BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Henry Adams (Colorado State University) DTSTART:20200525T140000Z DTEND:20200525T150000Z DTSTAMP:20260423T035932Z UID:OATS/6 DESCRIPTION:Title: Vie toris-Rips complexes and Borsuk-Ulam theorems\nby Henry Adams (Colorad o State University) as part of Online algebraic topology seminar\n\n\nAbst ract\nGiven a metric space X and a scale parameter r\, the Vietoris-Rips s implicial complex VR(X\;r) has X as its vertex set\, and contains a finite subset as a simplex if its diameter is at most r. Vietoris-Rips complexes were invented by Vietoris in order to define a (co)homology theory for me tric spaces\, and by Rips for use in geometric group theory. More recently \, they have found applications in computational topology for approximatin g of the shape of a dataset. I will explain how the Vietoris-Rips complexe s of the circle\, as the scale parameter r increases\, obtain the homotopy types of the circle\, the 3-sphere\, the 5-sphere\, the 7-sphere\, ...\, until they are finally contractible. Only very little is understood about the homotopy types of the Vietoris-Rips complexes of the n-sphere. Knowing the homotopy connectivities of Vietoris-Rips complexes of spheres allows one to prove generalizations of the Borsuk-Ulam theorem for maps from the n-sphere into k-dimensional Euclidean space with k > n. Joint work with Jo hn Bush and Florian Frick.\n LOCATION:https://researchseminars.org/talk/OATS/6/ END:VEVENT END:VCALENDAR