We describe the small-time heat kernel asymptotics of real powers $\operatorname {\Delta }^r$, $r \in (0,1)$ of a non-negative self-adjoint generalized Laplacian $\operatorname {\Delta }$ acting on the sections of a Hermitian vector bundle $\mathcal {E}$ over a closed oriented manifold M. First, we treat separately the asymptotic on the diagonal of $M \times M$ and in a compact set away from it. Logarithmic terms appear only if n is odd and r is rational with even denominator. We prove the non-triviality of the coefficients appearing in the diagonal asymptotics, and also the non-locality of some of the coefficients. In the special case $r=1/2$, we give a simultaneous formula by proving that the heat kernel of $\operatorname {\Delta }^{1/2}$ is a polyhomogeneous conormal section in $\mathcal {E} \boxtimes \mathcal {E}^* $ on the standard blow-up space $\operatorname {M_{heat}}$ of the diagonal at time $t=0$ inside $[0,\infty )\times M \times M$.

We consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small parameter goes to zero and show the stochastic averaging theorem for such equations. We find the explicit limits for the solutions as the small parameter goes to zero.

We present a review on the accuracy of asymptotic models for the scattering problem of electromagnetic waves in domains with thin layer. These models appear as first order approximations of the electromagnetic field. They are obtained thanks to a multiscale expansion of the exact solution with respect to the thickness of the thin layer, that makes possible to replace the thin layer by approximate conditions. We present the advantages and the drawbacks of several approximations together with numerical validations and simulations. The main motivation of this work concerns the computation of electromagnetic field in biological cells. The main difficulty to compute the local electric field lies in the thinness of the membrane and in the high contrast between the electrical conductivities of the cytoplasm and of the membrane, which provides a specific behavior of the electromagnetic field at low frequencies.

Perelman established a differential Li-Yau-Hamilton $\left( \text{LHY} \right)$ type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds. As an application of the $\text{LHY}$ inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. The conditions are satisfied by asymptotically flatmanifolds. We also prove a long time existence result for the Kähler-Ricci flow on complete nonnegatively curved Kähler manifolds.

The coefficients in asymptotics of regularized traces and associated trace (spectral) distributions for Schrödinger operators with short and long range potentials are computed. A kernel expansion for the Schrödinger semigroup is derived, and a connection with non-commutative Taylor formulas is established.

We study the existence of L2 holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of L2 holomorphic sections is bounded below under weak curvature conditions. We also give criteria for a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. As applications we prove the stability of the previous Moishezon pseudoconcave manifolds under perturbation of complex structures as well as weak Lefschetz theorems.