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SUMMARY:Three Inverse Problems for Persistent Homology
DTSTART;VALUE=DATE-TIME:20240515T101500
DTEND;VALUE=DATE-TIME:20240515T111500
DTSTAMP;VALUE=DATE-TIME:20240619T093729Z
UID:8535807f7bc8631e4fa1172429d8b697d7995d3911defd1e502d923c
CATEGORIES:Conferences - Seminars
DESCRIPTION:David Beers\, University of Oxford\nHow much information is lo
st when we apply the persistence map PH? We investigate this question by s
tudying the level sets of PH in three different settings. In the first\, w
e observe that path components of space of Morse functions on a given mani
fold with the same PH (from the sublevel set filtration) are orbits of Mor
se functions under the identity component of the diffeomorphism group. Whe
n barcode endpoints are distinct and the manifold is an orientable surface
\, this observation allows us to compute the homotopy type of path compone
nts of level sets of PH\, by leveraging the work of Maksymenko. The second
setting we study the persistence map in is that of continuous functions o
n the geometric realization of a tree. By establishing a homotopy equivale
nce between level sets of PH and a certain configuration space\, we are ab
le to compute the number of path components in level sets of PH in this se
tting\, for barcodes with distinct endpoints. Lastly\, we study the subspa
ce of point clouds with the same barcodes (from the Čech or Vietoris-Rips
filtration). Here we establish upper and lower bounds on the dimension of
this space\, and\, in the Vietoris-Rips case\, show that computing the di
mension of this space is challenging by connecting the problem to rigidity
theory. In particular\, we show that a point cloud being locally identifi
able under Vietoris-Rips persistence is equivalent to a certain graph bein
g rigid on the same point cloud.\n\n
LOCATION:MA B1 524 https://plan.epfl.ch/?room==MA%20B1%20524 https://epfl.
zoom.us/j/68364713927
STATUS:CONFIRMED
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