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# Topology

DMS Topology Seminar
Jan 27, 2023 01:00 PM
224 Parker Hall

Speaker: Will Brian, University of North Carolina, Charlotte

Title: Large metric spaces and partitions of the real line into Borel setsAbstract: I will sketch a proof that, assuming $0^dagger$ does not exist, if there is a partition of $R$ into $ℵ_ω$ Borel sets, then there is also a partition of $R$ into $ℵ_{ω+1}$ Borel sets. (And the same is true for any singular cardinal of countable cofinality in place of $ℵ_ω$.) This contrasts starkly with the situation for cardinals with uncountable cofinality and their successors, where the spectrum of possible sizes of partitions of R into Borel sets can (via forcing) be made completely arbitrary. The proof of this fact for $ℵ_ω$ uses the structure of a certain complete metric space of weight $ℵ_ω$, and the existence of a particular partition of that space into Polish spaces.

DMS Topology Seminar
Jan 18, 2023 01:00 PM
224 Parker Hall

Speaker: Arka Banerjee Title: Coarse cohomology of the complementAbstract: John Roe defined the notion of  Coarse cohomology of a metric space that measures the behavior at infinity of a space: more specifically, it measures the way in which uniformly large bounded sets fit together. In my talk I will give a brief introduction to this theory and define a new notion called "Coarse cohomology of the complement." Time permitting, I will discuss some related results and applications.

This talk is partly based on a joint work with Boris Okun.

DMS Topology Seminar
Nov 18, 2022 02:00 PM
224 Parker Hall and ZOOM

Speaker: Steven Clontz, University of South AlabamaTitle: Metrizability of Mahavier products indexed by partial orders

Abstract: Let $X$ be separable metrizable, and let $f\subseteq X^2$ be a non-trivial relation on $X$. For a given partial order $(P,\leq)$, the Mahavier product $M(X,f,P)\subseteq X^P$ (also known as a generalized inverse limit) collects functions such that $x(p)\in f(x(q))$ for all $p<q$. We will show that whenever $f$ satisfies condition $\Gamma$, $M(X,f,P)$ is separable metrizable if and only if $P$ is countable.

DMS Topology Seminar
Nov 11, 2022 02:00 PM
ZOOM

Speaker: Wlodek Kuperberg Title: Packing convex bodies in the plane and in space
DMS Topology Seminar
Nov 04, 2022 02:00 PM
224 Parker Hall and ZOOM

PLEASE note NEW DATE

Speaker: Michel Smith

Title: Inverse limits on Hausdorff arcs that are hereditarily indecomposable are metric

Abstract: I will conclude my argument that a hereditarily indecomposable inverse limit of Hausdorff arcs is metric.  Note that a Hausdorff arc is a linearly ordered compact connected Hausdorff space and so is not necessarily metric.

DMS Topology Seminar
Oct 21, 2022 02:00 PM
224 Parker Hall and ZOOM

Speaker: Michel Smith

Title: Inverse limits on Hausdorff arcs that are hereditarily indecomposable are metric

Abstract: A Hausdorff arc is a compact Hausdorff continuum with exactly two noncut points.   If $$X$$ is the inverse limit space of  Hausdorff arcs and $$M$$ is a hereditarily indecomposable subcontinuum of $$X$$, then $$M$$ is a metric continuum.

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

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 Set Theoretic Topology Seminar
Mar 06, 2020 02:00 PM
Parker Hall 246

Speaker: Professor Vladimir Tkachuk

Title: Character and tightness are discretely reflexive in compact spaces.

Abstract: We will show that tightness, character, sequentiality, and Frechet-Urysohn property are discretely reflexive in compact spaces, i.e., if $$X$$ is a compact space in which the closure of every discrete subset has a property $$P$$ from the list  $$tightness \leq \kappa, character\leq\kappa, sequentiality, Frechet-Urysohn property$$, then $$X$$ has the property $$P$$.

Last Updated: 01/17/2023