arXiv subjects

James Hughes

Publications and source records attributed to James Hughes.

14 recordsLinked to original sources

arXiv2025

Exact Lagrangian fillings of twist-spun torus links

We construct exact Lagrangian fillings of Legendrian torus links $\Lambda(k, n-k)$ that are fixed by a Legendrian loop that acts by $2\pi\ell/n$ rotation. Using these rotationally symmetric fillings, we produce fillings of the corresponding Legendrian twist-spun tori. Our construction is combinatorial in nature, relating symmetric weakly separated collections and plabic graphs to symmetric Legendrian weaves via the T-shift procedure of Casals, Le, Sherman-Bennett, and Weng. The main technical ingredient in this process is a necessary and sufficient condition for the existence of maximal weakly separated collections of $k$-element subsets of $\{1, \dots, n\}$ that are fixed by addition of $\ell$ modulo $n$.

math.SG↗

arXiv2025

Decompositions of augmentation varieties via weaves and rulings

The braid variety of a positive braid and the augmentation variety of a Legendrian link both admit decompositions coming from weaves and rulings, respectively. We prove that these decompositions agree under an isomorphism between the braid variety and the augmentation variety. We also prove that both decompositions coincide with a Deodhar decomposition and another decomposition coming from the microlocal theory of sheaves. Our proof relies on a detailed comparison between weaves and Morse complex sequences. Among other things, we show that the cluster variables of the maximal cluster torus of the augmentation variety can be computed from the Legendrian via Morse complex sequences.

math.SG↗

arXiv2025

Exact Lagrangian fillability of 3-braid closures

We determine when a Legendrian quasipositive 3-braid closure in standard contact $\mathbb{R}^3$ admits an orientable or non-orientable exact Lagrangian filling. Our main result provides evidence for the orientable fillability conjecture of Hayden and Sabloff, showing that a 3-braid closure is orientably exact Lagrangian fillable if and only if it is quasipositive and the HOMFLY bound on its maximum Thurston-Bennequin number is sharp. Of possible independent interest, we construct explicit Legendrian representatives of quasipositive 3-braid closures with maximum Thurston-Bennequin number.

math.SG↗

arXiv2025

Legendrian doubles, twist spuns, and clusters

Let $\lambda$ be a Legendrian link in standard contact $\mathbb{R}^3$, such that $L_1$, $L_2$ are two exact fillings of $\lambda$ and $\varphi$ is a Legendrian loop of $\lambda$. We study fillability and isotopy characterizations of Legendrian surfaces in standard contact $\mathbb{R}^5$ built from the above data by doubling or twist spinning; denoting them $\Lambda(L_1,L_2)$ or $\Sigma_\varphi(\lambda)$ respectively. In the case of doubles $\Lambda(L_1,L_2)$, if the sheaf moduli $\mathcal{M}_1(\lambda)$ admits a cluster structure, we introduce the notion of mutation distance and study its relationship with the isotopy class of the Legendrian surface. For twist spuns $\Sigma_\varphi(\lambda)$, when $\mathcal{M}_1(\lambda)$ admits a globally foldable cluster structure, we use the existence of a $\varphi$-symmetric filling of the Legendrian link to build a cluster structure on the sheaf moduli of the twist spun by folding. We then use that to motivate, and provide evidence for, conjectures on the number of embedded exact fillings of certain twist spuns. Further, we obstruct the exact fillability of certain twist spuns by analyzing fixed points of the cyclic shift action on Grassmanians.

math.SG↗

arXiv2024

Detection of subclinical atherosclerosis by image-based deep learning on chest x-ray

