A number of questions related to the length spectrum of surfaces are discussed and in particular the existence of pairs of surfaces which though not isometric are isospectral. Here by isospectral we mean that a pair of bodies have the same distribution of chord lengths. In the Euclidean setting, we study isospectral convex dodecagons found by Mallows and Clark in the 1970's. Starting from their idea, we give constructions for isospectral pairs of hyperbolic surfaces that have no common cover. Since the work of Mallows and Clark is probably unfamiliar to readers with a background in topology/hyperbolic geometry we include expository material on other related topics about the distribution of chord lengths.

We prove a general result implying the $L^2$ stability of Haar decompositions of $L^2({\bf R}^d)$ functions when the Haar functions are distorted by arbitrary, independent, affine changes of variable that are close to the identity. We apply our method to get fully $d$-dimensional generalizations of results of Aimar, Bernardis, Gorosito, Govil, and Zalik, on constructing frames of smooth functions which are, in many natural senses, arbitrarily close to the Haar functions. We also obtain a best-possible estimate on the $L^2$ sensitivity of dyadic averages of functions to small distortions caused by local affine changes of variable.

In this paper, we prove that a complete noncompact weighted manifold supporting the weighted Sobolev inequality has at least linear weighted volume growth. We also obtain vanishing results and finiteness theorems for the weighted $L^2_f$ $f$-harmonic 1-forms on a complete noncompact weighted manifold supporting the weighted Sobolev inequality.

In this paper we investigate two important properties of metric measure spaces satisfying the reduced curvature-dimension condition for negative values of the dimension parameter: the existence of a transport map between two suitable marginals and the so-called local-toglobal property.

The Hartree-Fock equation which is the Euler-Lagrange equation corresponding to the Hartree-Fock energy functional is used in many-electron problems. Since the Hartree-Fock equation is a system of nonlinear eigenvalue problems, the study of structures of sets of all solutions needs new methods different from that for the set of eigenfunctions of linear operators. In this paper we prove that the sets of all solutions to the Hartree-Fock equation associated with critical values of the Hartree-Fock energy functional less than the first energy threshold are unions of a finite number of compact connected real-analytic spaces. The result would also be a basis for the study of approximation methods to solve the equation.

It is known that the class of developable surfaces which have zero Gaussian curvature in three dimensional Euclidean space is preserved by the parallel transformations. A tangent developable surface is defined as a ruled developable surface by tangent lines to a space curve and it has singularities at least along the space curve, called the directrix or the the edge of regression. Also the class of tangent developable surfaces are invariant under the parallel deformations. In this paper the notions of tangent developable surfaces and their parallels are naturally generalized for frontal curves in general in Euclidean spaces of arbitrary dimensions. We study singularities appearing on parallels to tangent developable surfaces of frontal curves and give the classification of generic singularities on them for frontal curves in 3 or 4 dimensional Euclidean spaces. Given a surface in R 3 , a "parallel surface" or a "parallel" to the surface is simply defined as a surface which have the common family of normal affine lines with the original surface. In fact, given a surface (u, v) → f (u, v) with a unit normal ν(u, v), its parallels are given by the surfaces f (u, v) + rν(u, v) with the parameter r ∈ R. In general a line congruence, i.e., a two-dimensional family of affine lines in R 3 is called a system of rays if it forms a (possibly singular) Lagrangian surface in the space of affine lines in R 3 [1, 2]. The condition is equivalent to that the family is expressed in a parametric form as it is an integral map for the standard contact structure on the unit tangent bundle R 3 × S 2 of the Euclidean space R 3 . Parallel deformations are regarded as an important and interesting transformations of surfaces in differential geometry. For instance, constant mean curvature surfaces and positive constant Gaussian curvature surfaces are related as parallels to each other. Parallels are studied deeply from geometric point of view of differential geometric surface theory (see [21] p.185, [18] p.225 for instance). It is characteristic also that, in the process of taking parallels, the surfaces may have

The purpose of the paper is to study the behavior of the classes of functions of bounded lower oscillation under the change of measure given by the weight $w$. More specifically, we provide sharp upper and lower bounds for the norm of the inclusion $BLO\hookrightarrow BLO(w)$ in terms of $A_\infty$ constant of the weight. The results hold in the general context of probability spaces equipped with a treelike structure.

We study nonorientable maximal surfaces in Lorentz-Minkowski 3-space. We prove some existence results for surfaces of this kind with high genus and one end.

We compute Stokes matrices for generalised Airy equations and prove that they are regular unipotent (up to multiplication with the formal monodromy). This class of differential equations was defined by Katz and includes the classical Airy equation. In addition, it includes differential equations which are not rigid. Our approach is based on the topological computation of Stokes matrices of the enhanced Fourier-Sato transform of a perverse sheaf due to D'Agnolo, Hien, Morando and Sabbah.