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SUMMARY:Alexander Borisov (Binghamton University)
DTSTART:20260715T183000Z
DTEND:20260715T185500Z
DTSTAMP:20260710T111633Z
UID:CANT2026/38
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/38/
 ">A structure sheaf for Kirch topology on N</a>\nby Alexander Borisov (Bin
 ghamton University) as part of Combinatorial and additive number theory se
 minar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate C
 enter (4th floor).\n\nAbstract\nKirch topology on $\\mathbb N$ goes back t
 o a 1969 paper of Kirch. It can be defined by a basis of open sets that co
 nsists of all infinite arithmetic progressions $a+d\\mathbb N_0$\, such th
 at $\\gcd(a\,d)=1$ and $d$ is square-free. It is Hausdorff\, connected\, a
 nd locally connected. One can hope that in the classical imperfect analogy
  between arithmetic and geometry this can serve as an arithmetic analog of
  the usual topology on $\\mathbb C$. However\, the usual topology on $\\ma
 thbb C$ comes with a structure sheaf of complex-analytic functions. As far
  as I know\, no analog for Kirch topology has been proposed before me. I b
 elieve that I have stumbled upon just such a thing\, more by accident than
  by a conscious effort: locally LIP functions. These are functions from Ki
 rch-open sets to $\\mathbb Z$ such that for every point in the domain ther
 e is a Kirch-open neighborhood on which the function is "locally integer p
 olynomial" (LIP): its interpolation polynomial on every finite set has int
 eger coefficients. I will explain why this seems to be a natural object\, 
 what I know about it\, and what I hope to achieve.\n
LOCATION:https://researchseminars.org/talk/CANT2026/38/
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