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SUMMARY:Gilyoung Cheong (UC Irvine)
DTSTART:20211115T230000Z
DTEND:20211115T235000Z
DTSTAMP:20260423T041409Z
UID:UCLA_NTS/39
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCLA_NTS/39/
 ">Counting square-free numbers in arithmetic geometry</a>\nby Gilyoung Che
 ong (UC Irvine) as part of UCLA Number Theory Seminar\n\n\nAbstract\nWe de
 lve into an innocuous question about counting\n"square-free numbers" in va
 rious forms\, following the philosophy of\nWeil's three columns.\n\nExampl
 e 1. We count square-free integers.\n\nExample 2. We count square-free pol
 ynomials of a fixed degree over a\nfinite field.\n\nExample 3. We compute 
 the Betti numbers of the space of square-free\npolynomials of a fixed degr
 ee over complex numbers by quoting a theorem\nof Arnol'd in topology.\n\nB
 y viewing Example 3 as counting square-free 0-cycles on the affine line\no
 ver complex numbers\, we add one more example to this list\, using our\nma
 in result.\n\nExample 4. We compute the Betti numbers of the space of squa
 re-free\n0-cycles of a fixed degree on a punctured elliptic curve over com
 plex\nnumbers.\n\nWe briefly explain how Examples 3 and 4 can be obtained 
 by showing that\nthe mixed Hodge structure of the i-th singular cohomology
  group with\nrational coefficients is pure of some weight different from i
 . This is\njoint work with Yifeng Huang.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/39/
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