In this paper we investigate completely homogeneous Ltopological spaces. The smallest completely homogeneous Ltopology on a set X containing an Lset f is called the principal completely homogeneous Ltopology generated by f . Here we also study the principal completely homogeneous Ltopological spaces generated by an Lset and characterize completely homogeneous Alexandroff discrete Ltopological s...

We derive a formula for the η-invariants of equivariant Dirac operators on quotients of compact Lie groups, and for their infinitesimally equivariant extensions. As an example, we give some computations for spheres. Quotients M = G/H of compact Lie groups provide many important examples of Riemannian manifolds with non-negative sectional curvature. The primary characteristic classes and numbers...

θ. This definition is deficient as it depends on a choice of basis. A definition in R is that a rotation is a linear map g with an axis, a line L fixed by g, and on the orthogonal complement L⊥ of L the restriction of g is a two-dimensional rotation. For this to make sense, one must have understood that the two-dimensional definition is independent of basis, and that g does stabilize the orthog...

Let G be a compact connected Lie group with trivial center. Using the action of G on its Lie algebra, we define an operator norm | |G which induces a bi-invariant metric dG(x, y) = |Ad(yx−1)|G on G. We prove the existence of a constant β ≈ .23 (independent of G) such that for any closed subgroup H ( G, the diameter of the quotient G/H (in the induced metric) is ≥ β.

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain ows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprind zuk in 1964. We also prove several related hypotheses of Baker and Sprind zuk formulated in 1970...

This will be a course on arithmetic invariant theory. The goal of the course is to understand orbits of representations of algebraic groups over non-algebraically closed fields and eventually over Z. We will focus on representations that arise “naturally” from number theory, algebraic geometry, Vinberg theory, knot theory etc. 1 Let G be an algebraic group acting “integrally” on a vector space ...

Let G be a connected reductive group. To any irreducible G-variety one associates a certain linear group generated by reflections called the Weyl group. Weyl groups play an important role in the study of embeddings of homogeneous spaces. We establish algorithms for computing Weyl groups for homogeneous spaces and affine homogeneous vector bundles. For some special classes of G-varieties (affine...

We study curvature homogeneous spaces or locally homogeneous spaces whose curvature tensors are invariant by the action of “large” Lie subalgebras h of so(n). In this paper we deal with the cases of h = so(r)⊕ so(n− r) (2 ≤ r ≤ n− r), so(n− 2), and the Lie algebras of Lie groups acting transitively on spheres, and classify such curvature homogeneous spaces or locally homogeneous spaces.