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SUMMARY:Discrete surfaces with length and area and minimal fillings of the
  circle.
DTSTART:20191014T161500
DTEND:20191014T173000
DTSTAMP:20260929T050429Z
UID:836b5b6f9d1db2b7978a7d2dcaea63d6a981c4247a5f03dd57f2905c
CATEGORIES:Conferences - Seminars
DESCRIPTION:Marcos Cossarini (Paris Est Marne-La-Vallée)\nAbstract: We pr
 opose to imagine that every Riemannian metric on a surface is discrete at 
 the small scale\, made of curves called walls. The length of a curve is it
 s number of crossings with the walls\, and the area of the surface is the 
 number of crossings between the walls themselves. We show how to approxima
 te a Riemannian or self-reverse Finsler metric by a wallsystem.\nThis work
  is motivated by Gromov's filling area conjecture (FAC) that the hemispher
 e has minimum area among orientable Riemannian surfaces that fill isometri
 cally a closed curve of given length. (A surface fills its boundary curve 
 isometrically if the distance between each pair of boundary points measure
 d along the surface is not less than the distance measured along the bound
 ary.) We introduce a discrete FAC: every square-celled surface that fills 
 isometrically a 2n-cycle graph has at least n(n-1)/2 squares. This conject
 ure is equivalent to the FAC extended to surfaces with self-reverse Finsle
 r metric.\nIf the surface is a disk\, the discrete FAC follows from Steini
 tz's algorithm for transforming curves into pseudolines. This gives a new\
 , combinatorial proof that the FAC holds for disks with Riemannian or self
 -reverse Finsler metric.\nIf time allows\, we also discuss how to discreti
 ze a directed metric on a surface using a triangulation with directed edge
 s. The length of each edge is 1 in one way and 0 in the other way\, and th
 e area of the surface is the number of triangles. These discrete surfaces 
 are dual to Postnikov's plabic graphs.
LOCATION:MA B1 524 https://plan.epfl.ch/?request_locale=en&room=MA%2BB1%2B
 524&domain=places&dim_floor=1&lang=fr&dim_lang=fr&tree_groups=centres_nevr
 algiques%2Cacces%2Cmobilite_reduite%2Censeignement%2Ccommerces_et_servi
STATUS:CONFIRMED
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