We show that a smooth unknotted curve in ℝ³ satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve, and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r ². In the direction of lower bounds, we give a sequence of length one curves with r → 0 for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1/r. In particular, given any constant A, there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than A.

We show that the only compact positively curved Riemannian manifolds of even dimension acted on by a simple group with a codimension one orbit are the compact rank one symmetric spaces.

We will define a Ruelle–Selberg type zeta function for a certain lomathcal system over a Riemann surface whose genus is greater than or equal to three. Also we will investigate its property, especially their special values. As an application, we will show that a geometric analogue of BSD conjecture is true for a family of abelian varieties which has only semi-stable reductions defined over the complex number field.

We give a description of an affine mapping T involving contact pairs of two general convex bodies K and L, when T(K) is in a position of maximal volume in L. This extends the classical John's theorem of 1948, and is applied to the solution of a problem of Grünbaum; namely, any two convex bodies K and L in ℝ ⁿ have non-degenerate affine images K′ and L′ such that K′ ⊂ L′ ⊂ - n K′. As a corollary, we obtain that if L has a center of symmetry, then there are non-degenerate affine images K″ and L″ of K and L such that K″ ⊂ L″ ⊂ n K″. Other applications to volume ratios and distance estimates are given. In particular, the Banach-Mazur distance between the n-dimensional simplex and any centrally symmetric convex body is equal to n.

We develop a new approach for the study of “typical” Riemann surfaces with high genus. The method that we use is the construction of random Riemann surfaces from oriented cubic graphs. This construction enables us to get a control over the global geometry properties of compact Riemann surfaces. We use the theory of random regular graphs to show that almost all such surfaces have large first eigenvalues and large Cheeger constants. Moreover a closer analysis of the probability space of oriented cubic graphs shows that on a typical surface there is a large embedded hyperbolic ball.

A direct approach is used to establish both Ball and Barthe's reverse isoperimetric inequalities for the unit balls of subspaces of L _p . This approach has the advantage that it completely settles all the open uniqueness questions for these inequalities.