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SUMMARY:Jim Propp (University of Massachusetts Lowell)
DTSTART:20200623T140000Z
DTEND:20200623T153000Z
DTSTAMP:20260404T110828Z
UID:GROSCALIN/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/GROSC
 ALIN/1/">Packings in one\, two\, and three dimensions: a macro-meso-micros
 copic view</a>\nby Jim Propp (University of Massachusetts Lowell) as part 
 of CALIN seminar (combinatorics\, algorithms\, and interactions)\n\n\nAbst
 ract\nHexagonal close-packings are the most efficient way to pack unit dis
 ks in $\\R^2$. But what do we mean by the definite article "the" in the pr
 evious sentence? Are hexagonal close-packings the only optimal packings? I
 f we measure optimality by density\, the answer is\, No. In fact\, there a
 re 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 st
 udy deficiency\, and seek ways to quantify defects in a regular packing. A
 n obstacle here is that common kinds of defects inhabit disparate scales (
 e.g.\, point defects are infinitesimal compared to line defects\, which in
  turn are infinitesimal compared to the bulk). This suggests we turn to ex
 tensions of the real numbers that include infinitesimal elements (or rathe
 r\, as turns out to be more helpful\, infinite elements). We use a regular
 ization trick to make sense of these ideas (starting in one dimension). Th
 is enables us to sharpen our notion of optimal packing so that the optimal
  disk-packings are provably the hexagonal close-packings and no others. A 
 side-benefit is a natural but apparently new finitely additive\, non-Archi
 medean measure in Euclidean n-space\; it agrees with n-dimensional volume 
 when applied to finite regions\, but some infinite regions are "more infin
 ite" than others. For slides related to an earlier version of this talk\, 
 see <a href="http://jamespropp.org/brown18a.pdf">http://jamespropp.org/bro
 wn18a.pdf</a>\n
LOCATION:https://stable.researchseminars.org/talk/GROSCALIN/1/
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