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1. Sheaves of maximal intersection and multiplicities of stable log maps

   https://doi.org/10.1007/s00029-021-00671-0  

   Choi, Jinwon ; van Garrel, Michel ; Katz, Sheldon ; Takahashi, Nobuyoshi ( September 2021 , Selecta Mathematica)

   Abstract A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps. 

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2. Noncommutative Bohnenblust–Hille inequalities

   https://doi.org/10.1007/s00208-023-02680-0  

   Volberg, Alexander ; Zhang, Haonan ( July 2023 , Mathematische Annalen)

   Bohnenblust–Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree (Defant et al. in Math Ann 374(1):653–680, 2019). Such inequalities have found great applications in learning low-degree Boolean functions (Eskenazis and Ivanisvili in Proceedings of the 54th annual ACM SIGACT symposium on theory of computing, pp 203–207, 2022). Motivated by learning quantum observables, a qubit analogue of Bohnenblust–Hille inequality for Boolean cubes was recently conjectured in Rouzé et al. (Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum Boolean functions, 2022. arXiv preprintarXiv:2209.07279). The conjecture was resolved in Huang et al. (Learning to predict arbitrary quantum processes, 2022. arXiv preprintarXiv:2210.14894). In this paper, we give a new proof of these Bohnenblust–Hille inequalities for qubit system with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials. Using similar ideas, we also study learning problems of low degree quantum observables and Bohr’s radius phenomenon on quantum Boolean cubes.

    

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3. Birth–death dynamics for sampling: global convergence, approximations and their asymptotics

   https://doi.org/10.1088/1361-6544/acf988  

   Lu, Yulong ; Slepčev, Dejan ; Wang, Lihan ( September 2023 , Nonlinearity)

   Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth–death dynamics. We improve results in previous works (Liuet al2023Appl. Math. Optim.8748; Luet al2019 arXiv:1905.09863) and provide weaker hypotheses under which the probability density of the birth–death governed by Kullback–Leibler divergence or byχ²divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth–death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker–Planck equation and relies on kernel-based approximations of the measure. Using the technique of Γ-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelised dynamics converge on finite time intervals, to the pure birth–death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimisers of the energy corresponding to the kernelised dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelised dynamics towards the Gibbs measure.

    

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4. Stochastic Lagrangian dynamics of vorticity. Part 1. General theory for viscous, incompressible fluids

   https://doi.org/10.1017/jfm.2020.491  

   Eyink, Gregory L. ; Gupta, Akshat ; Zaki, Tamer A. ( October 2020 , Journal of Fluid Mechanics)

   null (Ed.)

   Prior mathematical work of Constantin & Iyer ( Commun. Pure Appl. Maths , vol. 61, 2008, pp. 330–345; Ann. Appl. Probab. , vol. 21, 2011, pp. 1466–1492) has shown that incompressible Navier–Stokes solutions possess infinitely many stochastic Lagrangian conservation laws for vorticity, backward in time, which generalize the invariants of Cauchy ( Sciences mathématiques et physique , vol. I, 1815, pp. 33–73) for smooth Euler solutions. We reformulate this theory for the case of wall-bounded flows by appealing to the Kuz'min ( Phys. Lett. A , vol. 96, 1983, pp. 88–90)–Oseledets ( Russ. Math. Surv. , vol. 44, 1989, p. 210) representation of Navier–Stokes dynamics, in terms of the vortex-momentum density associated to a continuous distribution of infinitesimal vortex rings. The Constantin–Iyer theory provides an exact representation for vorticity at any interior point as an average over stochastic vorticity contributions transported from the wall. We point out relations of this Lagrangian formulation with the Eulerian theory of Lighthill (Boundary layer theory. In Laminar Boundary Layers (ed. L. Rosenhead), 1963, pp. 46–113)–Morton ( Geophys. Astrophys. Fluid Dyn. , vol. 28, 1984, pp. 277–308) for vorticity generation at solid walls, and also with a statistical result of Taylor ( Proc. R. Soc. Lond. A , vol. 135, 1932, pp. 685–702)–Huggins ( J. Low Temp. Phys. , vol. 96, 1994, pp. 317–346), which connects dissipative drag with organized cross-stream motion of vorticity and which is closely analogous to the ‘Josephson–Anderson relation’ for quantum superfluids. We elaborate a Monte Carlo numerical Lagrangian scheme to calculate the stochastic Cauchy invariants and their statistics, given the Eulerian space–time velocity field. The method is validated using an online database of a turbulent channel-flow simulation (Graham et al. , J. Turbul. , vol. 17, 2016, pp. 181–215), where conservation of the mean Cauchy invariant is verified for two selected buffer-layer events corresponding to an ‘ejection’ and a ‘sweep’. The variances of the stochastic Cauchy invariants grow exponentially backward in time, however, revealing Lagrangian chaos of the stochastic trajectories undergoing both fluid advection and viscous diffusion. 

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5. Generalized Parking Function Polytopes

   https://doi.org/10.1007/s00026-023-00671-1  

   Hanada, Mitsuki ; Lentfer, John ; Vindas-Meléndez, Andrés R. ( November 2023 , Annals of Combinatorics)

   A classical parking function of lengthnis a list of positive integers$$(a_1, a_2, \ldots , a_n)$$$(a_1,a_2,\dots ,a_n)$whose nondecreasing rearrangement$$b_1 \le b_2 \le \cdots \le b_n$$$b_1\le b_2\le \cdots \le b_n$satisfies$$b_i \le i$$$b_i\le i$. The convex hull of all parking functions of lengthnis ann-dimensional polytope in$${\mathbb {R}}^n$$$R^n$, which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of$${\textbf{x}}$$$x$-parking functions for$${\textbf{x}}=(a,b,\dots ,b)$$$x=(a,b,\cdots ,b)$, which we refer to as$${\textbf{x}}$$$x$-parking function polytopes. We explore connections between these$${\textbf{x}}$$$x$-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of$${\textbf{x}}$$$x$-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.

    

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