arXiv subjects

Ryan E. Grady

Publications and source records attributed to Ryan E. Grady.

12 recordsLinked to original sources

arXiv2026

Sharp Tangential NEC Minimization and Israel Surface Layers in Schwarzschild Mass Interpolations

We study static, spherically symmetric metrics in Schwarzschild gauge whose mass function increases smoothly from $M_1$ to $M_2>M_1$ across an annulus $[R_1,R_2]$ outside the larger Schwarzschild radius. An elementary obstruction shows that tangential NEC violation is unavoidable in this class. This reduces the physical question to a quantitative one: what is the least possible violation, and what geometry is selected by near-minimization? We first determine the exact $L^1$-relaxation of the resulting nonlocal weighted positive-variation functional and solve the relaxed problem explicitly. Its unique minimizer is a normalized box profile, which yields the infimum of violations: the relaxed minimum is attained, while the smooth infimum is not. An exact deficit decomposition yields sharp quantitative stability, with a square-root rate at a nondegenerate critical optimizer and a linear rate at a strict constrained boundary optimizer; the same rates control the mass function and metric coefficients. We identify a variational phase transition between these regimes and prove reciprocal-width divergence in the thin-annulus limit. Every smooth minimizing sequence converges to one locally Lipschitz metric with a constant-density $p=-ρ$ bulk and two timelike surface layers, its Einstein tensors converge distributionally, and the singular terms agree with the Israel surface stress tensors. The negative tangential null energy concentrates on the inner layer, whose integrated negative pressure equals the sharp variational cost.

gr-qc↗

arXiv2026

Adjoint L-Infinity Actions and Conserved Charges in GR

In this work we compute the conserved currents and charges associated to the action of an infinitesimal isometry (Killing field) in Einstein--Cartan--Palatini gravity. We offer a new approach to these quantities through the formalism of $L_\infty$-algebras and the work of Ćirić, Giotopoulos, Radovanović, and Szabo, and Costello and Gwilliam. We demonstrate our approach by computing the entropy of the Schwarzchild and Kerr black holes. Along the way, we prove a purely algebraic result about the existence and utility of a higher (a full $\infty$) version of the adjoint action of an $L_\infty$-algebra.

math-ph↗

arXiv2023

Regularity via Links and Stein Factorization

Here, we introduce a new definition of regular point for piecewise-linear (PL) functions on combinatorial (PL triangulated) manifolds. This definition is given in terms of the restriction of the function to the link of the point. We show that our definition of regularity is distinct from other definitions that exist in the combinatorial topology literature. Next, we stratify the Jacobi set/critical locus of such a map as a poset stratified space. As an application, we consider the Reeb space of a PL function, stratify the Reeb space as well as the target of the function, and show that the Stein factorization is a map of stratified spaces.

math.AT↗

arXiv2023

K-Theory of Multi-parameter Persistence Modules: Additivity

Persistence modules stratify their underlying parameter space, a quality that make persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multi-parameter persistence modules. Namely, we show the $K$-theory of grid multi-parameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multi-parameter notions of zig-zag persistence. We compare our calculations for the specific group $K_0$ with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.

math.AT↗

arXiv2022

Zig-Zag Modules: Cosheaves and K-Theory

Persistence modules have a natural home in the setting of stratified spaces and constructible cosheaves. In this article, we first give explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig-zag filtrations (including monotone filtrations), and for augmented persistence modules (which encode the data of instantaneous events). We then identify an equivalence of categories between a particular notion of zig-zag modules and the combinatorial entrance path category on stratified $\mathbb{R}$. Finally, we compute the algebraic $K$-theory of generalized zig-zag modules and describe connections to both Euler curves and $K_0$ of the monoid of persistence diagrams as described by Bubenik and Elchesen.

math.AT↗

arXiv2021

Lax Functors, Cospans, and the Center Construction

The center construction is not (classically) functorial. In this note, we specialize a universal construction of Jacob Lurie to the category of rings and upgrade the classical center to a lax functor. In particular, we find lax functors to the Morita category and the category of cospans.

math.CT↗

arXiv2020

Quantizing Derived Mapping Stacks

In this review we discuss several topological and geometric invariants obtained by quantizing $σ$-models. More precisely, we don't quantize the entire mapping stack of fields, but rather only the substack of low energy fields. The theory restricted to this substack can be presented Lie theoretically and the problem is reduced to perturbative gauge theory. Throughout, we make extensive use of derived symplectic geometry and the BV formalism of Costello and Gwilliam. Finally, we frame the AJ Conjecture in knot theory as a question of quantizing character stacks.

math-ph↗

arXiv2019

A Glimpse of Arithmetic Dynamics

In this note, we offer a palatable introduction to the field of arithmetic dynamics. That is, we study the patterns that arise when iterating a polynomial map. This note is accessible to those who have taken an introductory proof based course and some linear algebra; the appendix utilizes abstract algebra.

math.HO↗

arXiv2018

Lie algebroids as $L_\infty$ spaces

In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid $L$-and the natural generalization to dg Lie algebroids-provides an (essentially unique) $L_\infty$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_\infty$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_\infty$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_\infty$ space. Finally, we show that a shifted-symplectic structure on a dg Lie algebroid $L$ produces a shifted-symplectic structure on the associated $L_\infty$ space.

math.DG↗

arXiv2017

Batalin-Vilkovisky quantization and the algebraic index

Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosov's deformation quantization of a symplectic manifold X and the BV quantization of a one-dimensional sigma model with target X. This model is a quantum field theory of AKSZ type and is quantized rigorously using Costello's homotopic theory of effective renormalization. We show that Fedosov's Abelian connections on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV integration produces a natural trace map on the deformation quantized algebra. This formulation allows us to exploit a (rigorous) localization argument in quantum field theory to deduce the algebraic index theorem via semi-classical analysis, i.e., one-loop Feynman diagram computations.

math.QA↗

arXiv2016

Parametrized L-infinity Spaces

Motivated by families of formal moduli problems, in this note we generalize the notion of L-infinity space by allowing sheaves of L-infinity algebras over any (reasonable) nilpotent dg manifold. We discuss various examples including those coming from Lie algebroids. Given a Lie algebroid, we show that there is an L-infinity space parametrized by (cochains of) the Lie algebroid; further, characteristic classes of this L-infinity space recover the primary invariants of the original algebroid.

math.DG↗

arXiv2006

Kazhdan's Property (T) for Graphs

D. A. Kahzdan first put forth property (T) in relation to the study of discrete subgroups of Lie groups of finite co-volume. Through a combinatorial approach, we define an analogue of property (T) for regular graphs. We then prove the basic combinatorial and metric properties of Kazhdan groups in this context. In particular, we use our methods to construct infinite families of expanders as in the classical case. Finally, we consider the combinatorial analogue of the group theoretic property $(τ)$ and prove its basic properties.

math.CO↗