Aims. To develop a deep-learning based system for recognition of subclinical atherosclerosis on a plain frontal chest x-ray. Methods and Results. A deep-learning algorithm to predict coronary artery calcium (CAC) score (the AI-CAC model) was developed on 460 chest x-ray (80% training cohort, 20% internal validation cohort) of primary prevention patients (58.4% male, median age 63 [51-74] years) with available paired chest x-ray and chest computed tomography (CT) indicated for any clinical reason and performed within 3 months. The CAC score calculated on chest CT was used as ground truth. The model was validated on an temporally-independent cohort of 90 patients from the same institution (external validation). The diagnostic accuracy of the AI-CAC model assessed by the area under the curve (AUC) was the primary outcome. Overall, median AI-CAC score was 35 (0-388) and 28.9% patients had no AI-CAC. AUC of the AI-CAC model to identify a CAC>0 was 0.90 in the internal validation cohort and 0.77 in the external validation cohort. Sensitivity was consistently above 92% in both cohorts. In the overall cohort (n=540), among patients with AI-CAC=0, a single ASCVD event occurred, after 4.3 years. Patients with AI-CAC>0 had significantly higher Kaplan Meier estimates for ASCVD events (13.5% vs. 3.4%, log-rank=0.013). Conclusion. The AI-CAC model seems to accurately detect subclinical atherosclerosis on chest x-ray with elevated sensitivity, and to predict ASCVD events with elevated negative predictive value. Adoption of the AI-CAC model to refine CV risk stratification or as an opportunistic screening tool requires prospective evaluation.

cs.CV↗

arXiv2024

Legendrian loops and cluster modular groups

This work studies Legendrian loop actions on exact Lagrangian fillings of Legendrian links in $(\R^3, ξ_{\st})$. By identifying the induced action of Legendrian loops as generators of cluster modular groups, we establish the existence of faithful group actions on the exact Lagrangian fillings of several families of Legendrian positive braid closures, including all positive torus links. In addition, we leverage a Nielsen-Thurston-like classification of cluster automorphisms to provide new combinatorial and algebraic tools for proving that a Legendrian loop action has infinite order.

math.SG↗

arXiv2023

Augmentations, Fillings, and Clusters for 2-Bridge Links

We produce the first examples relating non-orientable exact Lagrangian fillings of Legendrian links to cluster theory, showing that the ungraded augmentation variety of certain max-tb representatives of Legendrian $2$-bridge links is isomorphic to a product of $A_n$-type cluster varieties. As part of this construction, we describe a surjective map from the set of (possibly non-orientable) exact Lagrangian fillings to cluster seeds, producing a product of Catalan numbers of distinct fillings. We also relate the ruling stratification of the ungraded augmentation variety to Lam and Speyer's anticlique stratification of acyclic cluster varieties, showing that the two coincide in this context. As a corollary, we apply a result of Rutherford to show that the cluster-theoretic stratification encodes the information of the lowest $a$-degree term of the Kauffman polynomial of the smooth isotopy class of the $2$-bridge links we study.

math.SG↗

arXiv2022

Lagrangian Fillings in A-type and their Kalman Loop Orbits

We compare two constructions of exact Lagrangian fillings of Legendrian positive braid closures, the Legendrian weaves of Casals-Zaslow, and the decomposable Lagrangian fillings, of Ekholm-Honda-Kálmán and show that they coincide for large families of Lagrangian fillings. As a corollary, we obtain an explicit correspondence between Hamiltonian isotopy classes of decomposable Lagrangian fillings of Legendrian $(2,n)$ torus links described by Ekholm-Honda-Kálmán and the weave fillings constructed by Treumann and Zaslow. We apply this result to describe the orbital structure of the Kálmán loop and give a combinatorial criteria to determine the orbit size of a filling. We follow our geometric discussion with a Floer-theoretic proof of the orbital structure, where an identity studied by Euler in the context of continued fractions makes a surprise appearance. We conclude by giving a purely combinatorial description of the Kálmán loop action on the fillings discussed above in terms of edge flips of triangulations.

math.SG↗

arXiv2022

Weave Realizability for D-type

We study exact Lagrangian fillings of Legendrian links of $D_n$-type in the standard contact 3-sphere. The main result is the existence of a Lagrangian filling, represented by a weave, such that any algebraic quiver mutation of the associated intersection quiver can be realized as a geometric weave mutation. The method of proof is via Legendrian weave calculus and a construction of appropriate 1-cycles whose geometric intersections realize the required algebraic intersection numbers. In particular, we show that in $D$-type, each cluster chart of the moduli of microlocal rank-1 sheaves is induced by at least one embedded exact Lagrangian filling. Hence, the Legendrian links of $D_n$-type have at least as many Hamiltonian isotopy classes of Lagrangian fillings as cluster seeds in the $D_n$-type cluster algebra, and their geometric exchange graph for Lagrangian disk surgeries contains the cluster exchange graph of $D_n$-type.

