Igor Rivin
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Igor Rivin has a total of 23 co-authors. Here are the top co-authors by number of shared publications:
- Ilya Kapovich
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA
3 shared publications - Alexander Kolpakov
University of Austin, Austin TX, USA
2 shared publications - Bojan Mohar
Department of Mathematics Simon Fraser University Burnaby British Columbia Canada
2 shared publications - Louis Theran
School of Mathematics and Statistics, University of St Andrews, St Andrews, Scotland
2 shared publications - Paul Schupp
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA
2 shared publications - Vladimir Shpilrain
Department of Mathematics , 14770 The City College of New York , New York , NY 10031 , USA
2 shared publications
Latest Publications
- Open AccessMonte-Carlo Irreducibility and Imprimitivity Detection of Polynomials over $\mathbb{Q}$We study fast Monte-Carlo methods for testing irreducibility and detecting arithmetic imprimitivity of polynomials over $\mathbb{Q}$. Building on the subset-sum criterion of Pemantle-Peres-Rivin, we develop a probabilistic irreducibility test whose expected running time, measured in the number of primes examined, is logarithmic in the degree for generic inputs. Unlike the standard modular irreducibility test, the method aggregates information from modular factorizations rather than discarding unsuccessful trials. We show that failure of this test, when combined with a standard modular irreducibility certificate, is a strong indicator of non-generic algebraic structure. In particular, it often signals arithmetic imprimitivity of the Galois action. We present an explicit and efficient Monte-Carlo algorithm for detecting such imprimitivity via subfield extraction, yielding constructive algebraic certificates in the imprimitive case. To our knowledge, this is the first practical algorithm for detecting arithmetic imprimitivity of polynomials over $\mathbb{Q}$ in high degree. We further show that the subset-sum data produced by the Pemantle--Peres--Rivin test provides a warm start for polynomial factorization by sharply restricting the possible degrees of rational factors, significantly accelerating subsequent lifting procedures. The proposed methods are orders of magnitude faster in practice than known deterministic algorithms, and are effective in degrees far beyond the reach of current deterministic techniques.Open AccessMonte-Carlo Irreducibility and Imprimitivity Detection of Polynomials over $\mathbb{Q}$We study fast Monte-Carlo methods for testing irreducibility and detecting arithmetic imprimitivity of polynomials over $\mathbb{Q}$. Building on the subset-sum criterion of Pemantle-Peres-Rivin, we develop a probabilistic irreducibility test whose expected running time, measured in the number of primes examined, is logarithmic in the degree for generic inputs. Unlike the standard modular irreducibility test, the method aggregates information from modular factorizations rather than discarding unsuccessful trials. We show that failure of this test, when combined with a standard modular irreducibility certificate, is a strong indicator of non-generic algebraic structure. In particular, it often signals arithmetic imprimitivity of the Galois action. We present an explicit and efficient Monte-Carlo algorithm for detecting such imprimitivity via subfield extraction, yielding constructive algebraic certificates in the imprimitive case. To our knowledge, this is the first practical algorithm for detecting arithmetic imprimitivity of polynomials over $\mathbb{Q}$ in high degree. We further show that the subset-sum data produced by the Pemantle--Peres--Rivin test provides a warm start for polynomial factorization by sharply restricting the possible degrees of rational factors, significantly accelerating subsequent lifting procedures. The proposed methods are orders of magnitude faster in practice than known deterministic algorithms, and are effective in degrees far beyond the reach of current deterministic techniques.
- Open Access
- 25 September 2025
- Published by The Open Journal in Journal of Open Source Software
- Vol. 10 (113) , 8855
Graphem - JAX: Node Influence Maximization via Geometric Embeddings
Computing classical centrality measures such as betweenness and closeness is computationally expensive on large graphs. Graphem-JAX provides an efficient force layout algorithm that embeds a graph into a low-dimensional space in such a way that the radial distance from the origin serves as a proxy for various centrality measures. Our method shows strong correlations with degree, PageRank, and paths-based centralities on multiple common graph families such as Erdős-Rényi random graphs, Watts-Strogatz "small world" graphs, and a few other. We also provide benchmarks on real world datasets such as Social Circles (Facebook) and Wikipedia Vote Network. Moreover in practice, Graphem-JAX allows one to find high-influence nodes in a network, and provides a fast and scalable alternative to the standard greedy algorithm.
