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SUMMARY:Journée de Rham 2017 :  Flavors of bicycle mathematics
DTSTART:20170308T171500
DTEND:20170308T181500
DTSTAMP:20260428T190722Z
UID:02621b629be87c0066c6be12114d75c84730260437a6c5223f5c09e6
CATEGORIES:Conferences - Seminars
DESCRIPTION:Prof. Sergei Tabachnikov\, Pennsylvania State University\nThis
  talk concerns a naive model of bicycle motion: a bicycle is a segment of 
 fixed length that can move so that the velocity of the rear end is always 
 aligned with the segment. Surprisingly\, this simple model is quite rich. 
 Here is a sampler of the problems that I plan to discuss.\n\n1) The trajec
 tory of the front wheel and the initial position of the bicycle uniquely d
 etermine its motion and its terminal position\; the monodromy map sending 
 the initial position to the terminal one arises. This  circle mapping is 
 a Moebius transformation\, a remarkable fact that has various geometrical 
 and dynamical consequences. Moebius transformations belong to one of the t
 hree types: elliptic\, parabolic and hyperbolic. I shall outline a proof o
 f a 100 years old  conjecture: if the front wheel track of a unit length 
 bicycle is an oval with area at least Pi then the respective monodromy is 
 hyperbolic.\n\n2) The rear wheel track and a choice of the direction of mo
 tion uniquely determine the front wheel track\; changing the direction to 
 the opposite\, yields another front track. These two front tracks are rela
 ted by the bicycle (Backlund\, Darboux) correspondence which defines a dis
 crete time dynamical system on the space of curves. What do pairs of curve
 s in the bicycle correspondence have in common? It turns out\, infinitely 
 many quantities (the perimeter length\, the total curvature squared\,…) 
 I shall explain that the bicycle correspondence\nis closely related with a
 nother\, well studied\, completely integrable dynamical system\, the filam
 ent (a.k.a binormal\, smoke ring\, local induction) equation. \n\n3) Give
 n the rear and front tracks of a bicycle\, can one tell which way the bicy
 cle went? Usually\, one can\, but sometimes one cannot. The description of
  these ambiguous tire tracks is an open problem\, intimately related with 
 Ulam's problem in flotation theory (in dimension two): is the round ball t
 he only body that floats in equilibrium in all positions? ​
LOCATION:CO 2 https://plan.epfl.ch/theme/generalite_thm_v2?request_locale=
 fr&room=CO%202&domain=places&dim_floor=1&lang=fr&dim_lang=fr&tree_groups=c
 entres_nevralgiques%2Cacces%2Cmobilite_reduite%2Censeignement%2Cco
STATUS:CONFIRMED
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