arXiv subjects

Jesse Wolfson

Publications and source records attributed to Jesse Wolfson.

At least 19 recordsLinked to original sources

arXiv2026

Correctness, Artificial Intelligence, and the Epistemic Value of Mathematical Proof

We argue that it is neither necessary nor sufficient for a mathematical proof to have epistemic value that it be "correct", in the sense of formalizable in a formal proof system. We then present a view on the relationship between mathematics and logic that clarifies the role of formal correctness in mathematics. Finally, we discuss the significance of these arguments for recent discussions about automated theorem provers and applications of AI to mathematics.

math.HO↗

arXiv2025

Essential dimension relative to branched covers of degree at most n

We prove for various finite groups $G$ and integers $n\geq 1$ that there are families of equations with Galois group $G$ that cannot be simplified to a one-parameter family even after adjoining a root of a polynomial of degree at most $n$. In more geometric language, there are $G$-varieties $X$ with the following property: for any $G$-equivariant branched cover $\widetilde{X}\to X$ of degree $\leq n$, there is no dominant rational $G$-map $\widetilde{X}\dashrightarrow C$ to any $G$-curve $C$. The method of proof is new, and applies in cases where previous methods do not.

math.AG↗

arXiv2025

Solvable points on intersections of quadrics, cubics, and quartics

Let $k$ be a field of characteristic not 2 or 3. We establish polynomial lower bounds on the ambient dimension $N$ for an intersection $X\subset\mathbb{P}^N$ of quadrics, cubics and quartics to have a dense collection of solvable points, i.e. points in $X(k^{\mathsf{Sol}})$ where $k^{\mathsf{Sol}}/k$ is a solvable closure. Our method connects the classical theory of polar hypersurfaces, as redeveloped by Sutherland, to Fano varieties $\mathcal{F}(j,X)$ of $j$-dimensional linear subspaces on $X$, and we use this to obtain improved control on the arithmetic of $\mathcal{F}(j,X)$.

math.AG↗

arXiv2024

Lie's Third Theorem for Lie $\infty$-Algebras

We introduce the theory of local minimal models for Kan simplicial manifolds, which provide the appropriate generalization of minimal Kan simplicial sets to geometric contexts. We use this to obtain the first proof of Lie's third theorem for finite-type Lie $\infty$-algebras: Every finite-type, homologically and non-negatively graded $L_\infty$-algebra over $\mathbb{R}$ integrates to a finite-dimensional Lie $\infty$-group. As a corollary, our construction yields a new explicit finite-dimensional model for the string Lie 2-group.

math.RA↗

arXiv2024

Math and Dance: Notes from emerging interaction

Choreographer Reggie Wilson and mathematician Jesse Wolfson describe interactions of math and dance emerging from their 12+ year engagement with Black movement and music traditions as part of Wilson's research-to-performance/performance-to-research choreographic practice, with examples including fractals, braids and choreographic and mathematical notions of space, time and movement.

math.HO↗

arXiv2024

Fractals in Africanist Music

We investigate fractal structures in African and African diasporic music, building on hypotheses of choreographer Reggie Wilson and research on fractals in African material culture by Ron Eglash.

math.HO↗

arXiv2024

On Simplicial Principal Bundles in Descent Categories

Simplicial objects $\mathsf{sC}$ in descent categories $\mathsf{C}$, as introduced by Behrend and Getzler, provide a context in which to study higher stacks. In this note, we extend the construction of the canonical cocycle of a smooth principal $G$-bundle to the context of principal $G$-bundles in $\mathsf{sC}_{/X}$. As an application, we show how this specializes to $\mathsf{C}=\mathsf{Sets}$ to give a streamlined construction of $k$-invariants of reduced Kan complexes. We adapt this to give a similarly streamlined construction of minimal Kan complexes, with the goal of clarifying the role of the axiom of choice; Postnikov towers of minimal Kan complexes provide examples of towers of simplicial principal bundles of the type we consider.

math.AT↗

arXiv2024

Essential dimension via prismatic cohomology

For $X$ a smooth, proper complex variety we show that for $p\gg 0$, the restriction of the mod $p$ cohomology $H^i(X,\mathbb{F}_p)$ to any Zariski open has dimension at least $h^{0,i}_X$. The proof uses the prismatic cohomology of Bhatt-Scholze. We use this result to obtain lower bounds on the $p$-essential dimension of covers of complex varieties. For example, we prove the $p$-incompressibility of the mod $p$ homology cover of an abelian variety, confirming a conjecture of Brosnan for sufficiently large $p.$ By combining these techniques with the theory of toroidal compactifications of Shimura varieties, we show that for any Hermitian symmetric domain $X,$ there exist $p$-congruence covers that are $p$-incompressible.

math.AG↗

arXiv2024

Generalized Versality, Special Points, and Resolvent Degree for the Sporadic Groups

Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group $G$ has only been investigated when $G$ is a cylic group; an alternating group; a simple factor of a Weyl group of type $E_6$, $E_7$, or $E_8$; or $\operatorname{PSL}\left(2, \mathbb{F}_7\right)$. In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) $\operatorname{RD}_k^{\leq d}$-versality, which we connect to the existence of "special points" on varieties.

