arXiv subjects

Logan Hyslop

Publications and source records attributed to Logan Hyslop.

6 recordsLinked to original sources

arXiv2026

A Nilpotence Theorem for Rational Rigid 2-Rings of Moderate Growth

In this short note, we prove a general nilpotence theorem for a rational rigid 2-ring all of whose objects satisfy a certain ``moderate growth condition'' inspired from the theory of tensor categories. This applies in particular to the category of modules over a rational $E_{\infty}$-ring, to the derived category of any super-Tannakian category in characteristic zero, and conjecturally to Voevodsky's rational category of mixed motives over a field $DM_{\mathbb{Q}}$. In fact, we further prove that any such category has enough tt-fields, which can be chosen to be of the form Perf(L) for an even 2-periodic field L.

math.AG↗

arXiv2026

Special Values without Semi-Simplicity Via K-Theory

In this paper, motivated by studying special values of zeta functions attached to finite type F_p-schemes, we introduce a category of ``arithmetic C(S^1,R)-modules'' attached to any Dedekind ring R, and compute the 0th K-group of this category. Specializing to the case of R=Z_l for some prime l neq p (resp. R=Z_p), we prove that there is a natural functorial lift of the etale cohomology of perfect etale Z_l sheaves (resp. syntomic cohomology of perfect prismatic F-gauges) on a point to arithmetic C(S^1,Z_l)-modules (resp. arithmetic C(S^1,Z_p)-modules). This allows us to define a notion of the multiplicative Euler characteristic via a map from the K_0-group which makes sense without assuming Tate's semi-simplicity conjecture. In particular, we can remove this hypothesis from a theorem of Milne proving a cohomological formula for zeta values attached to smooth proper F_p-schemes. We also discuss extensions of these zeta value formulae to finite type F_p-schemes, and how recent progress in motivic homotopy theory allows us to prove some results without any assumptions on resolution of singularities or Tate's semi-simplicity conjecture.

math.AG↗

arXiv2026

Geometric Points in Tensor Triangular Geometry

In this paper, we study geometric points in tensor triangular geometry. In doing so, we construct a counter-example to Balmer's Nerves of Steel conjecture using free constructions in higher Zariski geometry. We then go on to introduce and discuss constructible spectra in the context of tensor triangular geometry. For tensor triangulated categories satisfying a mild enhancement condition, we use these spectra to construct geometric incarnations of (homological or triangular) primes via maps to "pointlike" tensor triangulated categories.

math.AT↗

arXiv2025

Towards the Nerves of Steel Conjecture

Given a local $\otimes$-triangulated category, and a fiber sequence $y\to 1 \to x$, one may ask if there is always a nonzero object $z$ such that either $z\otimes f$ or $z\otimes g$ is $\otimes$-nilpotent. The claim that this property holds for all local $\otimes$-triangulated categories is equivalent to Balmer's "nerves of steel conjecture" from arXiv:2001.00284. In the present paper, we will see how this property can fail if the category we start with is not rigid, discuss a large class of categories where the property holds, and ultimately prove that the nerves of steel conjecture is equivalent to a stronger form of this property.

math.CT↗

arXiv2023

Whitehead Filtrations for Computations in Topological Hochschild Homology

We discuss spectral sequences coming from Whitehead filtrations in the computation of topological Hochschild homology of ring spectra. Using cyclic invariance, this makes for simple computations of $THH$ of connective rings $R$ with coefficients in discrete ring spectra. In particular, we show how to use this to compute $THH(tmf,\mathbb{F}_2)$, and $THH(tmf,\mathbb{Z}_{(2)})$, where $tmf$ denotes the $\mathbb{E}_\infty$ ring spectrum of topological modular forms. Then, we obtain a description of $THH(\ell/v_1^n)$ in terms of $THH(\ell,\ell/v_1^n)$, where the latter can be computed by results of arXiv:0710.4368. We next explain how the methods of this computation generalize to give us information about $THH(cofib(x^k:\Sigma^{k|x|}R\to R))$ for $R$ and $cofib(x^k)$ suitably structured connective ring spectra, $k>1$, and $x\in \pi_{*}(R)$ an arbitrary element in positive degree. Finally, we examine the general framework to describe the topological Hochschild homology of 2-local connective self-conjugate K-theory, $ksc_2$.

math.AT↗

arXiv2020

Well-posedness for a modified nonlinear Schrodinger equation modeling the formation of rogue waves

The Cauchy problem for a higher order modification of the nonlinear Shcrodinger equation (MNLS) on the line is shown to be well-posed in Sobolev spaces with exponent $\ge 0$. This result is achieved by demonstrating that the associated integral operator is a contraction on a Bourgain space that has been adapted to the particular linear symbol present in the equation. the ctraction is proved by using microlocal analysis and a new trilinear estimate.

math.AP↗