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1. A CLASSIFICATION OF LAGRANGIAN PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES

   https://doi.org/10.1017/S1474748015000328  

   Bakker, Benjamin (September 2017, Journal of the Institute of Mathematics of Jussieu)

   Classically, an indecomposable class $$R$$ in the cone of effective curves on a K3 surface $$X$$ is representable by a smooth rational curve if and only if $$R^{2}=-2$$ . We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety $$M$$ deformation equivalent to a Hilbert scheme of $$n$$ points on a K3 surface, an extremal curve class $$R\in H_{2}(M,\mathbb{Z})$$ in the Mori cone is the line in a Lagrangian $$n$$ -plane $$\mathbb{P}^{n}\subset M$$ if and only if certain intersection-theoretic criteria are met. In particular, any such class satisfies $$(R,R)=-\frac{n+3}{2}$$ , and the primitive such classes are all contained in a single monodromy orbit. 

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2. The symplectic isotopy problem for rational cuspidal curves

   https://doi.org/10.1112/s0010437x2200762x  

   Golla, Marco; Starkston, Laura (July 2022, Compositio Mathematica)

   We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology. 

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3. A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

   https://doi.org/10.1090/cams/39  

   Brenner, Susanne; Sung, Li-yeng; Tan, Zhiyu; Zhang, Hongchao (October 2024, Communications of the American Mathematical Society)

   We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in ${\mathbb{R}}^2$. It is based on an isoparametric $C^0$finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions.A priorianda posteriorierror estimates together with corroborating numerical results are presented. 

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4. Inference in high-dimensional linear regression via lattice basis reduction and integer relation detection

   https://doi.org/10.1109/TIT.2021.3113921  

   Gamarnik, D.; Kizildag, E.C.; Ilias Zadik, I. (January 2021, IEEE transactions on information theory)

   We consider the high-dimensional linear regression problem, where the algorithmic goal is to efficiently infer an unknown feature vector $$\beta^*\in\mathbb{R}^p$$ from its linear measurements, using a small number $$n$$ of samples. Unlike most of the literature, we make no sparsity assumption on $$\beta^*$$, but instead adopt a different regularization: In the noiseless setting, we assume $$\beta^*$$ consists of entries, which are either rational numbers with a common denominator $$Q\in\mathbb{Z}^+$$ (referred to as $Q-$$rationality); or irrational numbers taking values in a rationally independent set of bounded cardinality, known to learner; collectively called as the mixed-range assumption. Using a novel combination of the Partial Sum of Least Squares (PSLQ) integer relation detection, and the Lenstra-Lenstra-Lov\'asz (LLL) lattice basis reduction algorithms, we propose a polynomial-time algorithm which provably recovers a $$\beta^*\in\mathbb{R}^p$ enjoying the mixed-range assumption, from its linear measurements $$Y=X\beta^*\in\mathbb{R}^n$$ for a large class of distributions for the random entries of $$X$$, even with one measurement ($n=1$). In the noisy setting, we propose a polynomial-time, lattice-based algorithm, which recovers a $$\beta^*\in\mathbb{R}^p$$ enjoying the $Q-$rationality property, from its noisy measurements $$Y=X\beta^*+W\in\mathbb{R}^n$$, even from a single sample ($n=1$). We further establish that for large $$Q$$, and normal noise, this algorithm tolerates information-theoretically optimal level of noise. We then apply these ideas to develop a polynomial-time, single-sample algorithm for the phase retrieval problem. Our methods address the single-sample ($n=1$) regime, where the sparsity-based methods such as the Least Absolute Shrinkage and Selection Operator (LASSO) and the Basis Pursuit are known to fail. Furthermore, our results also reveal algorithmic connections between the high-dimensional linear regression problem, and the integer relation detection, randomized subset-sum, and shortest vector problems. 

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5. Soliton resolution for energy-critical wave maps in the equivariant case

   https://doi.org/10.1090/jams/1012  

   Jendrej, Jacek; Lawrie, Andrew (December 2024, Journal of the American Mathematical Society)

   We consider the equivariant wave maps equation ${\mathbb{R}}^{1+2}\to <\#comment/>{\mathbb{S}}^2$, in all equivariance classes $k\in <\#comment/>\mathbb{N}$. We prove that every finite energy solution resolves, continuously in time, into a superposition of asymptotically decoupling harmonic maps and free radiation. 

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