arXiv subjects

Jordan Ellenberg

Publications and source records attributed to Jordan Ellenberg.

9 recordsLinked to original sources

arXiv2026

Bruhat intervals that are large hypercubes

We study the question of finding big Bruhat intervals that are poset hypercubes in the symmetric group $S_n$. Using permutations suggested by AlphaEvolve (an evolutionary coding agent developed by Google DeepMind), we were led to an unusual situation in which the agent produced a pattern which performed well for the $n$ tested, and which we show works well for general $n$. When $n$ is a power of 2 we exhibit a hypercube of dimension $O(n\log n)$, matching the largest possible dimension up to a constant multiple. Furthermore, we give an exact characterization of the vertices of this hypercube: they are precisely the \emph{dyadically well-distributed} permutations -- a simple digitwise property that already appeared in connection with Monte Carlo integration and mathematical finance. The maximal dimension of a Bruhat interval that is an hypercube in $S_n$ gives a lower bound (and possibly is equal to) the maximal possible coefficient of the second-highest degree term in the Kazhdan--Lusztig $R$-polynomial in $S_n$. As a surprising consequence, we obtain a new lower bound of order $n\log n$ for the maximal number of frozen variables appearing in the cluster algebras attached to the open Richardson varieties in $S_n$, and a similar result for moduli spaces of embeddings of Bruhat graphs.

math.CO↗

arXiv2026

Averages of Arithmetic Functions over Conductors of Function Fields

For a finite group $G$ and a sufficiently large (but fixed) prime power $q$ coprime to $G$ we obtain asymptotics for the number of regular Galois extensions $L/ \mathbb{F}_q(t)$, with $\mathrm{Gal}(L/\mathbb{F}_q(t)) \cong G$, ramified at a single place of $\mathbb{F}_q(t)$, thus making progress on a positive characteristic analog of the Boston--Markin conjecture. We also obtain similar results for other arithmetic functions of the product of places of $\mathbb{F}_q(t)$ ramified in $L$, and for more general one-variable function fields over $\mathbb{F}_q$ in place of $\mathbb{F}_q(t)$. Some of our proofs make crucial use of a series of recent breakthroughs by Landesman--Levy, as well as a new `vanishing of stable homology in a given direction' result for representations of braid groups arising from braided vector spaces. Other inputs include a study of (rings of coinvariants of) braided vector spaces associated to racks with $2$-cocycles, a connection between convolution of arithmetic functions and direct sums of braided vector spaces, and a Goursat lemma for racks.

math.NT↗

arXiv2024

Certifying nontriviality of Ceresa classes of curves

The Ceresa cycle is a canonical algebraic $1$-cycle on the Jacobian of an algebraic curve. We construct an algorithm which, given a curve over a number field, often provides a certificate that the Ceresa cycle is non-torsion, without relying on the presence of any additional symmetries of the curve. Under the hypothesis that the Sato--Tate group is the whole of $\operatorname*{GSp}$, we prove that if the Ceresa class (the image of the Ceresa cycle in \'{e}tale cohomology) is non-torsion, then the algorithm will eventually terminate with a certificate attesting to this fact.

math.AG↗

arXiv2020

The Ceresa class: tropical, topological, and algebraic

The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in \'etale cohomology called the Ceresa class. Describing the Ceresa class explicitly for non-hyperelliptic curves is in general not easy. We present a "combinatorialization" of this problem, explaining how to define a Ceresa class for a tropical algebraic curve, and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over $\mathbb{C}(\!(t)\!)$, and show that the Ceresa class in each of these settings is torsion.

math.AG↗

arXiv2017

Approximate Gradient Coding via Sparse Random Graphs

Distributed algorithms are often beset by the straggler effect, where the slowest compute nodes in the system dictate the overall running time. Coding-theoretic techniques have been recently proposed to mitigate stragglers via algorithmic redundancy. Prior work in coded computation and gradient coding has mainly focused on exact recovery of the desired output. However, slightly inexact solutions can be acceptable in applications that are robust to noise, such as model training via gradient-based algorithms. In this work, we present computationally simple gradient codes based on sparse graphs that guarantee fast and approximately accurate distributed computation. We demonstrate that sacrificing a small amount of accuracy can significantly increase algorithmic robustness to stragglers.

stat.ML↗

arXiv2016

On $\ell$-torsion in class groups of number fields

For each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of $\mathbb{Q}$ of degree $d$, for any fixed $d \in \{2,3,4,5\}$ (with the additional restriction in the case $d=4$ that the field be non-$D_4$). For sufficiently large $\ell$ (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our argument, we develop a probabilistic "Chebyshev sieve," and give uniform, power-saving error terms for the asymptotics of quartic (non-$D_4$) and quintic fields with chosen splitting types at a finite set of primes.

math.NT↗

arXiv2010

Expander graphs, gonality and variation of Galois representations

We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric (genus and gonality) growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in one-parameter families of abelian varieties.

math.NT↗

arXiv2009

The Kakeya set and maximal conjectures for algebraic varieties over finite fields

Using the polynomial method of Dvir \cite{dvir}, we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties $W$ over finite fields $F$. For instance, given an $n-1$-dimensional projective variety $W \subset \P^n(F)$, we establish the Kakeya maximal estimate $$ \| \sup_{\gamma \ni w} \sum_{v \in \gamma(F)} |f(v)| \|_{\ell^n(W)} \leq C_{n,W,d} |F|^{(n-1)/n} \|f\|_{\ell^n(F^n)}$$ for all functions $f: F^n \to \R$ and $d \geq 1$, where for each $w \in W$, the supremum is over all irreducible algebraic curves in $F^n$ of degree at most $d$ that pass through $w$ but do not lie in $W$, and with $C_{n,W,d}$ depending only on $n, d$ and the degree of $W$; the special case when $W$ is the hyperplane at infinity in particular establishes the Kakeya maximal function conjecture in finite fields, which in turn strengthens the results of Dvir.

math.CO↗

arXiv2006

Local-global principles for representations of quadratic forms

We prove the local-global principle holds for the problem of representations of quadratic forms by quadratic forms, in codimension $\geq 7$. The proof uses the ergodic theory of $p$-adic groups, together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety.

math.NT↗