Igor Rivin
Temple University, Mathematics, Faculty Member
Research Interests:
We describe extensive computational experiments on spectral properties of random objects - random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere).... more
We describe extensive computational experiments on spectral properties of random objects - random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere). We look at bulk eigenvalue distribution, eigenvalue spacings, and locality properties of eigenvectors. We also look at the statistics of nodal domains of eigenvectors on these graphs. In all cases we discover completely new (at least to this author) phenomena. The author has tried to refrain from making specific conjectures, inviting the reader, instead, to meditate on the data.
In this paper, we combine a number of results (some due to the author, some not) to show that Zariski density is, in a strong sense, a generic property of subgroups of SL(n,Z) and Sp(2n,Z). We also extend previous results of the author on... more
In this paper, we combine a number of results (some due to the author, some not) to show that Zariski density is, in a strong sense, a generic property of subgroups of SL(n,Z) and Sp(2n,Z). We also extend previous results of the author on generic properties of elements of the special linear and symplectic groups to generic elements in arbitrary Zariski-dense (as subgroups of the ambient complex algebraic group) subgroups of these groups. Jointly with Ilya Kapovich, we also show that a generic free group automor-phism is hyperbolic. 1
Abstract. We analyze completely the convergence speed of the batch learning algorithm, and compare its speed to that of the memoryless learning algorithm and of learning with memory (as analyzed in [KR2001b]). We show that the batch... more
Abstract. We analyze completely the convergence speed of the batch learning algorithm, and compare its speed to that of the memoryless learning algorithm and of learning with memory (as analyzed in [KR2001b]). We show that the batch learning algorithm is never worse than the memoryless learning algorithm (at least asymptotically). Its performance vis-a-vis learning with full memory is less clearcut, and depends on certain probabilistic assumptions.
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Abstract. We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our experiments indicate that the level spacing distribution of a generic k-regular graph approaches that of the Gaussian Orthogonal Ensemble of... more
Abstract. We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our experiments indicate that the level spacing distribution of a generic k-regular graph approaches that of the Gaussian Orthogonal Ensemble of random matrix theory as we increase the number of vertices. A review of the basic facts on graphs and their spectra is included. 1.
Abstract. We study convex sets C of finite (but non-zero) volume inH n andE n. We show that the intersection C ∞ of any such set with the ideal boundary ofH n has Minkowski (and thus Hausdorff) dimension of at most (n−1)/2. and this bound... more
Abstract. We study convex sets C of finite (but non-zero) volume inH n andE n. We show that the intersection C ∞ of any such set with the ideal boundary ofH n has Minkowski (and thus Hausdorff) dimension of at most (n−1)/2. and this bound is sharp, at least in some dimensions n. We also show a sharp bound when C ∞ is a smooth submanifold of∂∞H n. In the hyperbolic case, we show that for any k≤(n−1)/2 there is a bounded section S of C through any prescribed p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of a convex body inE n, and give asymptotic estimates as 1≪k≪n..
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In this article we relate two different densities. Let Fk be the free group of finite rank k ≥ 2 and let α be the abelianization map from Fk onto Zk. We prove that if S ⊆ Zk is invariant under the natural action of SL(k, Z) then the... more
In this article we relate two different densities. Let Fk be the free group of finite rank k ≥ 2 and let α be the abelianization map from Fk onto Zk. We prove that if S ⊆ Zk is invariant under the natural action of SL(k, Z) then the asymptotic density of S in Zk and the annular density of its full preimage α−1(S) in Fk are equal. This implies, in particular, that for every integer t ≥ 1, the annular density of the set of elements in Fk that map to t-th powers of primitive elements in Zk is equal to 1 tkζ(k) , where ζ is the Riemann zeta-function. An element g of a group G is called a test element if every endomorphism of G which fixes g is an automorphism of G. As an application of the result above we prove that the annular density of the set of all test elements in the free group F (a, b) of rank two is 1 − 6 π2 . Equivalently, this shows that the union of all proper retracts in F (a, b) has annular density 6 π2 . Thus being a test element in F (a, b) is an “intermediate property” ...
