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# Topology - Continuum Theory

**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.

**Continuum Theory Seminar**

Feb 08, 2016 04:00 PM

Parker Hall 224

**Michel Smith**will continue (and possibly conclude) his presentation on the non-metric pseudo-arc

**Continuum Theory Seminar**

Feb 01, 2016 04:00 PM

Parker Hall 224

Speaker: **Michel Smith**

Topic: some properties of the non-metric pseudo-arc relevant to the Wood’s Conjecture

**Continuum Theory Seminar**

Jan 25, 2016 04:00 PM

Parker Hall 224

Speakers: **Michel Smith**

Topic: joint work with Jan Boronski--pseudo-circle and non-metric pseudo-arc and counter examples to the Wood's conjecture

**Joint Continuum Theory and Set Theoretic Topology Seminar**

Sep 28, 2015 04:00 PM

Parker Hall 247 +/-1

Speaker: David Lipham

Title: 2^c different subcontinua of (C(H*) (where H* = βH - H) none of which are in H* (and where C(X) is the hyperspace of subcontinua of X.)

Last Updated: 09/11/2015