We give a short and almost self-contained proof of generalizations of Kollár’s vanishing and torsion-free theorems. Although they are contained in Ambro's much more general results on embedded normal crossing pairs, we give an alternate and direct reduction argument to the mixed Hodge theory. In this sense, this paper gives a more readable account of the application to the log minimal model program for log canonical pairs.

In a recent paper in this Proceedings, H. Okamoto presented a parameterized family of continuous functions which contains Bourbaki’s and Perkins’s nowhere differentiable functions as well as the Cantor-Lebesgue singular function. He showed that the function changes it’s differentiability from ‘differentiable almost everywhere’ to ‘non-differentiable almost everywhere’ at a certain parameter value. However, differentiability of the function at the critical parameter value remained unknown. For this problem, we prove that the function is non-differentiable almost everywhere at the critical case.

We study group actions on a coarse space and the induced actions on the Higson corona from a dynamical point of view. Our main theorem states that if an action of an abelian group on a proper metric space satisfies certain conditions, the induced action has a fixed point in the Higson corona. As a corollary, we deduce a coarse version of Brouwer’s fixed point theorem.

In this paper we construct Löwner chains which enable us to derive quasiconformal extension criteria for typical classes of univalnet functions. This method also provides us explicit quasiconformal extensions.

Green function of the clamped-free boundary value problem for (-1)^M(d/dx)^2M on the interval (-1,1) is obtained. Its Green function is a reproducing kernel for a suitable set of Hilbert space and an inner product. By using the fact, the best constant of Sobolev inequality corresponding to this boundary value problem is obtained as a function of M. The best constant is the maximal value of the diagonal value G(y,y) of Green function G(x,y).

We construct some explicit Galois extensions of Q (or integral polynomials) with Galois group W(E₈) via Mordell-Weil lattices.