Two Regularizations for the Determinant of Laplace Operators.
Organizer
Speaker
Wen Shen
Time
Wednesday, January 14, 2026 10:00 AM - 11:30 AM
Venue
A3-3-201
Online
Zoom 442 374 5045
(BIMSA)
Abstract
We review two regularizations for the determinant of the Laplace operator on Riemannian manifolds and give formulas, under appropriate conditions, to compute them by resurgence theory, respectively. One regularization is the formal logarithmic derivative of the determinant, and the other is the exponential deformation of the determinant series. In our formulas, they can be viewed as the summation of the singularities along the analytic continuation of the theta series $\hat{\Theta}_{D_X}$ and its analogue. The series resembles the trace of the heat kernel, but is defined via the spectrum of the square root of the Laplacian. Furthermore, the 1-Gevrey asymptotic behavior of the second regularization at infinity is considered, whose asymptotic coefficients are determined by the trace of the heat kernel.
As applications, we compute the two regularizations of the determinant series on $S^1$ and compact Riemann surfaces with genus ≥2 . For the first regularization, the results give new insights on the Poisson summation formula and the Selberg trace formula, respectively; for the second regularization of the determinant series on $S^1$, we obtain the deformed Poisson summation formula. Finally, we establish the relationship between the two regularized determinants: they have the same derivatives if we let the deformation parameter tend to 0 in the exponentially deformed regularization.Based on joint works with Shanzhong Sun and David Sauzin, the talk will be delivered in Chinese.
As applications, we compute the two regularizations of the determinant series on $S^1$ and compact Riemann surfaces with genus ≥2 . For the first regularization, the results give new insights on the Poisson summation formula and the Selberg trace formula, respectively; for the second regularization of the determinant series on $S^1$, we obtain the deformed Poisson summation formula. Finally, we establish the relationship between the two regularized determinants: they have the same derivatives if we let the deformation parameter tend to 0 in the exponentially deformed regularization.Based on joint works with Shanzhong Sun and David Sauzin, the talk will be delivered in Chinese.