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DESCRIPTION: Towards a better understanding of arrangements of circles and 
 also to get rid of geometric difficulties\, we look at the more general set
 ting of ''arrangements of pseudocircles'' which was first introduced by Grü
 nbaum in the 1970's. An arrangement of pseudocircles is a collection of sim
 ple closed curves on the sphere or in the plane such that any two of the cu
 rves are either disjoint or intersect in exactly two points\, where the two
  curves cross. In his book\, Grünbaum conjectured that every digon-free arr
 angement of n pairwise intersecting pseudocircles contains at least $2n-4$ 
 triangular cells. We present arrangements to disprove this conjecture and g
 ive new bounds on the number of triangular cells for various classes of arr
 angements.   Furthermore\, we study the ''circularizability'' of arrangemen
 ts: it is clear that every arrangement of circles is an arrangement of pseu
 docircles\, however\, deciding whether an arrangement of pseudocircles is i
 somorphic to an arrangement of circles is computationally hard. Using a com
 puter program\, we have enumerated all combinatorially different arrangemen
 ts of up to $7$ pseudocircles. For the class of arrangements of $5$ pseudoc
 ircles and for the class of digon-free intersecting arrangements of $6$ pse
 udocircles\, we give a complete classification: we either provide a circle 
 representation or a non-circularizability proof. For these proofs we use in
 cidence theorems like Miquel's and arguments based on continuous deformatio
 n\, where circles of an assumed circle representation grow or shrink in a c
 ontrolled way.   This talk summarizes results from two articles\, which are
  both joint work with Stefan Felsner: * Arrangements of Pseudocircles: Tria
 ngles and Drawings\; short version in Proc. GD'17\; full version available 
 at arXiv (1708.06449) * Arrangements of Pseudocircles: On Circularizability
 \; short version in Proc. GD'18\; full version in DCG: Ricky Pollack Memori
 al Issue (doi:10.1007/s00454-019-00077-y) 
DTSTAMP:20190524T131600
DTSTART:20190701T160000
CLASS:PUBLIC
LOCATION:Freie Universität Berlin \n Institut für Informatik \n Takustr. 9 
 \n 14195 Berlin \n Room 005 (Ground Floor)
SEQUENCE:0
SUMMARY:Manfred Scheucher (Technische Universität Berlin): On Arrangements 
 of Pseudocircles
UID:95691268@/www.mi.fu-berlin.de
URL:https://www.mi.fu-berlin.de/en/facetsofcomplexity/monday/20190701-C-Sch
 eucher.html
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