arXiv subjects

Ryan Grady

Publications and source records attributed to Ryan Grady.

9 recordsLinked to original sources

arXiv2026

Functoriality of the KSGNS Construction for Intertwiners of Strict Positive $C^*$-Correspondences

We prove that the KSGNS construction can be viewed as an endofunctor on a category whose objects are positive $C^*$-correspondences from a fixed $C^*$-algebra and morphisms are given by intertwiners which account for automorphisms of the fixed $C^*$-algebra. Using this perspective, we provide a functorial perspective for strict positive equivariant $C^*$-correspondences of $C^*$-dynamical systems and show every strict positive equivariant $C^*$-correspondence of $C^*$-dynamical systems unitarily uniquely dilates under the KSGNS construction to an equivariant $C^*$-correspondence of the dynamical systems.

math.OA↗

arXiv2025

DGLA Actions: An Application in GR

This note serves two purposes: 1) define actions by differential graded Lie algebras, and 2) apply such differential graded Lie symmetry in general relativity (GR) to constrain the spacetime geometry on a neighborhood of infinity.

math-ph↗

arXiv2024

Causally Disjoint Discs: Another $\mathbb{E}_n$-operad

Motivated by (perturbative) quantum observables in Lorentzian signature we define a new operad: the operad of causally disjoint disks. In order to describe this operad we use the orthogonal categories of Benini, Schenkel, and Woike and the prefactorization functor of Benini, Carmona, Grant-Stuart, and Schenkel. Along the way we extend these constructions to the topological setting, i.e., (multi-)categories enriched over spaces.

math.QA↗

arXiv2023

Continued fractions and lines across the Stern--Brocot diagram

This paper concerns the relationships between continued fractions and the geometry of the Stern-Brocot diagram. Each rational number can be expressed as a continued fraction $[a_0; a_1, \ldots, a_n]$ whose terms $a_i$ are integers and are positive if $i \geq 1$. Select an index $i \in \{ 1, \ldots, n \}$ and replace $a_i$ with an integer $m$ to obtain a continued fraction expansion for an extended rational $\alpha_m \in \mathbb{Q} \cup \{ \infty \}$. This paper shows that the vertices of the Stern-Brocot diagram corresponding to the numbers $\{ \alpha_m \}_{m \in \mathbb{Z}}$ lie on a pair of (extended) Euclidean lines across the diagram. The slopes of these two lines differ only by a sign change and they meet at the point $L=\left([a_0; a_1, \ldots, a_{i-1}], 0\right) \in \mathbb{R}^2$. Moreover, as $\lvert m \rvert \to \infty$, the associated vertices move down these lines and converge to $L$. This paper concludes with a discussion which interprets this result in the context of 2-bridge link complements and Thurston's work on hyperbolic Dehn surgery.

math.GT↗

arXiv2020

Ultrahigh Doping of Graphene Using Flame-Deposited MoO3

The expected high performance of graphene-based electronics is often hindered by lack of adequate doping, which causes low carrier density and large sheet resistance. Many reported graphene doping schemes also suffer from instability or incompatibility with existing semiconductor processing. Here we report ultrahigh and stable p-type doping up to ~7x10^13 1/cm^2 (~2x10^21 1/cm^3}) of monolayer graphene grown by chemical vapor deposition. This is achieved by direct polycrystalline MoO3 growth on graphene using a rapid flame synthesis technique. With this approach, the metal-graphene contact resistance for holes is reduced to ~200 Ohm-um. We also demonstrate that flame-deposited MoO3 provides over 5x higher doping of graphene, as well as superior thermal and long-term stability, compared to electron-beam deposited MoO3.

cond-mat.mes-hall↗

arXiv2017

Homotopy RG Flow and the Non-Linear $\sigma$-model

The purpose of this note is two give a mathematical treatment to the low energy effective theory of the two-dimensional sigma model. Perhaps surprisingly, our low energy effective theory encodes much of the topology and geometry of the target manifold. In particular, we relate the $\beta$-function of our theory to the Ricci curvature of the target, recovering the physical result of Friedan.

math-ph↗

arXiv2014

L-infinity spaces and derived loop spaces

We develop further the approach to derived differential geometry introduced in Costello's work on the Witten genus. In particular, we introduce several new examples of L-infinity spaces, discuss vector bundles and shifted symplectic structures on L-infinity spaces, and examine in some detail the example of derived loop spaces. This paper is background for a companion paper, in which we define a quantum field theory on a derived stack, building upon Costello's definition of an effective field theory.

math.AG↗

arXiv2013

One-dimensional Chern-Simons theory and the $\hat{A}$ genus

We construct a Chern-Simons gauge theory for dg Lie and L-infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin-Vilkovisky formalism and Costello's renormalization techniques. Koszul duality and derived geometry allow us to encode topological quantum mechanics, a nonlinear sigma model of maps from a 1-manifold into a cotangent bundle T*X, as such a Chern-Simons theory. Our main result is that the partition function of this theory is naturally identified with the A-genus of X. From the perspective of derived geometry, our quantization construct a volume form on the derived loop space which can be identified with the A-class.

math.QA↗

arXiv2012

The $\hat{A}$ genus as a projective volume form on the derived loop space

In the present work, we extend our previous work with Gwilliam by realizing $\hat{A}(X)$ as the projective volume form associated to the BV operator in our quanitization of a one-dimensional sigma model. We also discuss the associated integration/expectation map. We work in the formalism of $L_\infty$ spaces, objects of which are computationally convenient presentations for derived stacks. Both smooth and complex geometry embed into $L_\infty$ spaces and we specialize our results in both of these cases.

math.QA↗