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Topology - Continuum Theory


DMS Topology Seminar
Sep 23, 2022 02:00 PM
224 Parker Hall and ZOOM


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Speaker: Henry Adams Title:  An introduction to Vietoris-Rips complexesAbstract: I will give an introduction to Vietoris-Rips complexes, Vietoris-Rips metric thickenings, and their uses in applied and computational topology. So, this talk will provide a complementary perspective to Krystyna Kuperberg's talk last week, though attendance last week is not assumed, since I wasn't there! If a dataset is sampled from some unknown underlying space (say a manifold), then as more and more samples are drawn, the Vietoris-Rips persistent homology of the dataset converges to the Vietoris-Rips persistent homology of the manifold. But little is known about Vietoris-Rips complexes of manifolds. An exception is the case of the circle: I will describe how as the scale parameter increases, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, ..., until finally they are contractible. Much less is known about Vietoris-Rips complexes of spheres. Throughout my talk, I'll advertise some open questions.


DMS Topology Seminar
Sep 16, 2022 02:00 PM
224 Parker Hall


Speaker: Dr. Krystyna Kuperberg 

Title: The Vietoris complexAbstract:  In 1927, Leopold Vietoris defined homology groups for arbitrary compact metric spaces. Soon after that Eduard Čech extended the definition to non-metric spaces. The chain complex defined by Vietoris gained a renewed attention some 60 years later in the work of Mikhael  (Misha) Gromov and became known as the Vietoris-Rips complex. The Vietoris complex is better suitable for applications than the Čech complex.I shall give the basic definitions of the Vietoris complex. No prior knowledge is required, although it might be helpful to know the very basics of simplicial homology for finite complexes, i.e., compact polyhedra in the Euclidean space.


DMS Topology Seminar
Sep 09, 2022 02:00 PM
224 Parker Hall


Speaker: Hannah Alpert Title: Quantum error-correcting codes and systolic freedom of manifolds

Abstract: Whenever we have homology with mod 2 coefficients, we can turn it into a quantum error-correcting code.  The question, "How good is the code?" is a question of how many cells of our manifold are needed to form a homology class.


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.

 


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Last Updated: 09/11/2015