arXiv subjects

Jordan S. Ellenberg

Publications and source records attributed to Jordan S. Ellenberg.

At least 19 recordsLinked to original sources

arXiv2026

Recent Progress around Cohen-Lenstra Heuristics

In 1983, Henri Cohen and Hendrik Lenstra proposed a conjecture about the distribution of the N-torsion of the class group of a random quadratic field, supported by what was at the time a large amount of computational evidence. The Cohen-Lenstra heuristics, which are still almost entirely unproven, have become one of the central foundational problems in arithmetic statistics. Recent years have seen a rapidly accelerated pace of development in Cohen-Lenstra problems. I will give a tour of these developments, including the work of Wood and her collaborators developing a fully fleshed out roster of generalized Cohen-Lenstra conjectures, with support from topology; Smith's theorems proving the Cohen--Lenstra conjectures for the 2-primary part of the class group, as part of more general theorems about Selmer groups in quadratic twists, leading to a resolution of the minimalist conjecture for elliptic curves; and recent work by Koymans and Pagano in the ell-primary case, expanding on Smith's work and proving Stevenhagen's conjecture on the negative Pell equation.

math.NT↗

arXiv2025

Smyth's conjecture and a non-deterministic Hasse principle

In a 1986 paper, Smyth proposed a conjecture about which integer-linear relations were possible among Galois-conjugate algebraic numbers. We prove this conjecture. The main tools (as Smyth already anticipated) are combinatorial rather than number-theoretic in nature. For instance, the question can be reinterpreted as a question about the possible eigenvalues of a specified linear combination of permutation matrices. What's more, we reinterpret Smyth's conjecture as a local-to-global principle for a "non-deterministic system of equations" where variables are interpreted as compactly supported K-valued random variables (for K a local or global field) rather than as elements of K.

math.NT↗

arXiv2025

Generative Modeling for Mathematical Discovery

We present a new implementation of the LLM-driven genetic algorithm {\it funsearch}, whose aim is to generate examples of interest to mathematicians and which has already had some success in problems in extremal combinatorics. Our implementation is designed to be useful in practice for working mathematicians; it does not require expertise in machine learning or access to high-performance computing resources. Applying {\it funsearch} to a new problem involves modifying a small segment of Python code and selecting a large language model (LLM) from one of many third-party providers. We benchmarked our implementation on three different problems, obtaining metrics that may inform applications of {\it funsearch} to new problems. Our results demonstrate that {\it funsearch} successfully learns in a variety of combinatorial and number-theoretic settings, and in some contexts learns principles that generalize beyond the problem originally trained on.

cs.LG↗

arXiv2024

PatternBoost: Constructions in Mathematics with a Little Help from AI

We introduce PatternBoost, a flexible method for finding interesting constructions in mathematics. Our algorithm alternates between two phases. In the first ``local'' phase, a classical search algorithm is used to produce many desirable constructions. In the second ``global'' phase, a transformer neural network is trained on the best such constructions. Samples from the trained transformer are then used as seeds for the first phase, and the process is repeated. We give a detailed introduction to this technique, and discuss the results of its application to several problems in extremal combinatorics. The performance of PatternBoost varies across different problems, but there are many situations where its performance is quite impressive. Using our technique, we find the best known solutions to several long-standing problems, including the construction of a counterexample to a conjecture that had remained open for 30 years.

math.CO↗

arXiv2023

Homological stability for generalized Hurwitz spaces and Selmer groups in quadratic twist families over function fields

We prove a version of the Bhargava-Kane-Lenstra-Poonen-Rains heuristics for Selmer groups of quadratic twist families of abelian varieties over global function fields. As a consequence, we derive a result towards the "minimalist conjecture" on Selmer ranks of abelian varieties in such families. More precisely, we show that the probabilities predicted in these two conjectures are correct to within an error term in the size of the constant field, $q$, which goes to $0$ as $q$ grows. Two key inputs are a new homological stability theorem for a generalized version of Hurwitz spaces parameterizing covers of punctured Riemann surfaces of arbitrary genus, and an expression of average sizes of Selmer groups in terms of the number of rational points on these Hurwitz spaces over finite fields.

math.NT↗

arXiv2023

Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle's conjecture for function fields

The purpose of this paper is to prove the upper bound in Malle's conjecture on the distribution of finite extensions of $\mathbb{F}_q(t)$ with specified Galois group. As in previous work of Ellenberg-Venkatesh-Westerland, our result is based upon computations of the homology of braid groups with certain (exponential) coefficients. However, the approach in this paper is new, relying on a connection between the cohomology of Hurwitz spaces and the cohomology of quantum shuffle algebras.

math.NT↗

arXiv2022

Heights on stacks and a generalized Batyrev-Manin-Malle conjecture

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack X, which specializes to the Baytev-Manin conjecture when X is a scheme and to Malle's conjecture when X is the classifying stack of a finite group.

math.NT↗

arXiv2022

Sparsity of Integral Points on Moduli Spaces of Varieties

Let $X$ be a quasi-projective variety over a number field, admitting (after passage to $\mathbb{C}$) a geometric variation of Hodge structure whose period mapping has zero-dimensional fibers. Then the integral points of $X$ are sparse: the number of such points of height $\leq B$ grows slower than any positive power of $B$. For example, homogeneous integral polynomials in a fixed number of variables and degree, with discriminant divisible only by a fixed set of primes, are sparse when considered up to integral linear substitutions.

