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Igor Rivin

Igor Rivin appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.GTmath.GRmath.NTmath.PRmath-phmath.HOmath.MPcond-mat.stat-mechhep-thmath.COmath.DSSymbolic ComputationApplicationsComputational GeometryMachine Learningmath.AGmath.APmath.CAmath.CVmath.DGmath.FAmath.MGmath.SP

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Published work

26 published item(s)

preprint2026arXiv

Probing Structural Mathematical Reasoning in Language Models with Algebraic Trapdoors

We introduce a benchmark suite for evaluating structural mathematical reasoning in language models, built on subgroup-construction problems in SL(3, Z) with cryptographic-style verifier-prover asymmetry. Each instance presents a finitely generated subgroup as a list of integer matrices and asks for an arithmetic invariant -- index, surjection-at-prime, or membership -- that the construction-time information (N, K) pins down in O(1) closed form, but that the solver, lacking that information, must derive by either Aschbacher-classification analysis or by a membership query in SL(3, Z) of unknown decidability. The benchmark therefore distinguishes models with internalized algebraic priors (Aschbacher classes, McLaughlin's theorem, Property (T), the congruence subgroup property) from models that rely on general-purpose computation. We report empirical results across five representative reasoning traces from two state-of-the-art models. The headline result: on the index variant, one model spent 152 minutes of reasoning, explicitly identified the kernel-side membership question as the bottleneck, attempted constructive verification, and abstained with "DON'T KNOW" rather than commit to its computed cokernel candidate -- demonstrating calibrated meta-cognition on the open-decidability boundary that the benchmark was designed to probe. We argue that the benchmark exposes a four-way classification of model behavior (commit-correct, commit-wrong, abstain-correct, abstain-wrong) that standard answer-key scoring conflates.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Bibliometric Analysis of Senior US Mathematics Faculty

We introduce a methodology to analyze citation metrics across fields of Mathematics. We use this methodology to collect and analyze the MathSciNet profiles of Full Professors of Mathematics at all 131 R1, research oriented US universities. The data recorded was citations, field, and time since first publication. We perform basic analysis and provide a ranking of US math departments, based on age corrected and field adjusted citations.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Data Analysis of the Responses to Professor Abigail Thompson's Statement on Mandatory Diversity Statements

An opinion piece by Abigail Thompson in the Notices of the American Mathematical Society has engendered a lot of discussion, including three open letters with over 1400 signatures. We analyze the professional profiles of signatories of these three letters, and, in particular, their citation records. We find that when restricting to R1 math professors, the means of their citations and citations per year are ordered $μ(A) < μ(B) < μ(C)$. The significance of these findings are validated using a one-sided permutation test.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Random space and plane curves

We study random knots, which we define as a triple of random periodic functions (where a random function is a random trigonometric series, \[f(θ) = \sum_{k=1}^\infty a_k \cos (k θ) +b_k (\sin k θ),\] with $a_k, b_k$ are independent gaussian random variables with mean $0$ and variance $σ(k)^2$ - our results will depend on the functional dependence of $σ$ on $k.$ In particular, we show that if $σ(k) = k^α,$ with $α< -3/2,$ then the probability of getting a knot type which admits a projection with $N$ crossings, decays at least as fast as $1/N.$ The constant $3/2$ is significant, because having $α< -3/2$ is exactly the condition for $f(θ)$ to be a $C^1$ function, so our class is precisely the class of random \emph{tame} knots. We also find some suprising experimental observations on the zeros of Alexander polynomials of random knots (with slowly and non-decaying coefficients), and even more surprising observations on their coefficients. Our observations persist in other models of random knots, making it likely that the results are universal.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Galois Groups of Generic Polynomials

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 $\ll \frac{\log^{Ω(d)}N}{N}.$ 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.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Generic thinness in finitely generated subgroups of $\textrm{SL}_n(\mathbb Z)$

We show that for any $n\geq 2$, two elements selected uniformly at random from a \emph{symmetrized} Euclidean ball of radius $X$ in $\textrm{SL}_n(\mathbb Z)$ will generate a thin free group with probability tending to $1$ as $X\rightarrow \infty.$ This is done by showing that the two elements will form a ping-pong pair, when acting on a suitable space, with probability tending to $1$. On the other hand, we give an upper bound less than $1$ for the probability that two such elements will form a ping-pong pair in the usual Euclidean ball model in the case where $n>2$.

