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SUMMARY:Michelle Manes (U. Hawaii)
DTSTART:20220519T210000Z
DTEND:20220519T220000Z
DTSTAMP:20260423T024731Z
UID:UCSD_NTS/62
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCSD_NTS/62/
 ">Iterating Backwards in Arithmetic Dynamics</a>\nby Michelle Manes (U. Ha
 waii) as part of UCSD number theory seminar\n\nLecture held in APM 6402 an
 d online.\n\nAbstract\nIn classical real and complex dynamics\, one studie
 s topological and analytic properties of orbits of points under iteration 
 of self-maps of $\\mathbb R$ or $\\mathbb C$ (or more generally self-maps 
 of a real or complex manifold). In arithmetic dynamics\, a more recent sub
 ject\, one likewise studies properties of orbits of self-maps\, but with a
  number theoretic flavor. Many of the motivating problems in arithmetic dy
 namics come via analogy with classical problems in arithmetic geometry: ra
 tional and integral points on varieties correspond to rational and integra
 l points in orbits\; torsion points on abelian varieties correspond to per
 iodic and preperiodic points of rational maps\; and abelian varieties with
  complex multiplication correspond to post-critically finite rational maps
 .\n\nThis analogy focuses on forward iteration\, but sometimes surprising 
 and interesting results can be found by thinking instead about pre-images 
 of rational points under iteration. In this talk\, we will give some backg
 round and motivation for the field of arithmetic dynamics in order to desc
 ribe some of these "backwards iteration" results\, including uniform bound
 edness for rational pre-images and open image results for Galois represent
 ations associated to dynamical systems.\n\nA pre-talk for graduate student
 s will describe some of the motivating results in arithmetic geometry.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/62/
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