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SUMMARY:Nicolau Saldanha (Pontifícia Universidade Católica\, Rio de Jane
 iro)
DTSTART:20211109T133000Z
DTEND:20211109T143000Z
DTSTAMP:20260423T035537Z
UID:Geometry/35
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Geometry/35/
 ">The homotopy type of spaces of locally convex curves in the sphere</a>\n
 by Nicolau Saldanha (Pontifícia Universidade Católica\, Rio de Janeiro) 
 as part of Pangolin seminar\n\n\nAbstract\nA smooth curve $\\gamma: [0\,1]
  \\to S^2$ is locally convex if its geodesic curvature is positive at ever
 y point. J. A. Little showed that the space of all locally positive curves
  $\\gamma$ with $\\gamma(0) = \\gamma(1) = e_1$ and $\\gamma'(0) = \\gamma
 '(1) = e_2$ has three connected components. Our first aim is to describe t
 he homotopy type of these spaces. One of the connected components is known
  to be contractible. The two other connected components are homotopically 
 equivalent to $(\\Omega S^3) \\vee S^2 \\vee S^6 \\vee S^{10} \\vee \\cdot
 s$ and $(\\Omega S^3) \\vee S^4 \\vee S^8 \\vee S^{12} \\vee \\cdots$\, re
 spectively: we describe the equivalence.\n\nMore generally\, a smooth curv
 e $\\gamma: [0\,1] \\to S^n$ is locally convex if \\[ \\det(\\gamma(t)\, \
 \gamma'(t)\, \\ldots\, \\gamma^{(n)}(t)) > 0 \\] for all $t$. A motivation
  for considering this space comes from linear ordinary differential equati
 ons. Again\, we would like to know the homotopy type of the space of local
 ly convex curves with prescribed initial and final jets. We present severa
 l partial results.\n\nIncludes joint work with E. Alves\, V. Goulart\, B. 
 Shapiro\, M. Shapiro\, C. Zhou and P. Zuhlke\n
LOCATION:https://researchseminars.org/talk/Geometry/35/
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