- 25 September 2025
- Published by The Open Journal in Journal of Open Source Software
- Vol. 10 (113) , 8855
Open AccessGraphem - JAX: Node Influence Maximization via Geometric Embeddings
Computing classical centrality measures such as betweenness and closeness is computationally expensive on large graphs. Graphem-JAX provides an efficient force layout algorithm that embeds a graph into a low-dimensional space in such a way that the radial distance from the origin serves as a proxy for various centrality measures. Our method shows strong correlations with degree, PageRank, and paths-based centralities on multiple common graph families such as Erdős-Rényi random graphs, Watts-Strogatz "small world" graphs, and a few other. We also provide benchmarks on real world datasets such as Social Circles (Facebook) and Wikipedia Vote Network. Moreover in practice, Graphem-JAX allows one to find high-influence nodes in a network, and provides a fast and scalable alternative to the standard greedy algorithm.
- Open Access
- 30 August 2025
- Published by Association for Mathematical Research in Journal of Experimental Mathematics
- Vol. 1 (2) , 375-386
Random Graphs from Random Matrices
We use the order complex corresponding to a symmetric matrix (introduced 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.
- 30 August 2025
- Published by Association for Mathematical Research in Journal of Experimental Mathematics
- Vol. 1 (2) , 375-386
Open AccessRandom Graphs from Random Matrices
We use the order complex corresponding to a symmetric matrix (introduced 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.
- Open AccessComputing Periodic Billiard Orbits in $L^p$ Balls via Newton's Method and Smale's $α$-CriterionWe present a computational method for finding and verifying periodic billiard orbits in $L^{p}$ balls ($p>2$) using Newton's method applied to a variational formulation. The orbits are verified with Smale's alpha-criterion, which provides a rigorous certificate of existence. We implement efficient batched computations in JAX and present systematic results for various $p$ and bounce counts $N$. Our experiments reveal striking patterns in the critical-point structure, including a predominance of specific Morse signatures and rotation numbers that depend on the parity and primality of $N$. Notably, our method routinely finds many more than the two periodic orbits per rotation number guaranteed by Birkhoff's theorem -- a large-scale run with five bounces in the $L^{3}$ ball produced 8,927 distinct certified orbits from 30,000 random seeds, uncovering power-law growth and intricate clustering visualised with UMAP.Open AccessComputing Periodic Billiard Orbits in $L^p$ Balls via Newton's Method and Smale's $α$-CriterionWe present a computational method for finding and verifying periodic billiard orbits in $L^{p}$ balls ($p>2$) using Newton's method applied to a variational formulation. The orbits are verified with Smale's alpha-criterion, which provides a rigorous certificate of existence. We implement efficient batched computations in JAX and present systematic results for various $p$ and bounce counts $N$. Our experiments reveal striking patterns in the critical-point structure, including a predominance of specific Morse signatures and rotation numbers that depend on the parity and primality of $N$. Notably, our method routinely finds many more than the two periodic orbits per rotation number guaranteed by Birkhoff's theorem -- a large-scale run with five bounces in the $L^{3}$ ball produced 8,927 distinct certified orbits from 30,000 random seeds, uncovering power-law growth and intricate clustering visualised with UMAP.
- Open Access
- 13 December 2024
- Published by Association for Computing Machinery (ACM)
- p. 75-81
Discovering temporal and personality aspects in meager and highly variable text samples
- 13 December 2024
- Published by Association for Computing Machinery (ACM)
- p. 75-81
Open AccessDiscovering temporal and personality aspects in meager and highly variable text samples
- Open Access
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 $\mu(A) < \mu(B) < \mu(C)$. The significance of these findings are validated using a one-sided permutation test.Open AccessData 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 $\mu(A) < \mu(B) < \mu(C)$. The significance of these findings are validated using a one-sided permutation test. - Open Access
Random Graphs from Random Matrices
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.
Open AccessRandom Graphs from Random Matrices
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.
- Open Access
Experiments with the Census
In this paper we study the manifolds in the census of "small" 3-manifolds as available in SnapPy. We compare our results with the statistics of random 3-manifolds obtained using the Dunfield Thurston and Rivin models.
Open AccessExperiments with the Census
In this paper we study the manifolds in the census of "small" 3-manifolds as available in SnapPy. We compare our results with the statistics of random 3-manifolds obtained using the Dunfield Thurston and Rivin models.
- Open Access
Fear Universality and Doubt in Asset price movements
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.
Open AccessFear Universality and Doubt in Asset price movements
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.