math.NT↗

arXiv2022

Rational points on solvable curves over $\mathbb{Q}$ via non-abelian Chabauty

We study the Selmer varieties of smooth projective curves of genus at least two defined over $\mathbb{Q}$ which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that Kim's non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov-Tschinkel and Poonen on unramified correspondences, we deduce that any cover of $\mathbf{P}^1$ with solvable Galois group, and in particular any superelliptic curve over $\mathbb{Q}$, has only finitely many rational points over $\mathbb{Q}$.

math.NT↗

arXiv2021

Nonvanishing of hyperelliptic zeta functions over finite fields

Fixing $t \in \mathbb{R}$ and a finite field $\mathbb{F}_q$ of odd characteristic, we give an explicit upper bound on the proportion of genus $g$ hyperelliptic curves over $\mathbb{F}_q$ whose zeta function vanishes at $\frac{1}{2} + it$. Our upper bound is independent of $g$ and tends to $0$ as $q$ grows.

math.NT↗

arXiv2020

Geometry, Inference, Complexity, and Democracy

Decisions about how the population of the United States should be divided into legislative districts have powerful and not fully understood effects on the outcomes of elections. The problem of understanding what we might mean by "fair districting" intertwines mathematical, political, and legal reasoning; but only in recent years has the academic mathematical community gotten directly involved in the process. I'll report on recent progress in this area, how newly developed mathematical tools have affected real political decisions, and what remains to be done. This survey represents the content of a lecture presented by the author in the Current Events Bulletin session of the Joint Mathematics Meetings in January 2020.

physics.soc-ph↗

arXiv2019

Convergence rates for ordinal embedding

We prove optimal bounds for the convergence rate of ordinal embedding (also known as non-metric multidimensional scaling) in the 1-dimensional case. The examples witnessing optimality of our bounds arise from a result in additive number theory on sets of integers with no three-term arithmetic progressions. We also carry out some computational experiments aimed at developing a sense of what the convergence rate for ordinal embedding might look like in higher dimensions.

math.ST↗

arXiv2019

Algebraic structures on cohomology of configuration spaces of manifolds with flows

Let PConf^n M be the configuration space of ordered n-tuples of distinct points on a smooth manifold M admitting a nowhere-vanishing vector field. We show that the ith cohomology group with coefficients in a field H^i(PConf^n M, k) is an N-module, where N is the category of noncommutative finite sets introduced by Pirashvili and Richter. Studying the representation theory of N, we obtain new polynomiality results for the cohomology groups H^i(PConf^n M, k). In the case of unordered configuration space Conf^n M = (PConf^n M)/S_n and rational coefficients, we show that cohomology dimension in fixed degree is nondecreasing.

math.AT↗

arXiv2019

Detection of Planted Solutions for Flat Satisfiability Problems

We study the detection problem of finding planted solutions in random instances of flat satisfiability problems, a generalization of boolean satisfiability formulas. We describe the properties of random instances of flat satisfiability, as well of the optimal rates of detection of the associated hypothesis testing problem. We also study the performance of an algorithmically efficient testing procedure. We introduce a modification of our model, the light planting of solutions, and show that it is as hard as the problem of learning parity with noise. This hints strongly at the difficulty of detecting planted flat satisfiability for a wide class of tests.

math.ST↗

arXiv2017

Sumsets as unions of sumsets of subsets

Let $S$ and $T$ be subsets of $\mathbf{F}_q^n$. We show there are subsets $S'$ of $S$ and $T'$ of $T$ such that $S+T$ is the union of $S+T'$ and $S'+T$, with $|S'| + |T'|$ bounded by $c^n$ with $c < q$. The proof relies on the method of Croot-Lev-Pach and Ellenberg-Gijswijt on the cap set problem, together with a result of Meshulam on linear spaces of low-rank matrices. The result is a modest generalization of the recent bounds on (single-colored and multi-colored) sum-free sets by the author and others.

math.CO↗

arXiv2016

New bounds on curve tangencies and orthogonalities

We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves are tangent. In particular, if no three curves are mutually tangent at a common point, then there are $O(n^{3/2})$ curve-curve tangencies. Second, given a family of algebraic plane curves and a set of $n$ curves from this family, we show that either there are $O(n^{3/2})$ points where two or more curves are orthogonal, or the family of curves has certain special properties. We obtain these bounds by transforming the arrangement of plane curves into an arrangement of space curves so that tangency (or orthogonality) of the original plane curves corresponds to intersection of space curves. We then bound the number of intersections of the corresponding space curves. For the case of curve-curve tangency, we use a polynomial method technique that is reminiscent of Guth and Katz's proof of the joints theorem. For the case of orthogonal curve intersections, we employ a bound of Guth and the third author to control the number of two-rich points in space curve arrangements.

math.CO↗

arXiv2016

Homology of FI-modules

We prove an explicit and sharp upper bound for the Castelnuovo-Mumford regularity of an FI-module V in terms of the degrees of its generators and relations. We use this to refine a result of Putman on the stability of homology of congruence subgroups, extending his theorem to previously excluded small characteristics and to integral homology while maintaining explicit bounds for the stable range.

math.RT↗

arXiv2016

On large subsets of $F_q^n$ with no three-term arithmetic progression

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of $F_q^n$ with no three terms in arithmetic progression by $c^n$ with $c < q$. For $q=3$, the problem of finding the largest subset with no three terms in arithmetic progression is called the `cap problem'. Previously the best known upper bound for the cap problem, due to Bateman and Katz, was $O(3^n / n^{1+ε})$.

math.CO↗