We study the Morava $K$-theory of the exceptional Lie groups at the prime 2, and of certain projective Lie groups at a variety of primes.

In this paper we shall define certain loop groups which act on simply connected $H$-surfaces in space forms preserving conformality, and obtain a criterion for these group actions to be equivariant.

We generalize the Gauss-Bonnet theorem for Alexandrov surfaces and show that we can define the Gaussian curvature almost everywhere on an Alexandrov surface.

Let $X$ be a codimension two nonsingular subvariety of a nonsingular quadric $L^2$ of dimension $n\ge 5$. We classify such subvarieties when they are scrolls. We also classify them when the degree $d\le 10$. Both results were known when $n=4$.

In this paper we consider Jørgensen's inequalities for classical Schottky groups of the real type. The infimum of Jørgensen's numbers for groups of types II, V and VII are 16, 4 $(1+\sqrt{2}{)}^2$ and 4 $(1+\sqrt{2}{)}^2$, respectively, each of which is the best possible for Jørgensen's inequality.

We use N. Tanaka's theory of normal Cartan connections associated with simple graded Lie algebras to study Cartan's equivalence problem of single third order ordinary differential equations under contact transformations. As a result we obtain the complete structure equation with two differential inv riants, which is applied on general Legendre Grassmann bundles of three-dimensional contact manifolds.