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1. 3D Mirror Symmetry for Instanton Moduli Spaces

   https://doi.org/10.1007/s00220-023-04831-5  

   Koroteev, Peter ; Zeitlin, Anton M. ( September 2023 , Communications in Mathematical Physics)

   We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$$C^2$($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$${\text{Hilb}}^k\left[C^2\right]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$$C_{ħ}^{\times }$-action. First, we find a two-parameter family$$X_{k,l}$$$X_{k,l}$of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$${\text{Hilb}}^k\left[C^2\right]$is obtained via direct limit$$l\longrightarrow \infty $$$l⟶\infty$and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$$ħ$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$$P^2$with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

    

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2. Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

   https://doi.org/10.1007/s00454-022-00426-4  

   Greenfeld, Rachel ; Tao, Terence ( January 2023 , Discrete & Computational Geometry)

   We construct an example of a group$$G = \mathbb {Z}^2 \times G_0$$$G=Z^2\times G_0$for a finite abelian group $$G_0$$$G_0$, a subsetEof $$G_0$$$G_0$, and two finite subsets$$F_1,F_2$$$F_1,F_2$of G, such that it is undecidable in ZFC whether$$\mathbb {Z}^2\times E$$$Z^2\times E$can be tiled by translations of$$F_1,F_2$$$F_1,F_2$. In particular, this implies that this tiling problem isaperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings ofEby the tiles$$F_1,F_2$$$F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$$Z^2$). A similar construction also applies for$$G=\mathbb {Z}^d$$$G=Z^d$for sufficiently large d. If one allows the group$$G_0$$$G_0$to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.

    

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3. Stochastic dual dynamic programming for multistage stochastic mixed-integer nonlinear optimization

   https://doi.org/10.1007/s10107-022-01875-8  

   Zhang, Shixuan ; Sun, Xu Andy ( August 2022 , Mathematical Programming)

   In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization withnon-Lipschitzianvalue functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a$$(T+1)$$$(T+1)$-stage stochastic MINLP satisfyingL-exact Lipschitz regularization withd-dimensional state spaces, to obtain an$$\varepsilon $$$\epsilon$-optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by$${\mathcal {O}}((\frac{2LT}{\varepsilon })^d)$$$O\left({\left(\frac{2LT}{\epsilon }\right)}^d\right)$, and is lower bounded by$${\mathcal {O}}((\frac{LT}{4\varepsilon })^d)$$$O\left({\left(\frac{\mathrm{LT}}{4\epsilon }\right)}^d\right)$for the general case or by$${\mathcal {O}}((\frac{LT}{8\varepsilon })^{d/2-1})$$$O\left({\left(\frac{\mathrm{LT}}{8\epsilon }\right)}^{d/2-1}\right)$for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity dependspolynomiallyon the number of stages. We further show that the iteration complexity dependslinearlyonT, if all the state spaces are finite sets, or if we seek a$$(T\varepsilon )$$$\left(T\epsilon \right)$-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale withT. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization.

    

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4. Green function estimates on complements of low-dimensional uniformly rectifiable sets

   https://doi.org/10.1007/s00208-022-02379-8  

   David, Guy ; Feneuil, Joseph ; Mayboroda, Svitlana ( March 2022 , Mathematische Annalen)

   It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$$L_{\beta ,\gamma }=-\text{div}{D}^{d+1+\gamma -n}\nabla$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$$\Omega \subset R^n$with a uniformly rectifiable boundary$$\Gamma $$$\Gamma$of dimension$$d < n-1$$$d<n-1$, the now usual distance to the boundary$$D = D_\beta $$$D=D_{\beta }$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$$D_{\beta }{\left(X\right)}^{-\beta }={\int }_{\Gamma }{|X-y|}^{-d-\beta }d\sigma \left(y\right)$for$$X \in \Omega $$$X\in \Omega$, where$$\beta >0$$$\beta >0$and$$\gamma \in (-1,1)$$$\gamma \in (-1,1)$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$$L_{\beta ,\gamma }$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$$D^{1-\gamma }$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$$|D\nabla (ln(\frac{G}{D^{1-\gamma }})){|}^2$satisfies a Carleson measure estimate on$$\Omega $$$\Omega$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).

    

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5. Steady Radiating Gravity waves: An Exponential Asymptotics Approach

   https://doi.org/10.1007/s42286-023-00081-z  

   Kataoka, Takeshi ; Akylas, T. R. ( January 2024 , Water Waves)

   The radiation of steady surface gravity waves by a uniform stream$$U_{0}$$$U_0$over locally confined (width$$L$$$L$) smooth topography is analyzed based on potential flow theory. The linear solution to this classical problem is readily found by Fourier transforms, and the nonlinear response has been studied extensively by numerical methods. Here, an asymptotic analysis is made for subcritical flow$$D/\lambda > 1$$$D/\lambda >1$in the low-Froude-number ($$F^{2} \equiv \lambda /L \ll 1$$$F^2\equiv \lambda /L\ll 1$) limit, where$$\lambda = U_{0}^{2} /g$$$\lambda =U_0^2/g$is the lengthscale of radiating gravity waves and$$D$$$D$is the uniform water depth. In this regime, the downstream wave amplitude, although formally exponentially small with respect to$$F$$$F$, is determined by a fully nonlinear mechanism even for small topography amplitude. It is argued that this mechanism controls the wave response for a broad range of flow conditions, in contrast to linear theory which has very limited validity.

    

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