Spheres and Minima

We write down a one-dimensional integral formula and compute large-n asymptotics for the expectation of the absolute value of the smallest component of a unit vector in n-dimensional Euclidean space. The method is general, and allows to write the mean over the sphere of an homogeneous function in terms of an expectation of a function of independent, identically distributed Gaussians. We also write down an asymptotic formula for the minimum of a large number of identical independent positive random variables

Surface area and other measures of ellipsoids

We begin by studying the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of the semi-axes. We write down an explicit formula as an integral over the unit sphere in n-dimensions and use this formula to derive convexity properties of the surface area, to give sharp estimates for the surface area of a large-dimensional ellipsoid, to produce asymptotic formulas for the surface area and the \emph{isoperimetric ratio} of an ellipsoid in large dimensions, and to give an expression for the surface in terms of the Lauricella hypergeometric function. We then write down general formulas for the volumes of projections of ellipsoids, and use them to extend the above-mentioned results to give explicit and approximate formulas for the higher integral mean curvatures of ellipsoids. Some of our results can be expressed as ISOPERIMETRIC results for higher mean curvatures.Comment: Supercedes preprint math.MG/030638

An extended correction to ``Combinatorial Scalar Curvature and Rigidity of Ball Packings,'' (by D. Cooper and I. Rivin)

It has been pointed out to the author by David Glickenstein that the proof of the (closely related) Lemmas 1.2 and 3.2 in the title paper is incorrect. The statements of both Lemmas are correct, and the purpose of this note is to give a correct argument. The argument is of some interest in its own right.Comment: Correction to Math. Research Letters articl

Walks on Free Groups and other Stories -- twelve years later

We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their abelianization (or equivalently, their integer homology class. We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions modulo an arbitrary prime of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.Comment: 45pp, appeared in the Schupp volume of the Illinois Journal of Mathematics, published version of arXiv:math/991107

Growth in free groups (and other stories)

We start by studying the distribution of (cyclically reduced) elements of the free groups with respect to their abelianization. We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions $\mod p$ ($p$ -- an arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.Comment: 28 Pages, 1998 Preprin

Combinatorial optimization in geometry

We study the moduli space of euclidean structures with cone points on a surface, and describe a decomposition into cells each of which corresponds to a given combinatorial type of Delaunay tessellation. We use some of the ideas to study hyperbolic structures on three-dimensional manifoldsComment: 27 pages, 1996 preprin

On some mean matrix inequalities of dynamical interest

Let A be an n by n matrix with determinant 1. We show that for all n > 2 there exist dimensional strictly positive constants C_n such that the average over the orthogonal group of log rho(A X) d X > C_n log ||A||, where ||A|| denotes the operator norm of A (which equals the largest singular value of A), rho denotes the spectral radius, and the integral is with respect to the Haar measure on O_n The same result (with essentially the same proof) holds for the unitary group U_n in place of the orthogonal group. The result does not hold in dimension 2. We also give a simple proof that the average value over the unit sphere of log ||A u|| is nonnegative, and vanishes only when A is orthogonal.Comment: 11 pages; revision shows notes that it is essentially necessary to use Haar measure (class

Symmetrized Chebyshev Polynomials

We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials, and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative. We further show that a Central Limit Theorem holds for our polynomials.Comment: Enhancement of math.CA/030121

Golden-Thompson from Davis

We give a very short proof of the Golden-Thompson inequalit

Large Galois groups with applications to Zariski density

We introduce the first provably efficient algorithm to check if a finitely generated subgroup of an almost simple semi-simple group over the rationals is Zariski-dense. We reduce this question to one of computing Galois groups, and to this end we describe efficient algorithms to check if the Galois group of a polynomial $p$ with integer coefficients is "generic" (which, for arbitrary polynomials of degree $n$ means the full symmetric group $S_n,$ while for reciprocal polynomials of degree $2n$ it means the hyperoctahedral group $C_2 \wr S_n.$). We give efficient algorithms to verify that a polynomial has Galois group $S_n,$ and that a reciprocal polynomial has Galois group $C_2 \wr S_n.$ We show how these algorithms give efficient algorithms to check if a set of matrices $\mathcal{G}$ in $\mathop{SL}(n, \mathbb{Z})$ or $\mathop{Sp}(2n, \mathbb{Z})$ generate a \emph{Zariski dense} subgroup. The complexity of doing this in$\mathop{SL}(n, \mathbb{Z})$ is of order $O(n^4 \log n \log \|\mathcal{G}\|)\log \epsilon$ and in $\mathop{Sp}(2n, \mathbb{Z})$ the complexity is of order $O(n^8 \log n\log \|\mathcal{G}\|)\log \epsilon$ In general semisimple groups we show that Zariski density can be confirmed or denied in time of order $O(n^14 \log \|\mathcal{G}\|\log \epsilon),$ where $\epsilon$ is the probability of a wrong "NO" answer, while $\|\mathcal{G}\|$ is the measure of complexity of the input (the maximum of the Frobenius norms of the generating matrices). The algorithms work essentially without change over algebraic number fields, and in other semi-simple groups. However, we restrict to the case of the special linear and symplectic groups and rational coefficients in the interest of clarity.Comment: 25 page