Igor Rivin

A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere

We describe a characterization of convex polyhedra in H 3 in terms of their dihedral angles, developed by Rivin. We also describe some geometric and combinatorial consequences of that theory. One of these consequences is a combinatorial characterization of convex polyhedra in E 3 all of whose vertices lie on the unit sphere. That resolves a problem posed by Jakob Steiner in 1832

Combinatorial optimization in geometry

In this paper we extend and unify the results of [Rivin, Ann. of Math. 143 (1996)] and [Rivin, Ann. of Math. 139 (1994)]. As a consequence, the results of [Rivin, Ann. of Math. 143 (1996)] are generalized from the framework of ideal polyhedra in H^3 to that of singular Euclidean structures on surfaces, possibly with an infinite number of singularities (by contrast, the results of [Rivin, Ann. of Math. 143 (1996)] can be viewed as applying to the case of non-singular structures on the disk, with a finite number of distinguished points). This leads to a fairly complete understanding of the moduli space of such Euclidean structures and thus, by the results of [Penner, Comm. Math. Phys. 113 (1987) 299-339; Epstein, Penner, J. Differential Geom. 27 (1988) 67-80; Naatanen, Penner, Bull. London Math. Soc. 6 (1991) 568-574] the author [Rivin, Ann. of Math. 139 (1994); Rivin, in: Lecture Notes in Pure and Appl. Math., Vol. 156, 1994], and others, further insights into the geometry and topology of the Riemann moduli space. The basic objects studied are the canonical Delaunay triangulations associated to the aforementioned Euclidean structures. The basic tools, in addition to the results of [Rivin, Ann. of Math. 139 (1994)] and combinatorial geometry are methods of combinatorial optimization-linear programming and network flow analysis; hence the results mentioned above are not only effective but also efficient. Some applications of these methods to three-dimensional topology are also given (to prove a result of Cassons).

Simple Curves on Surfaces

We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L6g+2b+2c−6. This answers a long-standing open question.

A norm on homology of surfaces and counting simple geodesics

We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.

A dynamic programming solution to the n -queens problem

Rivin, I. and R. Zabih, A dynamic programming solution to the n-queens problem, Information Processing Letters 41 (1992) 253-256. The n-queens problem is to determine in how many ways n queens may be placed on an n-by-n chessboard so that no two queens attack each other under the rules of chess. We describe a simple O( f(n)8”) solution to this problem that is based on dynamic programming, where f(n) is a low-order polynomial. This appears to be the first nontrivial upper bound for the problem.

Harmonic mean, random polynomials and stochastic matrices

Motivated by a problem in learning theory, we are led to study the dominant eigenvalue of a class of random matrices. This turns out to be related to the roots of the derivative of random polynomials (generated by picking their roots uniformly at random in the interval [0,1], although our results extend to other distributions). We produce sharp results on the statistical properties of the smallest critical point. This, in turn, requires the study of the statistical behavior of the harmonic mean of i.i.d. random variables, and we produce a number of limiting distributions and laws of large numbers.

The Schläfli formula in Einstein manifolds with boundary

We give a smooth analogue of the classical Schlafli formula, relating the variation of the volume bounded by a hypersurface moving in a general Einstein manifold and the integral of the variation of the mean curvature. We extend it to variations of the metric in a Riemannian Einstein manifold with boundary, and apply it to Einstein cone-manifolds, to isometric deformations of Euclidean hypersurfaces, and to the rigidity of Ricci-flat manifolds with umbilic boundaries. Resume. On donne un analogue regulier de la formule classique de Schlafli, reliant la variation du volume borne par une hypersurface se deplacant dans une variete d’Einstein a l’integrale de la variation de la courbure moyenne. Puis nous l’etendons aux variations de la metrique a l’interieur d’une variete d’Einstein riemannienne a bord. On l’applique aux cone-varietes d’Einstein, aux deformations isometriques d’hypersurfaces de En, et a la rigidite des varietes Ricci-plates a bord ombilique. Let M be a Riemannian (m + 1)-dimensional space-form of constant curvature K, and (Pt)t∈[0,1] a one-parameter family of polyhedra in M bounding compact domains, all having the same combinatorics. Call Vt the volume bounded by Pt, θi,t and Wi,t the dihedral angle and the (m− 1)-volume of the codimension 2 face i of Pt. The classical Schlafli formula (see [Mil94] or [Vin93]) is ∑ i Wi,t dθi,t dt = mK dVt dt . (1) This formula has been extended and used on several occasions recently; see for instance [Hod86], [Bon]. We give a smooth version of this formula, for 1-parameter families of hypersurfaces in (Riemannian of Lorentzian) Einstein manifolds. Then we extend it to variations of an Einstein metric inside a manifold with boundary (a much more general process in dimension above 3). Finally, we give three applications: to the variation of the volume of Einstein cone-manifolds, to isometric deformations of hypersurfaces in the Euclidean space, and to the rigidity of Ricci-flat manifolds with umbilic boundaries. The reader can find the details in [RS98]. Throughout this paper, M is an Einstein manifold of dimension m + 1 ≥ 3, and D is its Levi-Civita connection. When dealing with a hypersurface Σ (resp. with Received by the editors July 31, 1998. 1991 Mathematics Subject Classification. Primary 53C21; Secondary 53C25.

The Distribution of Zeros of the Derivative of a Random Polynomial

In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure μ on \(\mathbb{C}\). We conjecture that the zero set of f ′ always converges in distribution to μ as n → ∞. We prove this for measures with finite one-dimensional energy. When μ is uniform on the unit circle this condition fails. In this special case the zero set of f ′ converges in distribution to that of the IID Gaussian random power series, a well known determinantal point process.

Walks on graphs and lattices – effective bounds and applications

Abstract We continue the investigations started in [Rivin, Technical Report math., 1999, Rivin, Duke Math. J., 2006]. We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite group Γ. We consider all walks of length N on G, starting from νi and ending at νj . To each such walk w we assign the element of Γ equal to the product of the elements along the walk. The set of all walks of length N from νi to νj thus induces a probability distribution FN,i,j on Γ. In [Rivin, Technical Report math., 1999] we give necessary and sufficient conditions for the limit as N goes to infinity of FN,i,j to exist and to be the uniform density on Γ (a detailed argument is presented in [Rivin, Duke Math. J., 2006]). The convergence speed is then exponential in N. In this paper we consider (G, Γ), where Γ is a group possessing Kazhdans property T (or, less restrictively, property τ with respect to representations with finite image), and a family of homomorphisms ψk : Γ → Γ k with finite image. Each FN,i,j induces a distribution on Γ k (by push-forward under ψk ). Our main result is that, under mild technical assumptions, the exponential rate of convergence of to the uniform distribution on Γ k does not depend on k. As an application, we prove effective versions of the results of [Rivin, Duke Math. J., 2006] on the probability that a random (in a suitable sense) element of SL(n, ℤ) or Sp(n, ℤ) has irreducible characteristic polynomial, generic Galois group, etc.

Tight bounds on transition to perfect generalization in perceptrons

Sudden transition to perfect generalization in binary perceptrons is investigated. Building on recent theoretical works of Gardner and Derrida (1989) and Baum and Lyuu (1991), we show the following: for > c = 1.44797 , if n examples are drawn from the uniform distribution on {1, 1}n and classified according to a target perceptron wt {1, 1}n as positive if wt x 0 and negative otherwise, then the expected number of nontarget perceptrons consistent with the examples is 2(n); the same number, however, grows exponentially 2(n) if c. Numerical calculations for n up to 1,000,002 are reported.