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The ${\rm SL}_3$-skein algebra $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ of a punctured oriented surface $\mathfrak{S}$ is a quantum deformation of the coordinate algebra of the ${\rm SL}_3$-character variety of $\mathfrak{S}$. When $\bar{q}$ is a root of unity, we prove the Unicity Theorem for representations of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$, in particular the existence and uniqueness of a generic irreducible representation. Furthermore, we show that the center of $\mathscr{S}_{\bar{q}}(\frak{S})$ is generated by the peripheral skeins around punctures and the central elements contained in the image of the Frobenius homomorphism for $\mathscr{S}_{\bar{q}}(\frak{S})$, a surface generalization of Frobenius homomorphisms of quantum groups related to ${\rm SL}_3$. We compute the rank of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ over its center, hence the dimension of the generic irreducible representation., Comment: v2: 56 pages, reorganized, streamlined, added some details

Recent experiments have shown that metachronal waves (MCWs) can emerge from a chain of symmetrically beating nematodes aligned at the edge of sessile droplets. Our study, employing a coupled elastohydrodynamic model of active filaments, elucidates that a misalignment caused by a tilt against the bounding wall disrupts the synchronization and generates a constant time lag between adjacent filaments, giving rise to MCWs. The MCWs, enhancing the fluid circulation, achieve their maximum thermodynamic efficiency over the same range of tilt angles observed in the nematode experiments., Comment: 12 page, 8 figures

In 1980's H Verlinde suggested to construct and use a quantization of Teichm\"uller spaces to construct spaces of conformal blocks for the Liouville conformal field theory. This suggestion led to a mathematical formulation by Fock in 1990's, called the modular functor conjecture, based on the Chekhov-Fock quantum Teichm\"uller theory. In 2000's Teschner combined the Chekhov-Fock version and the Kashaev version of quantum Teichm\"uller theory to construct a solution to a modified form of the conjecture. We embark on a direct approach to the conjecture based on the Chekhov-Fock(-Goncharov) theory. We construct quantized trace-of-monodromy along simple loops via Bonahon and Wong's quantum trace maps developed in 2010's, and investigate algebraic structures of them, which will eventually lead to construction and properties of quantized geodesic length operators. We show that a special recursion relation used by Teschner is satisfied by the quantized trace-of-monodromy, and that the quantized trace-of-monodromy for disjoint loops commute in a certain strong sense., Comment: 74 pages

Mesh-like structures, such as mucus gel or cytoskeleton networks, are ubiquitous in biological systems. These intricate structures are composed of cross-linked, semi-flexible bio-filaments, crucial to numerous biological processes. In many biological systems, active self-propelled particles like motor proteins or bacteria navigate these intricate polymer networks. In this study, we develop a computational model of three-dimensional cubic-topological, swollen polymer networks of semi-flexible filaments. We perform Langevin dynamics simulations to investigate the diffusion of active tracer particles navigating through these networks. By analyzing various physical observables, we investigate the effects of mesh-to-particle size ratio, P\'eclet number of active particles, and bending stiffness of the polymer networks upon active trapped-and-hopping diffusion of the tracer. When the tracer size is equal to or larger than the mesh size, the polymer stiffness substantially enhances trapping while suppressing the hopping process. Notably, the mean trapped time exhibits an exponential growth law to the bending stiffness with an activity-dependent slope. An analytic theory based on the mean first-passage time of active particles in a harmonic potential is developed. Our findings deepen the comprehension of the intricate interplay between the polymer's bending stiffness, tracer size, and the activity of tracer particles. This knowledge can shed light on important biological processes, such as motor-driven cargo transport or drug delivery, which hinge on the behavior of active particles within biological gels.

Nonlinear optical signals in optically driven quantum systems can reveal coherences and thereby open up the possibility for manipulation of quantum states. While the limiting cases of ultrafast and continuous-wave excitation have been extensively studied, the time-dynamics of finite pulses bear interesting phenomena. In this paper, we explore the nonlinear optical probe signals of a two-level system excited with a laser pulse of finite duration. In addition to the prominent Mollow peaks, the probe spectra feature several smaller peaks for certain time delays. Similar features have been recently observed for resonance fluorescence signals [arxiv:2305.15827 (2023)]. We discuss that the emergent phenomena can be explained by a combination of Mollow triplet physics and perturbed free induction decay effects, providing an insightful understanding of the underlying physics.

This work presents a general chemical reaction network theory for olfactory sensing processes that employ G-protein-coupled receptors as olfactory receptors (ORs). The theory is applicable to general mixtures of odorants and an arbitrary number of ORs. Reactions of ORs with G-proteins, both in the presence and the absence of odorants, are explicitly considered. A unique feature of the theory is the definition of an odor activity vector consisting of strengths of odorant-induced signals from ORs relative to those due to background G-protein activity in the absence of odorants. It is demonstrated that each component of the odor activity defined this way reduces to a Michaelis-Menten form capable of accounting for cooperation or competition effects between different odorants. The main features of the theory are illustrated for a two-odorant mixture. Known and potential mixture effects, such as suppression, shadowing, inhibition, and synergy are quantitatively described. Effects of relative values of rate constants, basal activity, and G-protein concentration are also demonstrated.

For a punctured surface $\mathfrak{S}$, the author and Scarinci (arXiv:2112.13329) have recently constructed a quantization of a moduli space of Lorentzian metrics on the 3-manifold $\mathfrak{S} \times \mathbb{R}$ of constant sectional curvature $\Lambda \in \{-1,0,1\}$. The invariance of this quantization under the action of the mapping class group ${\rm MCG}(\mathfrak{S})$ of $\mathfrak{S}$ yields families of unitary representations of ${\rm MCG}(\mathfrak{S})$ on a Hilbert space, with key ingredients being three versions of the quantum dilogarithm functions depending on $\Lambda$. In this survey article, we review and elaborate on this result., Comment: 32 pages; to appear in the proceedings of the conference in honor of Igor Frenkel's 70th birthday / v2: minor updates

The interaction of a resonant light field with a quantum two-level system is of key interest both for fundamental quantum optics and quantum technological applications employing resonant excitation. While emission under resonant continuous-wave excitation has been well-studied, the more complex emission spectrum of dynamically dressed states, a quantum two-level system driven by resonant pulsed excitation, has so far been investigated in detail only theoretically. Here, we present the first experimental observation of the complete resonance fluorescence emission spectrum of a single quantum two-level system, in form of an excitonic transition in a semiconductor quantum dot, driven by finite Gaussian pulses. We observe multiple emerging sidebands as predicted by theory with an increase of their number and spectral detuning with excitation pulse intensity and a dependence of their spectral shape and intensity on the pulse length. Detuning-dependent measurements provide additional insights into the emission features. The experimental results are in excellent agreement with theoretical calculations of the emission spectra, corroborating our findings.