math.SG↗

arXiv2003

Length-Based Attacks for Certain Group Based Encryption Rewriting Systems

In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes use of the property that in the given group the word problem has a polynomial time solution, while the conjugacy problem has no known polynomial solution. An example is the braid group from topology in which the word problem is solvable in polynomial time while the only known solutions to the conjugacy problem are exponential. The attack in this paper is based on having a canonical representative of each string relative to which a length function may be computed. Hence the term length attack. Such canonical representatives are known to exist for the braid group.

cs.CR↗

arXiv1994

Skyrmion Quantization and the Decay of the Delta

We present the complete solution to the so-called ``Yukawa problem'' of the Skyrme model. This refers to the perceived difficulty of reproducing---purely from soliton physics---the usual pseudovector pion-nucleon coupling, echoed by pion coupling to the higher spin/isospin baryons $(I=J=3/2 , 5/2 , \cdots , N_c/2 )$ in a manner fixed by large-$N_c$ group theory. The solution involves surprisingly elegant interplay between the classical and quantum properties of a new configuration, the ``new improved skyrmion''. This is the near-hedgehog obtained by minimizing the usual skyrmion mass functional augmented by an all-important isorotational kinetic term. The numerics are pleasing: a $Δ$ decay width within a few MeV of its measured value, and furthermore, the higher-spin baryons $(I=J \ge 5/2 )$ with widths so large ($Γ> 800 MeV$) that these undesirable large-$N_c$ artifacts effectively drop out of the spectrum, and pose no phenomenological problem. Beyond these specific results, we ground the Skyrme model in the Feynman Path Integral, and set up a transparent collective coordinate formalism that makes maximal use of the $1/N_c$ expansion. This approach elucidates the connection between skyrmions on the one hand, and Feynman diagrams in an effective field theory on the other.

hep-ph↗

arXiv1994

Next-to-leading Order Parton Model Calculations in the Massless Schwinger Model

We carry out next-to-leading order (NLO) parton model calculations for the standard hard ``QCD'' processes in the massless Schwinger model. The asymptotic expansion of the exact result for the deep inelastic cross section is used to infer the NLO distribution function. These distribution functions are then used to calculate the NLO Drell-Yan parton model cross section and it is found to agree with the corresponding term in the expansion of the exact result for the Drell-Yan cross section. Finally, by using the bosonization formula and the exact solutions we study the interference between different partonic processes.

hep-ph↗

arXiv1994

Lepton-antilepton Annihilation in the Massless Schwinger Model

We evaluate the next-to-leading order radiative corrections to a process in the massless Schwinger model that is the analogue of the $e^+e^-$ inclusive annihilation into hadrons process in QCD. We then carry out an asymptotic expansion of the calculable exact result and find that it agrees with the perturbative answer. However, we also find that higher orders in the asymptotic expansion involve terms that depend on the logarithm of the strong coupling constant.

hep-ph↗

arXiv1993

Soliton quantization and internal symmetry

We apply the method of collective coordinate quantization to a model of solitons in two spacetime dimensions with a global $U(1)$ symmetry. In particular we consider the dynamics of the charged states associated with rotational excitations of the soliton in the internal space and their interactions with the quanta of the background field (mesons). By solving a system of coupled saddle-point equations we effectively sum all tree-graphs contributing to the one-point Green's function of the meson field in the background of a rotating soliton. We find that the resulting one-point function evaluated between soliton states of definite $U(1)$ charge exhibits a pole on the meson mass shell and we extract the corresponding S-matrix element for the decay of an excited state via the emission of a single meson using the standard LSZ reduction formula. This S-matrix element has a natural interpretation in terms of an effective Lagrangian for the charged soliton states with an explicit Yukawa coupling to the meson field. We calculate the leading-order semi-classical decay width of the excited soliton states discuss the consequences of these results for the hadronic decay of the $Δ$ resonance in the Skyrme model.

hep-th↗