[Turkmath:5578] seminar at IMBM by Lejla Smajlovic

[Özlem Ejder]

Dear all,

Lejla Smajlovic (Sarajevo University) will be giving a talk at IMBM on
Thursday, April 21.  Below you may find the details of her talk.  If
you would like to attend the talk at IMBM, please send an email to
ozlemejderff[at]gmail.com.

There is also the possibility to join the talk over zoom:

https://boun-edu-tr.zoom.us/j/93856938710

best,
Özlem

*Time:* Thursday, April 21, 2022, 14:00, IMBM Seminar Room

*Title:* Super-zeta functions and regularized determinants

*Abstract:* A question to define determinants of Laplacians, or related
spectral operators, on Riemannian manifolds is a very important in
mathematical physics and related areas. The classical approach, as
explained by Hawking (1977) relies on regularization by starting with
the trace of a heat kernel. Unfortunately, there are many instances
when such heat kernels are not of trace class, such as when the
hyperbolic Riemann surfaces has finite volume yet is not compact. Many
authors have succeeded in defining regularized traces of heat kernels
in this setting and developed zeta regularized products. However, in
doing so, one does not see very clearly the underlying sequence of
eigenvalues. The purpose of this talk is to describe a different
approach to defining determinants of Laplacians, or related spectral
operators using super-zeta functions. Super-zeta functions are the
Hurwitz-type zeta functions associated to the sequence of zeros of a
certain zeta function; this terminology was introduced in a series of
papers by A. Voros. When the underlying zeta function is the Selberg
zeta function of a cofinite Fuchsian group, then its super-zeta
function carries information about the spectrum of the Laplacian
operator and of the Lax-Phillips scattering operator. We describe how
to define the regularized determinant of those operators in terms of
the derivative of the meromorphic continuation of the associated
super-zeta function at zero. We also explain the formal relation of
our regularized determinant to the sequence of eigenvalues and
resonances, thus showing why our regularized determinant is a natural
extension of the classical determinant (i.e. of the product of
finitely many eigenvalues). As the time permits, we will briefly
discuss the more general setting of hyperbolic manifolds with cusps.
The talk is based on the joint work with Joshua Friedman and Jay
Jorgenson.
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