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1. Positroid Catalan numbers

   https://doi.org/10.1090/cams/33  

   Galashin, Pavel; Lam, Thomas (April 2024, Communications of the American Mathematical Society)

   Given a (bounded affine) permutation $f$, we study thepositroid Catalan number $C_f$defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class ofrepetition-free permutationsand show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated $q,t$-polynomials coincide with thegeneralized $q,t$-Catalan numbersthat recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov–Rozansky homology of Coxeter links. 

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2. The geometric distribution of Selmer groups of elliptic curves over function fields

   https://doi.org/10.1007/s00208-022-02429-1  

   Feng, Tony; Landesman, Aaron; Rains, Eric M. (September 2022, Mathematische Annalen)

   Abstract Fix a positive integernand a finite field$${\mathbb {F}}_q$$ $F_q$. We study the joint distribution of the rank$${{\,\mathrm{rk}\,}}(E)$$ $\mathrm{rk}\left(E\right)$, then-Selmer group$$\text {Sel}_n(E)$$ ${\text{Sel}}_n\left(E\right)$, and then-torsion in the Tate–Shafarevich group Equation missing<#comment/>asEvaries over elliptic curves of fixed height$$d \ge 2$$ $d\ge 2$over$${\mathbb {F}}_q(t)$$ $F_q\left(t\right)$. We compute this joint distribution in the largeqlimit. We also show that the “largeq, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains. 

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3. Casimir energy and modularity in higher-dimensional conformal field theories

   https://doi.org/10.1007/JHEP07(2023)028  

   Luo, Conghuan; Wang, Yifan (July 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> An important problem in Quantum Field Theory (QFT) is to understand the structures of observables on spacetime manifolds of nontrivial topology. Such observables arise naturally when studying physical systems at finite temperature and/or finite volume and encode subtle properties of the underlying microscopic theory that are often obscure on the flat spacetime. Locality of the QFT implies that these observables can be constructed from more basic building blocks by cutting-and-gluing along a spatial slice, where a crucial ingredient is the Hilbert space on the spatial manifold. In Conformal Field Theory (CFT), thanks to the operator-state correspondence, we have a non-perturbative understanding of the Hilbert space on a spatial sphere. However it remains a challenge to consider more general spatial manifolds. Here we study CFTs in spacetime dimensionsd >2 on the spatial manifoldT²× ℝ^d−3which is one of the simplest manifolds beyond the spherical topology. We focus on the ground state in this Hilbert space and analyze universal properties of the ground state energy, also commonly known as the Casimir energy, which is a nontrivial function of the complex structure moduliτof the torus. The Casimir energy is subject to constraints from modular invariance on the torus which we spell out using PSL(2,ℤ) spectral theory. Moreover we derive a simple universal formula for the Casimir energy in the thin torus limit using the effective field theory (EFT) from Kaluza-Klein reduction of the CFT, with exponentially small corrections from worldline instantons. We illustrate our formula with explicit examples from well-known CFTs including the criticalO(N) model ind= 3 and holographic CFTs ind ≥3. 

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4. Scaling the Mixing Efficiency of Sediment‐Stratified Turbulence

   https://doi.org/10.1029/2022GL099025  

   Tu, Junbiao; Fan, Daidu; Liu, Zhiyu; Smyth, William (June 2022, Geophysical Research Letters)

   Abstract The flux Richardson numberR_f, also called the mixing efficiency of stratified turbulence, is important in determining geophysical flow phenomena such as ocean circulation and air‐sea transports. MeasuringR_fin the field is usually difficult, thus parameterization ofR_fbased on readily observed properties is essential. Here, estimates ofR_fin a strongly turbulent, sediment‐stratified estuarine flow are obtained from measurements of covariance‐derived turbulent buoyancy fluxes (B) and spectrally fitted values of the dissipation rate of turbulent kinetic energy (ε). We test scalings forR_fin terms of the buoyancy Reynolds number (Re_b), the gradient Richardson number (Ri), and turbulent Froude number (Fr_t). Neither theRe_b‐based nor theRi‐based scheme is able to describe the observed variations inR_f, but theFr_t‐based parameterization works well. These findings support further use of theFr_t‐ based parameterization in turbulent oceanic and estuarine environments. 

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5. Variation of canonical height for\break Fatou points on ℙ ¹

   https://doi.org/10.1515/crelle-2022-0078  

   DeMarco, Laura; Mavraki, Niki Myrto (December 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let f : ℙ 1 → ℙ 1 {f:\mathbb{P}^{1}\to\mathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ⁢ ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K . For each point a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ⁢ ( ℚ ¯ ) {t\in X(\overline{\mathbb{Q}})} outside a finite set, induces a Weil height on the curve X ; i.e., we prove the existence of a ℚ {\mathbb{Q}} -divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ⁢ ( a t ) - h D ⁢ ( t ) {t\mapsto\hat{h}_{f_{t}}(a_{t})-h_{D}(t)} is bounded on X ⁢ ( ℚ ¯ ) {X(\overline{\mathbb{Q}})} for any choice of Weil height associated to D . We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ⁢ ( a t ) {t\mapsto\hat{\lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ⁢ ( ℂ v ) {X(\mathbb{C}_{v})} , at each place v of the number field K . These results were known for polynomial maps f and all points a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} without the stability hypothesis,[21, 14],and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} . [32, 29].Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {\tilde{f}:X\times\mathbb{P}^{1}\dashrightarrow X\times\mathbb{P}^{1}} over K ; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k , where the local canonical height λ ^ f , γ ⁢ ( a ) {\hat{\lambda}_{f,\gamma}(a)} can be computed as an intersection number. 

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