https://jamathr.org/index.php/jamr/issue/feedJournal of the Association for Mathematical Research2023-07-21T11:27:44-07:00Managing Editor - JAMRjamr_ed@amathr.orgOpen Journal Systems<p>The Journal of the Association for Mathematical Research (JAMR) is a Diamond Open Access journal publishing research articles in all branches of mathematics at the level of the best specialized journals. There are no strict page limits. A published article may be accompanied, when appropriate, by other media, including: links to github or other repositories for code or data, related notes and videos relevant to the article. </p>https://jamathr.org/index.php/jamr/article/view/Vol-1Issue-1Paper-1Isometric Immersions with Controlled Curvatures2023-07-21T11:04:24-07:00Misha Gromovjamr_ed@amathr.org<div class="page" title="Page 3"> <div class="layoutArea"> <div class="column"> <p>We δ-approximate strictly short (e.g. constant) maps between Riemannian manifolds f<sub>0</sub> : X<sup>m</sup> →Y<sup>N</sup> for N ≫ m<sup>2</sup>/2 by C<sup>∞</sup>-smooth isometric immersions f<sub>δ</sub> : X<sup>m</sup> →Y<sup>N</sup> <span style="font-size: 0.875rem; font-family: 'Noto Sans', 'Noto Kufi Arabic', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;">with curvatures curv(f<sub>δ</sub>)< √3/δ, for δ →0.</span></p> </div> </div> </div>2023-07-21T00:00:00-07:00Copyright (c) 2023 Misha Gromovhttps://jamathr.org/index.php/jamr/article/view/Vol-1Issue-1Paper-2Kähler-type Embeddings of Balls into Symplectic Manifolds2023-07-21T11:07:54-07:00Michael Entovjamr_ed@amathr.orgMisha Verbitskyjamr_ed@amathr.org<div class="page" title="Page 3"> <div class="layoutArea"> <div class="column"> <p>Consider a symplectic embedding of a disjoint union of domains (lying in the standard symplectic R<sup>2n</sup>) into a symplectic manifold M. We say that such an embedding is Kähler- type, or respectively tame, if it is holomorphic with respect to some (not a priori fixed, Kähler-type) integrable complex structure on M compatible with the symplectic form, or respectively tamed by it. Assume that M is either of the following: a complex projective space (with the standard symplectic form); an even-dimensional torus, or a K3 surface, equipped with an irrational Kähler-type symplectic form. Then any two Kähler-type embeddings of a disjoint union of balls into M can be mapped into each other by a symplectomorphism acting trivially on homology. If the embeddings are holomorphic with respect to complex structures compatible with the symplectic form and lying in the same connected component of the space of Kähler-type complex structures on M, then the symplectomorphism can be chosen to be smoothly isotopic to the identity. For certain M and certain disjoint unions of balls we describe precisely the obstructions to the existence of Kähler-type embeddings of the balls into M. In particular, symplectic volume is the only obstruction for the existence of Kähler-type embeddings of l<sup>n</sup> equal balls (for any l) into CP<sup>n</sup> with the standard symplectic form and of any number of possibly different balls into a torus or a K3 surface, equipped with an irrational symplectic form. We also show that symplectic volume is the only obstruction for the existence of tame embeddings of disjoint unions of equal balls, polydisks, or parallelepipeds, into a torus equipped with a generic Kähler-type symplectic form. For balls and parallelepipeds the same is true for K3 surfaces.</p> </div> </div> </div>2023-07-21T00:00:00-07:00Copyright (c) 2023 Michael Entov, Misha Verbitsky