math.AG↗

arXiv2023

On the normally ordered tensor product and duality for Tate objects

This paper generalizes the normally ordered tensor product from Tate vector spaces to Tate objects over arbitrary exact categories. We show how to lift bi-right exact monoidal structures, duality functors, and construct external Homs. We list some applications: (1) Pontryagin duality uniquely extends to n-Tate objects in locally compact abelian groups; (2) Adeles of a flag can be written as ordered tensor products; (3) Intersection numbers can be interpreted via these tensor products.

math.QA↗

arXiv2022

Alexander Ostrowski's "On Dirichlet Series and Algebraic Differential Equations"

This is an English translation of Ostrowski's article "Über Dirichletsche Reihen und algebraische Differentialgleichungen" published German in Math. Zeit. vol. 8, 1920, pp. 241-298. In this article, Ostrowski proves a conjecture of Hilbert that the two variable function $ζ(x,s)=\sum_{n\ge 1} \frac{x^n}{n^s}$ cannot be written as a composition of analytic functions of one variable and algebraic functions of any number of variables.

math.HO↗

arXiv2021

A higher Kac-Moody extension for two-dimensional gauge groups

Let $Γ$ be a finite dimensional Lie group and consider the smooth double loop group, i.e. the Fréchet Lie group of smooth maps from the 2-torus to $Γ$. For a finite dimensional Hilbert space V, let H denote the Hilbert space of vector valued $L^2$-functions on the 2-torus. The purpose of this paper is to construct a higher central extension of the smooth double loop group from the representation of the smooth double loop group on H induced by a smooth action of $Γ$ on V. This higher central extension comes from an action of the smooth double loop group on a 2-category and yields a group cohomology class of degree 3 on the smooth double loop group. We show by a concrete computation that this group cohomology class is non-trivial in general. We relate our higher central extension to the Kac-Moody extension of the smooth single loop group as a higher dimensional analogue of the latter. More generally, given a group G acting on a bipolarised Hilbert space, we apply higher category theory to construct a group cohomology class of degree 3 on G. As a second motivating example, we use these ideas to introduce a higher central extension of the group of invertible smooth functions on the noncommutative 2-torus.

math.KT↗

arXiv2021

A Generalized Contou-Carrère Symbol and its Reciprocity Laws in Higher Dimensions

We generalize the theory of Contou-Carrère symbols to higher dimensions. To an $(n+1)$-tuple $f_0,\dots,f_n \in A((t_1))\cdots((t_n))^{\times}$, where $A$ denotes a commutative algebra over a field $k$, we associate an element $(f_0,\dots,f_n) \in A^{\times}$, compatible with the higher tame symbol for $k = A$, and earlier constructions for $n = 1$, by Contou-Carrère, and $n = 2$ by Osipov--Zhu. Our definition is based on the notion of \emph{higher commutators} for central extensions of groups by spectra, thereby extending the approach of Arbarello--de Concini--Kac and Anderson--Pablos Romo. Following Beilinson--Bloch--Esnault for the case $n=1$, we allow $A$ to be arbitrary, and do not restrict to artinian $A$. Previous work of the authors on Tate objects in exact categories, and the index map in algebraic $K$-theory is essential in anchoring our approach to its predecessors. We also revisit categorical formal completions, in the context of stable $\infty$-categories. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic $K$-theory, and conclude the article by proving a version of Parshin--Kato reciprocity for higher Contou-Carrère symbols.

math.AG↗

arXiv2021

Tschirnhaus transformations after Hilbert

Let RD(n) denote the minimum d for which there exists a formula for the roots of the general degree n polynomial using only algebraic functions of d or fewer variables. In 1927, Hilbert sketched how the 27 lines on a cubic surface could be used to construct a 4-variable formula for the general degree 9 polynomial (implying $RD(9)\le 4$). In this paper, we turn Hilbert's sketch into a general method. We show this method produces best-to-date upper bounds on RD(n) for all n, improving earlier results of Hamilton, Sylvester, Segre and Brauer.

math.AG↗

arXiv2020

Problems in Arithmetic Topology

We present a list of problems in arithmetic topology posed at the June 2019 PIMS/NSF workshop on "Arithmetic Topology". Three problem sessions were hosted during the workshop in which participants proposed open questions to the audience and engaged in shared discussions from their own perspectives as working mathematicians across various fields of study. Participants were explicitly asked to provide problems of various levels of difficulty, with the goal of capturing a cross-section of exciting challenges in the field that could help guide future activity. The problems, together with references and brief discussions when appropriate, are collected below into three categories: 1) topological analogues of arithmetic phenomena, 2) point counts, stability phenomena and the Grothendieck ring, and 3) tools, methods and examples.

math.AT↗

arXiv2020

The Essential Dimension of Congruence Covers

Consider the algebraic function $Φ_{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Kronecker and Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $Φ_{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties.

math.AG↗

arXiv2020

Resolvent degree, Hilbert's 13th Problem and geometry

We develop the theory of resolvent degree, introduced by Brauer \cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilbert's 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.

math.AG↗

arXiv2019

Derived $\ell$-adic zeta functions

We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map $\mathbb{S} \to K(Var_k)$ induced by the inclusion of $0$-dimensional varieties is not surjective on $π_1$ for a wide range of fields $k$. The methods used in this paper should generalize to lifting other motivic measures to maps of $K$-theory spectra.

math.AG↗