We compute the structure of the Lie algebras associated to twoexamples of branch groups, and show that one has finite width whilethe other, the "Gupta-Sidki group", has unbounded width(Corollary~
ef{cor:gamma:rk}). This answers a question by Sidki.We then draw a general result relating the growth of a branch group,of its Lie algebra, of its graded group ring, and of a naturalhomogeneous space we call emph{parabolic space}, namely thequotient of the group by the stabilizer of an infinite ray. Thegrowth of the group is bounded from below by the growth of itsgraded group ring, which connects to the growth of the Lie algebraby a product-sum formula, and the growth of the parabolic space isbounded from below by the growth of the Lie algebraFinally we use this information to explicitly describe the normalsubgroups of $G$, the "Grigorchuk group". All normal subgroupsare characteristic, and the number $b_n$ of normal subgroups of$G$ of index $2^n$ is odd and satisfies${limsup,liminf}b_n/n^{log_2(3)}={5^{log_2(3)},frac29}$.

We compute the structure of the Lie algebras associated to twoexamples of branch groups, and show that one has finite width whilethe other, the \"Gupta-Sidki group\", has unbounded width(Corollary~
ef{cor:gamma:rk}). This answers a question by Sidki.We then draw a general result relating the growth of a branch group,of its Lie algebra, of its graded group ring, and of a naturalhomogeneous space we call emph{parabolic space}, namely thequotient of the group by the stabilizer of an infinite ray. Thegrowth of the group is bounded from below by the growth of itsgraded group ring, which connects to the growth of the Lie algebraby a product-sum formula, and the growth of the parabolic space isbounded from below by the growth of the Lie algebraFinally we use this information to explicitly describe the normalsubgroups of $G$, the \"Grigorchuk group\". All normal subgroupsare characteristic, and the number $b_n$ of normal subgroups of$G$ of index $2^n$ is odd and satisfies${limsup,liminf}b_n/n^{log_2(3)}={5^{log_2(3)},frac29}$.

The permutation $pi$ of $1,2,ldots,n$ that satisfies> $$0 < { pi(1) alpha } < { pi(2) alpha } < cdots < { pi(n)alpha} < 1$$ ($alpha$ is any irrational) has been studied from a combinatorialviewpoint (S—s, Boyd) and from an analytic viewpoint (Schoi§engeier). I will present some results on algebraic properties of this permutation, the most significant being a mysterious appearance in the study of nearly periodic binary words (a.k.a., Sturmian words) with the representation theory of symmetric groups.

The Hilbert series of the graded ring associated to a projective variety contains a lot of information about the variety, {it e.g./}, dimension, degree, arithmetic genus etc. If the projective variety is nice then the Hilbert series can be (uniquely) writtenin the form $h(t)=f(t)/(1-t)^n$, where $f(t)$ is a polynomial with non-negative integer coefficients and $f(1)
ot =0$.In this talk we consider the Hilbert series of determinantal varieties (including symmetric and skew-symmetric determinantal varieties). These varieties arise naturally in many branches of mathematics, {it e.g.}, in classical invariant theory. We give an interpretation of the coefficients of the numerator $f(t)$ of the Hilbert series as dimensions of representations of certain compact Lie groups. (The presented work is joint work with Enright and is an extension of previous work by Enright and Willenbring.)