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preprint2015arXiv

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 $O(n^4 \log n \log \|\mathcal{G}\|)\log ε$ and in $\mathop{Sp}(2n, \mathbb{Z})$ the complexity is of order $O(n^8 \log n\log \|\mathcal{G}\|)\log ε$ In general semisimple groups we show that Zariski density can be confirmed or denied in time of order $O(n^14 \log \|\mathcal{G}\|\log ε),$ where $ε$ 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.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Four Random Permutations Conjugated by an Adversary Generate $S_n$ with High Probability

We prove a conjecture dating back to a 1978 paper of D.R.\ Musser~\cite{musserirred}, namely that four random permutations in the symmetric group $\mathcal{S}_n$ generate a transitive subgroup with probability $p_n > ε$ for some $ε> 0$ independent of $n$, even when an adversary is allowed to conjugate each of the four by a possibly different element of $§_n$ (in other words, the cycle types already guarantee generation of $\mathcal{S}_n$). This is closely related to the following random set model. A random set $M \subseteq \mathbb{Z}^+$ is generated by including each $n \geq 1$ independently with probability $1/n$. The sumset $\text{sumset}(M)$ is formed. Then at most four independent copies of $\text{sumset}(M)$ are needed before their mutual intersection is no longer infinite.

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preprint2014arXiv

Spectral Experiments+

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 \emph{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.

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preprint2014arXiv

Statistics of Random 3-Manifolds occasionally fibering over the circle

We study random elements of subgroups (and cosets) 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), the dilatation of the monodromy, the injectivity radius, and the bottom eigenvalue of the Laplacian on these mapping tori. We also study mapping tori of punctured surface bundles, and various invariants of their Dehn fillings. We also study corresponding questions in the Dunfield-Thurston model of random Heegard splittings of fixed genus, and give a number of new and improved results in that setting.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Topological Designs

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is large, both bounds are the best known for large genus. In genus one and two, we solve the problem exactly. Our methods generalize to variants in which the allowed number of pairwise intersections is odd, even, or bounded, and to surfaces with boundary components.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Geodesics with one self-intersection, and other stories

In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to L raised to the dimension of the Teichmuller space of S. Since closed geodesics with one double point fall into a finite number of orbits under the mapping class group of S, we get the same asympotic estimate for the number of such geodesics of length bounded by L. We also use our (elementary) methods to do a more precise study of geodesics with a single double point on a punctured torus, including an extension of McShane's identity to such geodesics. In the second part of the paper we study the question of when a covering of the boundary of an oriented surface S can be extended to a covering of the surface S itself, we obtain a complete answer to that question, and also to the question of when we can further require the extension to be a \emph{regular} covering of S. We also analyze the question (first raised by K. Bou-Rabee) of the minimal index of a subgroup in a surface group which does not contain a given element. We give a (conjecturally) sharp result graded by the depth of an element in the lower central series, as well as "ungraded" results.

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preprint2011arXiv

Rigidity of Fibering

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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preprint2011arXiv

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 mu on the complex numbers. We conjecture that the zero set of f' always converges in distribution to mu as n goes to infinity. We prove this for measures with finite one-dimensional energy. When mu is uniform on the unit circle this condition fails. In this special case the zero set of f' converges in distribution to that the IID Gaussian random power series, a well known determinantal point process.

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preprint2011arXiv

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.

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preprint2009arXiv

On extension of coverings

We address the question of when a covering of the boundary of a surface can be extended to a covering of the surface (equivalently: when is there a branched cover with a prescribed monodromy). If such an extension is possible, when can the total space be taken to be connected? When can the extension be taken to be regular? We give necessary and sufficient conditions for both finite and infinite covers (infinite covers are our main focus). In order to prove our results, we show group-theoretic results of independent interests, such as the following extension (and simplification) of the theorem of Ore}: every element of the infinite symmetric group is the commutator of two elements which, together, act transitively

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preprint2006arXiv

Counting Reducible Matrices, Polynomials, and Surface and Free Group Automorphisms

We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a random (in the appropriate sense) element of the mapping class group of a closed surface is pseudo-Anosov, and that a random automorphism of a free group is strongly irreducible (aka irreducible with irreducible powers). We also give a necessary condition for all powers of an algebraic integers to be of the same degree, and give a simple proof (in the Appendix) that the distribution of cycle structures modulo a prime p for polynomials with a restricted coefficient is the same as that for general polynomials.

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preprint2003arXiv

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.

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preprint2000arXiv

On the Schlafli differential formula

he celebrated formula of Schlafli 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. We apply our results to extend the classical Euclidean inequalities of Aleksandrov to other 3-dimensional constant curvature spaces. We also obtain rigidity results for Ricci-flat manifolds with umbilic boundaries and existence results for foliations of Einstein manifolds by hypersurfaces.

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preprint1992arXiv

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.

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