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SUMMARY:The dynamics of classifying geometric structures - Bill Goldman (U
 niversity of Maryland\, College Park)
DTSTART:20170623T080000Z
DTEND:20170623T090000Z
UID:TALK73043@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:The general theory of locally homogeneous geometric  structure
 s (flat Cartan connections)&nbsp\;originated with Ehresmann.  Their classi
 fication is analogous to the classification of Riemann  surfaces by the Ri
 emann moduli space.  In general\, however\, the analog of the moduli space
  is intractable\,  but leads to a rich class of dynamical systems.  &nbsp\
 ;  <br><br>For example\, classifying Euclidean geometries  on the torus le
 ads to the usual action of the SL(2\,Z)&nbsp\;  on the upper half-plane. T
 his action is dynamically trivial\,  with a quotient space the familiar mo
 dular curve.&nbsp\;  In contrast\, the classification of other simple geom
 etries on&nbsp\;  on the torus leads to the standard linear action of SL(2
 \,Z) on R^2\,&nbsp\;  with chaotic dynamics and a pathological quotient sp
 ace.  &nbsp\;  <br><br>This talk describes such dynamical systems\,  and w
 e combine Teichmueller theory to understand the geometry  of the moduli sp
 ace when the topology is enhanced with a&nbsp\;  conformal structure. In j
 oint work with Forni\, we prove the  corresponding extended Teichmueller f
 low is strongly mixing.  &nbsp\;  <br><br>Basic examples arise when &nbsp\
 ;the moduli space &nbsp\;is described by the nonlinear symmetries of cubic
  equations like Markoff&rsquo\;s equation x^2 + y^2 + z^2 = x y z. &nbsp\;
 Here both trivial and chaotic dynamics arise simultaneously\,&nbsp\;relati
 ng to possibly singular hyperbolic-geometry  structures on surfaces. (This
  represents joint work with McShane-Stantchev- Tan.)
LOCATION:Seminar Room 1\, Newton Institute
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