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DTSTART;TZID=America/Toronto:20260129T143000
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URL:https://uwaterloo.ca/pure-mathematics/events/differential-geometry-work
 ing-seminar-173
SUMMARY:Differential Geometry Working Seminar
CLASS:PUBLIC
DESCRIPTION:VIKTOR MAJEWSKI\, UNIVERSITY OF WATERLOO\n\n_Dirac Operators on
  Orbifold Resolutions_\n\nIn this talk we discuss Dirac operators along de
 generating families of\nRiemannian manifolds that converge\, in the Gromov
 -Hausdorff sense\, to\na Riemannian orbifold. Such degenerations arise nat
 urally when\nanalysing the boundary of Teichmüller spaces of special Riem
 annian\nmetrics as well as moduli spaces appearing in gauge theory and\nca
 librated geometry. Here sequences of smooth geometric structures on\nRiema
 nnian manifolds may converge to an orbifold limit. To understand\nand cont
 rol these degenerations\, we introduce smooth Gromov-Hausdorff\nresolution
 s of orbifolds\, that are\, smooth families (X_t\,g_t)\, which\ncollapse t
 o the orbifold (X_0\,g_0) as t goes to 0.\n\nThe central analytic problem 
 addressed in this paper is to understand\nthe behaviour of Dirac operators
  along such resolutions\, in particular\nin collapsing regimes where class
 ical elliptic estimates fail. We\ndevelop a uniform Fredholm theory for th
 e family of Dirac operators on\nthe Gromov-Hausdorff resolution. Using wei
 ghted function spaces\,\nadiabatic analysis\, and a decomposition of X_t i
 nto asymptotically\nconical fibred (ACF)\, conically fibred (CF) and conic
 ally fibred\nsingular (CFS)\, we obtain uniform realisations of the model 
 operators\nand prove a linear gluing exact sequence relating global and lo
 cal\n(co)kernels. As a consequence\, we construct uniformly bounded right\
 ninverses for D_t\, and derive an index additivity formula.\n\nThe theory 
 developed here provides the analytic foundation for\nnonlinear gluing prob
 lems in gauge theory and special holonomy\ngeometry\, including torsion-fr
 ee G-structures\, instantons\, and\ncalibrated submanifolds of Riemannian 
 manifolds close to an orbifold\nlimit.\n\nMC 5403
DTSTAMP:20260503T080904Z
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