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BEGIN:VEVENT
SUMMARY:Alex McCearly (The Ohio State University)
DTSTART;VALUE=DATE-TIME:20210928T200000Z
DTEND;VALUE=DATE-TIME:20210928T210000Z
DTSTAMP;VALUE=DATE-TIME:20231211T001736Z
UID:TDGA/1
DESCRIPTION:Title: The
Functoriality of Persistent Homology\nby Alex McCearly (The Ohio Stat
e University) as part of Topology\, Geometry\, & Data Analysis (TGDA) Semi
nar\n\n\nAbstract\nThe pipeline that takes a filtration to its persistence
diagram is functorial: it takes morphisms of filtrations to morphisms of
persistence diagrams. We will analyze this structure\, focusing on the one
-parameter setting. We will start with an overview of the categories of fi
ltrations and persistence diagrams followed by some examples of morphisms
between filtrations arising in practice and what the induced morphisms bet
ween persistence diagrams can tell us.\n
LOCATION:https://researchseminars.org/talk/TDGA/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhengchao Wan (The Ohio State University)
DTSTART;VALUE=DATE-TIME:20211012T200000Z
DTEND;VALUE=DATE-TIME:20211012T210000Z
DTSTAMP;VALUE=DATE-TIME:20231211T001736Z
UID:TDGA/2
DESCRIPTION:Title: The
Gromov-Hausdorff distance between ultrametric spaces\nby Zhengchao Wa
n (The Ohio State University) as part of Topology\, Geometry\, & Data Anal
ysis (TGDA) Seminar\n\n\nAbstract\nThe Gromov-Hausdorff distance $(d_GH)$
is a natural distance between metric spaces. However\, computing $d_GH$ is
NP-hard\, even in the case of finite ultrametric spaces. We identify a on
e parameter family $\\{d_{GH\,p}\\}_{p\\in[1\,\\infty]}$ of Gromov-Hausdor
ff type distances on the collection of ultrametric spaces such that $d_{GH
\,1}=d_{GH}$. The extreme case when $p=\\infty$\, which we also denote by
$u_{GH}$\, turns out to be an ultrametric on the collection of ultrametric
spaces. We discuss various geometric and topological properties of $d_{GH
\,p}$ as well as some of its structural results. These structural results
in turn allow us to study the computational aspects of the distance. In pa
rticular\, we establish that (1) $u_{GH}$ is computationally tractable and
(2) when $p < \\infty$\, although $d_{GH\,p}$ is NP-hard to compute\, we
identify a fixed-parameter tractable algorithm for computing the exact val
ue of $d_{GH\,p}$ between finite ultrametric spaces.\n
LOCATION:https://researchseminars.org/talk/TDGA/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qingsong Wang (The Ohio State University)
DTSTART;VALUE=DATE-TIME:20211005T200000Z
DTEND;VALUE=DATE-TIME:20211005T210000Z
DTSTAMP;VALUE=DATE-TIME:20231211T001736Z
UID:TDGA/3
DESCRIPTION:Title: The
Persistent Topology of Optimal Transport Based Metric Thickenings\nby
Qingsong Wang (The Ohio State University) as part of Topology\, Geometry\
, & Data Analysis (TGDA) Seminar\n\n\nAbstract\nA metric thickening of a g
iven metric space X is any metric space admitting an isometric embedding o
f X. Thickenings have found use in applications of topology to data analys
is\, where one may approximate the shape of a dataset via the persistent h
omology of an increasing sequence of spaces. We introduce two new families
of metric thickenings\, the p-Vietoris–Rips and p-Čech metric thicken
ings for any p between 1 and infinity\, which include all measures on X wh
ose p-diameter or p-radius is bounded from above\, equipped with an optima
l transport metric. These families recover the previously studied Vietoris
–Rips and Čech metric thickenings when p is infinity. As our main contr
ibution\, we prove a stability theorem for the persistent homology of p-Vi
etoris–Rips and p-Čech metric thickenings\, which is novel even in the
case p is infinity. In the specific case p equals 2\, we prove a Hausmann-
type theorem for thickenings of manifolds\, and we derive the complete lis
t of homotopy types of the 2-Vietoris–Rips thickenings of the sphere as
the scale increases. This is joint work with Henry Adams\, Facundo Mémoli
and Michael Moy.\n
LOCATION:https://researchseminars.org/talk/TDGA/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mario Gómez Flores (The Ohio State University)
DTSTART;VALUE=DATE-TIME:20211109T210000Z
DTEND;VALUE=DATE-TIME:20211109T220000Z
DTSTAMP;VALUE=DATE-TIME:20231211T001736Z
UID:TDGA/4
DESCRIPTION:Title: Cur
vature Sets Over Persistence Diagrams\nby Mario Gómez Flores (The Ohi
o State University) as part of Topology\, Geometry\, & Data Analysis (TGDA
) Seminar\n\n\nAbstract\nWe study an invariant of compact metric spaces in
spired by the Curvature Sets defined by Gromov. The (n\,k)-Persistence Set
of X is the collection of k-dimensional VR persistence diagrams of any su
bset of X with n or less points. This research seeks to provide a cheaper
persistence-like invariant for metric spaces\, as the computation of the V
R complex becomes prohibitive once the input reaches a certain size. I'll
focus on the case n=2k+2\, where we can find a geometric formula to calcul
ate the VR persistence diagram of a space with n points. We explore the ap
plication of this formula to the characterization of persistence sets of s
everal spaces\, including circles\, higher dimensional spheres\, and surfa
ces with constant curvature. We also show that persistence sets can detect
the homotopy type of a certain family of graphs.\n
LOCATION:https://researchseminars.org/talk/TDGA/4/
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