An inclusion is said to be neutral to uniform fields if upon insertion into a homogeneous medium with a uniform field it does not perturb the uniform field at all. It is said to be weakly neutral if it perturbs the uniform field mildly. Such inclusions are of interest in relation to invisibility cloaking and effective medium theory. There have been some attempts lately to construct or to show existence of such inclusions in the form of core-shell structure or a single inclusion with the imperfect bonding parameter attached to its boundary. In this talk we review recent progress in such attempts. We also discuss the over-determined problem for confocal ellipsoids which is closely related with the neutral inclusion problem, and its equivalent formulation in terms of Newtonian potentials. We will also talk about recent applications to stress estimate on a biological body.

Our purpose here is to adapt the results of Geodesic circle foliations for Reeb flows or Hamiltonian flows on contact manifolds. Consequently, all periods are exactly the same if the contact manifold is connected and all orbits on the contact manifold are closed. As an application to quantum mechanics, we examine the spectrum of semiclassical Shrödinger operators. Then we have one of the semiclassical analogies of the Helton-Guillemin theorem.

3次元全空間における定常外力付き圧縮性Navier-Stokes方程式の定常解の安定性について考察する。外力が小さいときの定常解の存在及び摂動問題の可解性についてはShibata-Tanaka(2003)による結果が既に得られている。本講演では定常解周りの摂動の正則性と時間大域的な挙動に関して得られた結果を述べる。またマッハ数と呼ばれる無次元数をゼロに近づけたときの特異極限問題について最近得られた結果を紹介する。

Continuum limits of Laplace operators on general lattices are considered, and it is shown that these operators converge to elliptic operators on the Euclidean space in the sense of the generalized norm resolvent convergence. We then study operators on the hexagonal lattice, which does not apply the above general theory, but we can show its Laplace operator converges to the continuous Laplace operator in the continuum limit. We also study discrete operators on the square lattice corresponding to second order strictly elliptic operators with variable coefficients, and prove the generalized norm resolvent convergence in the continuum limit. This talk is based on a joint work with Keita Mikami (RIKEN) and Shu Nakamura (Gakushuin University).