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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. Ambient Prime Geodesic Theorems on Hyperbolic 3-Manifolds

   https://doi.org/10.1093/imrn/rnab048  

   Dever, Lindsay; Milićević, Djordje (October 2021, International Mathematics Research Notices)

   Abstract We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy. 

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3. Special Lagrangian submanifolds of log Calabi-Yau manifolds

   Collins, Tristan C.; Jacob, A.; Lin, Y.-S. (January 2021, Duke mathematical journal)

   null (Ed.)

   We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D is a smooth divisor in the linear system of K_Y with D^2=d, then X=Y/D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface. 

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4. Proof of the elliptic expansion moonshine conjecture of Căldăraru, He, and Huang

   https://doi.org/10.1090/proc/16032  

   Hong, Letong; Mertens, Michael; Ono, Ken; Zhang, Shengtong (January 2022, Proceedings of the American Mathematical Society)

   Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a “Moonshine Conjecture at Landau-Ginzburg points” [arXiv:2107.12405, 2021] for Klein’s modular j j -function at j = 0 j=0 and j = 1728. j=1728. The conjecture asserts that the j j -function, when specialized at specific flat coordinates on the moduli spaces of versal deformations of the corresponding CM elliptic curves, yields simple rational functions. We prove this conjecture, and show that these rational functions arise from classical 2 F 1 _2F_1 -hypergeometric inversion formulae for the j j -function. 

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5. Ogg’s torsion conjecture: Fifty years later

   https://doi.org/10.1090/bull/1851  

   Balakrishnan, Jennifer; Mazur, Barry; Dogra, Netan (April 2025, Bulletin of the American Mathematical Society)

   Andrew Ogg’s mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our subject, and his influence has been such a gift to all of us. Ogg’s celebrated torsion conjecture—as it relates to modular curves—can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason for them to exist. We give a survey of Ogg’s torsion conjecture and the subsequent developments in our understanding of rational points on modular curves over the last fifty years. 

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