Trace, Regularity, Fredholmness: the story of non-compact boundary for first-order elliptic operatorsOberseminar Global Analysis and Operator Algebras

by Lashi Bandara (Deakin University)

Tuesday Jan 13, 2026, 2:15 PM → 3:45 PM Europe/Berlin

Endenicher Allee 60/1-008 (Mathezentrum)

The index theorem  for compact manifolds with boundary, established by
Atiyah-Patodi-Singer in 1975, is considered one of the most significant
mathematical achievements of the 20th century. An important and curious
fact is that local boundary conditions are topologically obstructed for
index formulae and non-local boundary conditions lie at the heart of
this theorem. Consequently, this has inspired the study of boundary
value problems for first-order elliptic differential operators by many
different schools, with a class of induced boundary operators taking
centre stage in establishing non-local boundary conditions.

The work of Bär and Ballmann from 2012 is a modern and comprehensive
framework that is useful to study  elliptic boundary value problems for
first-order elliptic operators on manifolds with compact and smooth
boundary. As in the work of Atiyah-Patodi-Singer, a fundamental
assumption in Bär-Ballmann is that the induced operator on the boundary
can be chosen self-adjoint. All  Dirac-type operators, which in
particular includes the Hodge-Dirac operator as well as the
Atiyah-Singer Dirac operator, are captured via this framework.

In contrast to the APS index theorem, which is essentially restricted to
Dirac-type operators, the earlier index theorem of Atiyah-Singer  from
1968 on closed manifolds is valid for general first-order elliptic
differential operators. There are important operators from both geometry
and physics which are more general than those captured by the
state-of-the-art for BVPs and index theory. A quintessential example is
the Rarita-Schwinger operator on 3/2-spinors, which arises in physics
for the study of the so-called delta baryons. A fundamental and
seemingly fatal obstacle to study BVPs for such operators is that the
induced operator on the boundary may no longer be chosen self-adjoint,
even if the operator on the interior is symmetric.

In recent work with Bär, we extend the Bär-Ballmann framework to
consider general first-order elliptic differential operators by
dispensing with the self-adjointness requirement for induced boundary
operators. Modulo a zeroth order additive term, we show every induced
boundary operator is a bi-sectorial operator via the ellipticity of the
interior operator. An essential tool at this level of generality is the
bounded holomorphic functional calculus, coupled with
pseudo-differential operator theory, semi-group theory as well as
methods connected to the resolution of the Kato square root problem.
This perspective also paves way for studying non-compact boundary,
Lipschitz boundary, as well as boundary value problems in the L^p setting.