We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coefficients between $-N$ and $N$ is NOT $S_d$ with probability $\ll \frac{\log^{\Omega(d)}N}{N}.$ Conditionally on NOTbeing the full symmetric... more
We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coefficients between $-N$ and $N$ is NOT $S_d$ with probability $\ll \frac{\log^{\Omega(d)}N}{N}.$ Conditionally on NOTbeing the full symmetric group, we have a hierarchy of possibilities each of which has polylog probability of occurring. These results also apply to random polynomials with only a subset of the coefficients allowed to vary. This settles a question going back to 1936.
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We take a look the changes of different asset prices over variable periods, using both traditional and spectral methods, and discover universality phenomena which hold (in some cases) across asset classes.
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In this note we show that the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of... more
In this note we show that the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also construct families of non-congurent simplices not determined by their codimension-2 areas.
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The Sharpe ratio is the most widely used risk metric in the quantitative finance community - amazingly, essentially everyone gets it wrong. In this note, we will make a quixotic effort to rectify the situation.
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We discuss some (numerical and theoretical) results about the coefficients and zeros of Tutte (dichromatic) polynomial of graphs of bounded degree whose size increases. We also discuss related results for Bollobás-Riordan polynomials.
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We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power... more
We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power sums.
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Let T be a once punctured torus, equipped with a complete hyperbolic metric. Herein, we describe a new approach to the study of the set S of all simple geodesics on T. We introduce a valuation on the homology H1(T,ZZ), which associates to... more
Let T be a once punctured torus, equipped with a complete hyperbolic metric. Herein, we describe a new approach to the study of the set S of all simple geodesics on T. We introduce a valuation on the homology H1(T,ZZ), which associates to each homology class h the length l(h) of the unique simple geodesic homologous to h, and show that l extends to a norm on H1(T,R). We analyze the boundary of the unit ball B(l) and the variation of the area of B(l) over the moduli space of T . These results are applied to obtain sharp asymptotic estimates on the number of simple geodesics of length less than L. Courbes simples dans les tores Résumé. Soit T un tore troué, muni d’une métrique hyperbolique complète, d’aire finie. Nous présentons une nouvelle approche de l’étude de l’ensemble S de toutes les géodésiques fermées simples (sans points doubles) de T . Nous introduisons une application sur l’homologie H1(T,ZZ), qui associe à chaque classe h ∈ H1(T,ZZ) indivisible la longueur l(h) de l’uniqu...
A. We study the surface area of an ellipsoid in E as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over Sn−1, use the formula to derive convexity properties of the surface... more
A. We study the surface area of an ellipsoid in E as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over Sn−1, use the formula to derive convexity properties of the surface area, to sharpen the estimates given [6], to produce asymptotic formulas in large dimensions, and to give an expression for the surface in terms of the Lauricella hypergeometric function.
We use the order complex corresponding to a symmetric matrix (defined by Giusti et al in 2015). In this note, we use it to define a class of models of random graphs, and show some surprising experimental results, showing sharp phase... more
We use the order complex corresponding to a symmetric matrix (defined by Giusti et al in 2015). In this note, we use it to define a class of models of random graphs, and show some surprising experimental results, showing sharp phase transitions.
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In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the... more
In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles. A proof of this has recently been given by F. Luo (see math.GT/0412208). In this paper we give a simple proof of this conjecture, prove much sharper regularity results, and then extend the method to apply to a large class of convex polytopes. The simplex argument works without change in dimensions greater than 3 (and for spherical simplices in all dimensions), so the bulk of this paper is concerned with the three-dimensional argument. The estimates relating the diameter of a polyhedron to the length of the systole of the polar polyhedron are of independent interest.
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In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the... more
In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles (``The continuity conjecture''), and furthermore, the limit at a boundary point is equal to 0 if and only if the point lies in the closure of the space of angles of Euclidean tetrahedra (``the Vanishing Conjecture''). A proof of the Continuity Conjecture was given by F. Luo -- Luo's argument uses Kneser's formula for the volume together with some delicate geometric estimates). In this paper we give a simple proof of both parts of Milnor's conjecture, prove much sharper regularity results, and then extend the method to apply to all convex polytopes. We also give a precise description of the boundary of the space of angles of convex polyhedra in and sharp estimates on the diameter of a polyhedron in ...
