Sep 26, 2018 02:00 PM
Parker Hall 328

Speaker: David Lipham

Abstract.  I plan to talk about "Singularities of meager composants and filament composants" in metric continua.  Given a continuum $$Y$$ and a point $$x$$ in $$Y$$,

• ​the meager composant of $$x$$ in $$Y$$ is the union of all nowhere dense subcontinua $$Y$$ containing $$x$$;
• the filament composant of $$x$$ in $$Y$$ is the union of all filament subcontinua of $$Y$$ containing $$x$$ (a subcontinuum $$L$$ is filament if there is a neighborhood of $$L$$ in which the component of $$L$$ is nowhere dense); and
• a meager/filament composant $$P$$ is said to be singular if there exists $$y$$ in $$Y-P$$ such that every connected subset of $$P$$ limiting to $$y$$ has closure equal to $$P$$ ($$y$$ is called a singularity of $$P$$).

To avoid trivial singularities, I will usually ​assume $$P$$ is dense in $$Y$$.

I will prove that each singular dense meager composant of a continuum $$Y$$ is homeomorphic to a traditional composant of an indecomposable continuum, even though $$Y$$ may be decomposable.  If $$Y$$ is homogeneous and has singular dense meager or filament composants, then I conjecture $$Y$$ must be indecomposable (based on some partial results in this direction).

DMS Continuum Theory Seminar
Mar 19, 2018 05:00 PM
Parker Hall 228

Speaker: Benjamin Vejnar (Charles University, Prague)

Topic: The complexity of the homeomorphism equivalence relation on some classes of metrizable compacta with respect to Borel reducibility.

DMS Topology Seminar
Jan 31, 2018 02:00 PM
Parker Hall 246

Speaker: Stu Baldwin

Title: Inverse Limits of Flexagons

Abstract: Flexagons were first introduced in 1939 by Arthur H. Stone when he was a graduate student at Princeton, and they were popularized by Martin Gardner in the December 1956 issue of Scientific American in an article entitled "Flexagons" which launched his well known "Mathematical Games" column, which appeared in that magazine for many years.  By folding strips of paper into various geometrical shapes, Stone created a variety of different flexagons, of which the most elegant are the "hexaflexagons" created by folding strips of equilateral triangles into a hexagonal shape and attaching the ends.

Mathematical studies of flexagons have concentrated on the combinatorial properties of flexagons created with a finite number of polygons.  Here, we consider an infinite version which can be created either using inverse limits or nested intersections of solid tori (viewed as a folded annulus cross the unit interval). If $n \ge 3$, then a strip of $3n$ equilateral triangles can be folded into a hexaflexagon which (after the ends are identified) is topologically an annulus if $n$ is even and a Möbius strip if $n$ is odd.  Of these, the most natural ones are created using $9(2^n)$ triangles, leading to the construction of a space (via either inverse limits or nested intersections) which (viewed as a subset of $\mathbb{R}^3$ in a natural way) mimics the properties of all of the hexaflexagons having finitely many triangles.  Some preliminary results on the properties of this space will be discussed.

(Paper toys will be provided to the audience as visual aids.)

Continuum Theory Seminar
Feb 22, 2016 04:00 PM
Parker Hall 224

Piotr Minc will continue today talking about compactifications of the ray.
Continuum Theory Seminar
Feb 15, 2016 04:00 PM
Parker Hall 224

Piotr Minc will talk today about the work he’s done on certain compactifications for METRIC spaces.

Last Updated: 09/11/2015