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SUMMARY:Elizaveta Vishnyakova (Department of Math. UFMG\, Belo Horizonte\,
  Brazil)
DTSTART:20260401T110000Z
DTEND:20260401T120000Z
DTSTAMP:20260430T090639Z
UID:PHK-cohomology-seminar/152
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PHK-cohomolo
 gy-seminar/152/">About Graded coverings of supermanifolds and their applic
 ations</a>\nby Elizaveta Vishnyakova (Department of Math. UFMG\, Belo Hori
 zonte\, Brazil) as part of Prague-Hradec Kralove seminar Cohomology in alg
 ebra\, geometry\, physics and statistics\n\nLecture held in ZOOM meeting.\
 n\nAbstract\nIn geometry\, the concept of a covering space is classical an
 d well established. A familiar example is the universal covering \\( p: \\
 mathbb{R} \\to S^1 \\)\, given by \\( t \\mapsto \\exp(it) \\). Analogous 
 constructions appear in algebra as well—for instance\, in the theory of 
 modules over rings\, where one encounters flat or torsion-free coverings. 
 Although they arise in different contexts\, these notions share a common u
 nderlying idea: an object from a given category is covered by an object be
 longing to a smaller (or different) category in such a way that certain un
 iversal properties are satisfied.\n\n\n\nIn their paper "Super Atiyah clas
 ses and obstructions to splitting of supermoduli space\," Donagi and Witte
 n introduced a construction of the first obstruction class to the splittin
 g of a supermanifold. Later\, we observed that the infinite prolongation o
 f their construction satisfies universal properties analogous to those fou
 nd in other covering theories. In other words\, this construction yields a
  covering of a supermanifold in the category of graded manifolds associate
 d with the nontrivial homomorphism \\( \\mathbb{Z} \\to \\mathbb{Z}_2 \\).
  Moreover\, the space of infinite jets can also be viewed as a covering of
  a (super)manifold in the category of graded manifolds corresponding to th
 e homomorphism \\( \\mathbb{Z} \\times \\mathbb{Z}_2 \\to \\mathbb{Z}_2 \\
 )\, given by \\( (m\, \\bar{n}) \\mapsto \\bar{n} \\). (For ordinary manif
 olds\, this homomorphism reduces to the trivial map \\( \\mathbb{Z} \\to 0
  \\).)\n\nOur talk is devoted to the current state of the theory of graded
  coverings\, including the general framework\, key examples\, and a presen
 tation of our recent results.\n\nThe seminar starts **30 minutes earlier t
 han usual**.\n\nWe shall open ZOOM meeting at 12.45 for virtual coffee and
  close ZOOM at 14.30\n
LOCATION:https://researchseminars.org/talk/PHK-cohomology-seminar/152/
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