BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.bham.ac.uk//v3//EN
BEGIN:VEVENT
CATEGORIES:Analysis seminar
SUMMARY:Index theory and boundary value problems for gener
al first-order elliptic differential operators - L
ashi Bandara\, Brunel University London
DTSTART:20220331T140000Z
DTEND:20220331T150000Z
UID:TALK4860AT
URL:/talk/index/4860
DESCRIPTION:Connections between index theory and boundary valu
e problems are an old topic\, dating back to the s
eminal work of Atiyah-Patodi-Singer in the mid-70s
where they proved the famed APS Index Theorem for
Dirac-type operators. From relative index theory
arising in the study of positive scalar curvature
metrics to a rigorous understanding of the chiral
anomaly for the electron in particle physics\, th
is index theorem has been a central tool to many a
spects of modern mathematics.\n\nAPS showed that l
ocal boundary conditions are topologically obstruc
ted for index theory. Therefore\, a central theme
emerging from the work of APS is the significance
of non-local boundary conditions for first-order e
lliptic differential operators. An important contr
ibution from APS was to demonstrate how their cruc
ial non-local boundary condition for the index the
orem could be obtained by a spectral projection as
sociated to a so-called adapted boundary operator
. In their application\, this was a self-adjoint f
irst-order elliptic differential operator. \n\nThe
work of APS generated tremendous amount of activi
ty in the topic from the mid-70s onwards\, culmina
ting with the Bär-Ballmann framework in 2010. This
is a comprehensive machine useful to study ellipt
ic boundary value problems for first-order ellipt
ic operators on measured manifolds with compact an
d smooth boundary. It also featured an alternative
and conceptual reformulation of the famous relati
ve index theorem from the point of view of boundar
y value problems. However\, as with other general
isations\, a fundamental assumption in their work
was that an adapted boundary operator can always b
e chosen self-adjoint.\n\nMany operators\, includi
ng all Dirac-type operators\, satisfy this require
ment. In particular\, this includes the Hodge-Dira
c operator as well as the Atiyah-Singer Dirac oper
ator. Recently\, there has been a desire to study
more general first-order elliptic operators\, with
the quintessential example being the Rarita-Schwi
nger operator on 3/2-spinors. This operator has ph
ysical significance\, arising in the study of the
delta baryon\, analogous to the way in which the
Atiyah-Singer Dirac operator arises in the study o
f the electron. However\, not only does the Rarit
a-Schwinger operator fail to be of Dirac-type\, it
can be shown that outside of trivial geometric si
tuations\, this operator can never admit a self-ad
joint adapted boundary operator.\n\nIn this talk\,
I will present work with Bär where we extend the
theory for first-order elliptic differential opera
tors to full generality. That is\, we make no assu
mptions on the spectral theory of the adapted boun
dary operator. The ellipticity of the original ope
rator allows us to show that\, modulo a lower orde
r additive perturbation\, the adapted boundary ope
rator is in fact bi-sectorial. Identifying the spe
ctral theory makes the problem tractable\, althoug
h departure from self-adjointness significantly c
omplicates the analysis. Therefore\, we employ a
mixture of methods coming from pseudo-differential
operator theory\, bounded holomorphic functional
calculus\, semi-group theory\, and maximal regular
ity to extend the Bär-Ballman framework to the ful
ly general situation.\n\nTime permitting\, I will
also talk about recent work on the relative index
theorem for general first-order elliptic different
ial operators\, possible harmonic-analytic perspec
tives of the APS index theorem\, as well as recen
t developments in the study of noncompact boundary
\, Lipschitz boundary\, and problems in L^p.
LOCATION:Aston Webb WG12
CONTACT:Andrew Morris
END:VEVENT
END:VCALENDAR