For more information about this meeting, contact Xiantao Li, Yuxi Zheng, Kris Jenssen, Jinchao Xu, Hope Shaffer.
|Title:||The heat trace, zeta-function, and resolvent of elliptic operators on conic manifolds|
|Seminar:||Computational and Applied Mathematics Colloquium|
|Speaker:||Thomas Krainer, Penn State Altoona|
|On smooth compact manifolds (with and without boundary),
the heat equation
method is at the core of many investigations in spectral geometry,
index theory, and
topology. In the late 1970's, Cheeger initiated such investigations
on manifolds with geometric
singularities. Probably the simplest singularities to consider are
Since the geometry is incomplete the Laplacian is typically not
essentially selfadjoint, and
`boundary conditions' associated with the singular locus are to be
taken into account.
In his seminal paper, Cheeger studied the heat kernel of the
Friedrichs Laplacian on manifolds
with cone-like singularities.
The topic has been taken up again in the late 1980's and
1990's, also motivated by
applications in mathematical physics. More recently, again driven by
physicists, the question
arose whether it was possible to understand the heat kernel of
generic selfadjoint extensions
of elliptic operators on conic manifolds. In 2008, Kirsten, Loya, and
Park studied the Laplacian
under strong assumptions on the geometry near the singularities and
showed that the heat
kernel and zeta-function exhibit in general rather unexpected `exotic' effects.
In this talk I plan to report on recent joint work with J.
Gil and G. Mendoza where we
were able to obtain a completely general structural result about the
resolvent, heat kernel,
and zeta-function of elliptic operators on conic manifolds. Analyzing
this structure numerically
in concrete applications is a challenging open problem.|
Room Reservation Information
|Date:||04 / 29 / 2011|
|Time:||03:35pm - 04:25pm|