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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.)


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Last Updated: 09/11/2015