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1. K-stability and birational models of moduli of quartic K3 surfaces

   https://doi.org/10.1007/s00222-022-01170-5  

   Ascher, Kenneth; DeVleming, Kristin; Liu, Yuchen (May 2023, Inventiones mathematicae)

   Abstract We show that the K-moduli spaces of log Fano pairs $$({\mathbb {P}}^3, cS)$$ ( P 3 , c S ) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O’Grady’s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $${\mathbb {P}}^3$$ P 3 . 

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2. On finite energy monopoles on $$C x \Sigma$$.

   Wang, Donghao (July 2019, ArXiv.org)

   Let $$X=\mathbb{C}\times\Sigma$$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $$X$$ with finite analytic energy. The spin bundle $$S^+\to X$$ splits as $$L^+\oplus L^-$$. When $$2-2g\leq c_1(S^+)[\Sigma]<0$$, the moduli space is in bijection with the moduli space of pairs $$((L^+,\bar{\partial}), f)$$ where $$(L^+,\bar{\partial})$$ is a holomorphic structure on $L^+$ and $$f: \mathbb{C}\to H^0(\Sigma, L^+,\bar{\partial})$$ is a polynomial map. Moreover, the solution has analytic energy $$-4\pi^2d\cdot c_1(S^+)[\Sigma]$$ if $$f$$ has degree $$d$$. When $$c_1(S^+)=0$$, all solutions are reducible and the moduli space is the space of flat connections on $$\bigwedge^2 S^+$$. We also estimate the decay rate at infinity for these solutions. 

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3. Counting points on smooth plane quartics

   https://doi.org/10.1007/s40993-022-00397-8  

   Costa, Edgar; Harvey, David; Sutherland, Andrew V. (March 2023, Research in Number Theory)

   Abstract We present efficient algorithms for counting points on a smooth plane quartic curve X modulo a prime p . We address both the case where X is defined over  $${\mathbb {F}}_p$$ F p and the case where X is defined over $${\mathbb {Q}}$$ Q and p is a prime of good reduction. We consider two approaches for computing $$\#X({\mathbb {F}}_p)$$ # X ( F p ) , one which runs in $$O(p\log p\log \log p)$$ O ( p log p log log p ) time using $$O(\log p)$$ O ( log p ) space and one which runs in $$O(p^{1/2}\log ^2p)$$ O ( p 1 / 2 log 2 p ) time using $$O(p^{1/2}\log p)$$ O ( p 1 / 2 log p ) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $$X/{\mathbb {Q}}$$ X / Q that compute $$\#X({\mathbb {F}}_p)$$ # X ( F p ) for good primes $$p\leqslant N$$ p ⩽ N in $$O(N\log ^3 N)$$ O ( N log 3 N ) time using O ( N ) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $${\mathbb {P}}^1$$ P 1 , which in combination with previous results addresses all curves of genus $$g\leqslant 3$$ g ⩽ 3 . Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest. 

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4. Moduli Spaces of Quadratic Maps: Arithmetic and Geometry

   https://doi.org/10.1093/imrn/rnae126  

   Ramadas, Rohini (June 2024, International Mathematics Research Notices)

   Abstract We establish an implication between two long-standing open problems in complex dynamics. The roots of the $$n$$th Gleason polynomial $$G_{n}\in{\mathbb{Q}}[c]$$ comprise the $$0$$-dimensional moduli space of quadratic polynomials with an $$n$$-periodic critical point. $$\operatorname{Per}_{n}(0)$$ is the $$1$$-dimensional moduli space of quadratic rational maps on $${\mathbb{P}}^{1}$$ with an $$n$$-periodic critical point. We show that if $$G_{n}$$ is irreducible over $${\mathbb{Q}}$$, then $$\operatorname{Per}_{n}(0)$$ is irreducible over $${\mathbb{C}}$$. To do this, we exhibit a $${\mathbb{Q}}$$-rational smooth point on a projective completion of $$\operatorname{Per}_{n}(0)$$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $$n$$, $$\operatorname{Per}_{n}(0)$$ itself has no $${\mathbb{Q}}$$-rational points. 

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5. Tractability of approximation by general shallow networks

   https://doi.org/10.1142/S0219530524400013  

   Mhaskar, Hrushikesh N; Mao, Tong (April 2024, Analysis and Applications)

   In this paper, we present a sharper version of the results in the paper Dimension independent bounds for general shallow networks; Neural Networks, \textbf{123} (2020), 142-152. Let $$\mathbb{X}$$ and $$\mathbb{Y}$$ be compact metric spaces. We consider approximation of functions of the form $$ x\mapsto\int_{\mathbb{Y}} G( x, y)d\tau( y)$$, $$ x\in\mathbb{X}$$, by $$G$$-networks of the form $$ x\mapsto \sum_{k=1}^n a_kG( x, y_k)$$, $$ y_1,\cdots, y_n\in\mathbb{Y}$$, $$a_1,\cdots, a_n\in\mathbb{R}$$. Defining the dimensions of $$\mathbb{X}$$ and $$\mathbb{Y}$$ in terms of covering numbers, we obtain dimension independent bounds on the degree of approximation in terms of $$n$$, where also the constants involved are all dependent at most polynomially on the dimensions. Applications include approximation by power rectified linear unit networks, zonal function networks, certain radial basis function networks as well as the important problem of function extension to higher dimensional spaces. 

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