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SUMMARY:Trace\, Regularity\, Fredholmness: the story of non-compact bounda
 ry for first-order elliptic operators [Oberseminar Global Analysis and Ope
 rator Algebras]
DTSTART:20260113T131500Z
DTEND:20260113T144500Z
DTSTAMP:20260314T230000Z
UID:indico-event-1028@math-events.uni-bonn.de
DESCRIPTION:Speakers: Lashi Bandara (Deakin University)\n\nThe index theor
 em  for compact manifolds with boundary\, established by\nAtiyah-Patodi-Si
 nger in 1975\, is considered one of the most significant\nmathematical ach
 ievements of the 20th century. An important and curious\nfact is that loca
 l boundary conditions are topologically obstructed for\nindex formulae and
  non-local boundary conditions lie at the heart of\nthis theorem. Conseque
 ntly\, this has inspired the study of boundary\nvalue problems for first-o
 rder elliptic differential operators by many\ndifferent schools\, with a c
 lass of induced boundary operators taking\ncentre stage in establishing no
 n-local boundary conditions.\n\nThe work of Bär and Ballmann from 2012 is
  a modern and comprehensive\nframework that is useful to study  elliptic b
 oundary value problems for\nfirst-order elliptic operators on manifolds wi
 th compact and smooth\nboundary. As in the work of Atiyah-Patodi-Singer\, 
 a fundamental\nassumption in Bär-Ballmann is that the induced operator on
  the boundary\ncan be chosen self-adjoint. All  Dirac-type operators\, whi
 ch in\nparticular includes the Hodge-Dirac operator as well as the\nAtiyah
 -Singer Dirac operator\, are captured via this framework.\n\nIn contrast t
 o the APS index theorem\, which is essentially restricted to\nDirac-type o
 perators\, the earlier index theorem of Atiyah-Singer  from\n1968 on close
 d manifolds is valid for general first-order elliptic\ndifferential operat
 ors. There are important operators from both geometry\nand physics which a
 re more general than those captured by the\nstate-of-the-art for BVPs and 
 index theory. A quintessential example is\nthe Rarita-Schwinger operator o
 n 3/2-spinors\, which arises in physics\nfor the study of the so-called de
 lta baryons. A fundamental and\nseemingly fatal obstacle to study BVPs for
  such operators is that the\ninduced operator on the boundary may no longe
 r be chosen self-adjoint\,\neven if the operator on the interior is symmet
 ric.\n\nIn recent work with Bär\, we extend the Bär-Ballmann framework t
 o\nconsider general first-order elliptic differential operators by\ndispen
 sing with the self-adjointness requirement for induced boundary\noperators
 . Modulo a zeroth order additive term\, we show every induced\nboundary op
 erator is a bi-sectorial operator via the ellipticity of the\ninterior ope
 rator. An essential tool at this level of generality is the\nbounded holom
 orphic functional calculus\, coupled with\npseudo-differential operator th
 eory\, semi-group theory as well as\nmethods connected to the resolution o
 f the Kato square root problem.\nThis perspective also paves way for study
 ing non-compact boundary\,\nLipschitz boundary\, as well as boundary value
  problems in the L^p setting.\n\n\nhttps://math-events.uni-bonn.de/event/1
 028/
LOCATION:Endenicher Allee 60/1-008 (Mathezentrum)
URL:https://math-events.uni-bonn.de/event/1028/
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