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SUMMARY:Jim Propp (University of Massachusetts Lowell)
DTSTART:20200623T140000Z
DTEND:20200623T153000Z
DTSTAMP:20260404T121640Z
UID:GROSCALIN/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GROSCALIN/1/
 ">Packings in one\, two\, and three dimensions: a macro-meso-microscopic v
 iew</a>\nby Jim Propp (University of Massachusetts Lowell) as part of CALI
 N seminar (combinatorics\, algorithms\, and interactions)\n\n\nAbstract\nH
 exagonal close-packings are the most efficient way to pack unit disks in $
 \\R^2$. But what do we mean by the definite article "the" in the previous 
 sentence? Are hexagonal close-packings the only optimal packings? If we me
 asure optimality by density\, the answer is\, No. In fact\, there are far 
 too many density-optimal packings to classify in any meaningful way. This 
 suggests that density is too coarse a notion to capture everything that we
  mean (or should mean!) by "efficient". To study efficiency\, we study def
 iciency\, and seek ways to quantify defects in a regular packing. An obsta
 cle here is that common kinds of defects inhabit disparate scales (e.g.\, 
 point defects are infinitesimal compared to line defects\, which in turn a
 re infinitesimal compared to the bulk). This suggests we turn to extension
 s of the real numbers that include infinitesimal elements (or rather\, as 
 turns out to be more helpful\, infinite elements). We use a regularization
  trick to make sense of these ideas (starting in one dimension). This enab
 les us to sharpen our notion of optimal packing so that the optimal disk-p
 ackings are provably the hexagonal close-packings and no others. A side-be
 nefit is a natural but apparently new finitely additive\, non-Archimedean 
 measure in Euclidean n-space\; it agrees with n-dimensional volume when ap
 plied to finite regions\, but some infinite regions are "more infinite" th
 an others. For slides related to an earlier version of this talk\, see <a 
 href="http://jamespropp.org/brown18a.pdf">http://jamespropp.org/brown18a.p
 df</a>\n
LOCATION:https://researchseminars.org/talk/GROSCALIN/1/
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