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Given a manifold M, it is natural to ask in how many ways it fibers (we mean fibering in a general way, where the base might be an orbifold -- this could be described as Seifert fibering)There are group-theoretic obstructions to the... more
Given a manifold M, it is natural to ask in how many ways it fibers (we mean fibering in a general way, where the base might be an orbifold -- this could be described as Seifert fibering)There are group-theoretic obstructions to the existence of even one fibering, and in some cases (such as Kahler manifolds or three-dimensional manifolds) the question reduces to a group-theoretic question. In this note we summarize the author's state of knowledge of the subject.
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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... more
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.
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The celebrated formula of Schläfli relates the variation of the dihedral angles of a smooth family of polyhedra in a space form and the variation of volume. We give a smooth analogue of this classical formula – our result relates the... more
The celebrated formula of Schläfli relates the variation of the dihedral angles of a smooth family of polyhedra in a space form and the variation of volume. We give a smooth analogue of this classical formula – our result relates 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. The argument is direct, and the classical polyhedral result (as well as results for Lorenzian space forms) is an easy corollary. We extend it to variations of the metric in a Riemannian Einstein manifold with boundary.
The celebrated formula of Schlafli relates the variation of the dihedral angles of asmooth family of polyhedra in a space form and the variation of volume. We give asmooth analogue of this classical formula -- our result relates the... more
The celebrated formula of Schlafli relates the variation of the dihedral angles of asmooth family of polyhedra in a space form and the variation of volume. We give asmooth analogue of this classical formula -- our result relates the variation of the volumebounded by a hypersurface moving in a general Einstein manifold and the integral of thevariation of the mean curvature. The argument is direct, and the classical polyhedralresult (as well as results for Lorenzian space forms) is an easy...
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We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups.
An extra large metric is a spherical cone metric with all cone angles greater than 2 pi and every closed geodesic longer than 2pi. We show that every two-dimensional extra large metric can be triangulated with vertices at cone points... more
An extra large metric is a spherical cone metric with all cone angles greater than 2 pi and every closed geodesic longer than 2pi. We show that every two-dimensional extra large metric can be triangulated with vertices at cone points only. The argument implies the same result for Euclidean and hyperbolic cone metrics, and can be modified to show a similar result for higher-dimensional extra-large metrics. The extra-large hypothesis is necessary.
I describe some deep-seated problems in higher mathematical education, and give some ideas for their solution -- I advocate a move away from the traditional introduction of mathematics through calculus, and towards computation and... more
I describe some deep-seated problems in higher mathematical education, and give some ideas for their solution -- I advocate a move away from the traditional introduction of mathematics through calculus, and towards computation and discrete mathematics.
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We obtain sharp bounds for the number of n--cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. We prove sharp estimates on both the sum of k-th... more
We obtain sharp bounds for the number of n--cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. We prove sharp estimates on both the sum of k-th powers of the coordinates and the Lk norm subject to the constraints that the sum of squares of the coordinates is fixed, and that the sum of the coordinates vanishes.
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ABSTRACT We discuss the question of how to pick a matrix uniformly (in an appropriate sense) at random from groups big and small. We give algorithms in some cases, and indicate interesting problems in others.
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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... more
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.
In this note we show that the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of... more
In this note we show that the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also construct families of non-congurent simplices not determined by their codimension-2 areas.
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We study random elements of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational,... more
We study random elements of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume (whenever the manifold is hyperbolic), and the dilatation.
We use the above results to show that a random (in terms of word length) element of the mapping class group of a surface is pseudo-Anosov, and that a a random free group automorphism is irreducible with irreducible powers (or strongly... more
We use the above results to show that a random (in terms of word length) element of the mapping class group of a surface is pseudo-Anosov, and that a a random free group automorphism is irreducible with irreducible powers (or strongly irreducible).