Related papers: Heat kernels, Bergman kernels, and cusp forms
This is a report on recent work, with Wen-Ching Winnie Li and Ling Long. In that work explicit formulas are given, involving hypergeometric character sums, for the traces of Hecke operators $T_p$ acting spaces of cusp forms $S_k(\Gamma)$ of…
We find a Gaussian off-diagonal heat kernel estimate for uniformly elliptic operators with measurable coefficients acting on regions $\Omega\subseteq\real^N$, where the order $2m$ of the operator satisfies $N<2m$. The estimate is expressed…
Surface heat transfer in convective and radiative environments is sometimes measured by recording the surface temperature history in a transient experiment and interpreting this surface temperature with the aid of a suitable model for…
Linear-scaling electronic structure methods based on the calculation of moments of the underlying electronic Hamiltonian offer a computationally efficient and numerically robust scheme to drive large-scale atomistic simulations, in which…
A diagramatic heat kernel expansion technique is presented. The method is especially well suited to the small-derivative expansion of the heat kernel, but it can also be used to reproduce the results obtained by the approach known as…
In this article, we prove exact estimates for the $W$-invariant Dunkl kernel and heat kernel, for the root system of type $A$ with arbitrary positive multiplicities. We apply the estimates of the $W$-invariant Dunkl heat kernel to compute…
We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present ageneral positive definite kernel setting using bilinear forms, and we provide new…
In the literature, the standard approach to finding bases of spaces of modular forms is via modular symbols and the homology of modular curves. By using the Eichler-Shimura isomorphism, a work by Wang shows how one can use a cohomological…
We give a physicist's derivation of the geometrical (in the spirit of Duplantier-Sheffield) KPZ relations, via heat kernel methods. It gives a covariant way to define neighborhoods of fractals in 2d quantum gravity, and shows that these…
The Lie group SU(2) endowed with its canonical subriemannian structure appears as a three-dimensional model of a positively curved subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related…
We study the heat kernel transform on a nilmanifold $ M $ of the Heisenberg group. We show that the image of $ L^2(M) $ under this transform is a direct sum of weighted Bergman spaces which are related to twisted Bergman and Hermite-Bergman…
By making full use of heat kernel estimates, we establish the integral tests on the zero-one laws of upper and lower bounds for the sample path ranges of symmetric Markov processes. In particular, these results concerning on upper rate…
The goal of this paper is to study the action of the heat operator on the Heisenberg group H^d, and in particular to characterize Besov spaces of negative index on H^d in terms of the heat kernel. That characterization can be extended to…
We construct the fundamental solution (the heat kernel) $p^{\kappa}$ to the equation $\partial_t=\mathcal{L}^{\kappa}$, where under certain assumptions the operator $\mathcal{L}^{\kappa}$ takes one of the following forms, \begin{align*}…
The heat kernel transform H_t for the Heisenberg group is studied in detail. The main result shows that the image of H_t is a direct sum of two weighted Bergman spaces whose associated weighted functions are of oscillatory nature, i.e.…
This paper provides explicit pointwise formulas for the heat kernel on compact metric measure spaces that belong to a $(\mathbb{N}\times\mathbb{N})$-parameter family of fractals which are regarded as projective limits of metric measure…
We study the heat kernel for an operator of Laplace type with a $\delta$-function potential concentrated on a closed surface. We derive the general form of the small $t$ asymptotics and calculate explicitly several first heat kernel…
In this paper we continue the analysis of spectral problems in the setting of complete manifolds with fibred boundary metrics, also referred to as $\phi$-metrics, as initiated in our previous work. We consider the Hodge Laplacian for a…
We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing "invariant versions" of iterated integrals of modular forms. The construction will be based on an extension of…
This paper studies strongly local symmetric Dirichlet forms on general measure spaces. The underlying space is equipped with the intrinsic metric induced by the Dirichlet form, with respect to which the metric measure space does not…