{"title":"Closed-String Mirror Symmetry for Log Calabi-Yau Surfaces","authors":"Hyunbin Kim","doi":"arxiv-2408.02592","DOIUrl":"https://doi.org/arxiv-2408.02592","url":null,"abstract":"This paper establishes closed-string mirror symmetry for all log Calabi-Yau\u0000surfaces with generic parameters, where the exceptional divisor are\u0000sufficiently small. We demonstrate that blowing down a $(-1)$-divisor removes a\u0000single geometric critical point, ensuring that the resulting potential remains\u0000a Morse function. Additionally, we show that the critical values are distinct,\u0000which implies that the quantum cohomology $QH^{ast}(X)$ is semi-simple.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A stable homotopy invariant for Legendrians with generating families","authors":"Hiro Lee Tanaka, Lisa Traynor","doi":"arxiv-2408.01587","DOIUrl":"https://doi.org/arxiv-2408.01587","url":null,"abstract":"We construct a stable homotopy type invariant for any Legendrian submanifold\u0000in a jet bundle equipped with a linear-at-infinity generating family. We show\u0000that this spectrum lifts the generating family homology groups. When the\u0000generating family extends to a generating family for an embedded Lagrangian\u0000filling, we lift the Seidel isomorphism to the spectrum level. As applications,\u0000we establish topological constraints on Lagrangian fillings arising from\u0000generating families, algebraic constraints on whether generating families admit\u0000fillings, and lower bounds on how many fiber dimensions are needed to construct\u0000a generating family for a Legendrian.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Koszul Duality for star-shaped partial Heegaard diagrams","authors":"Isabella Khan","doi":"arxiv-2408.01564","DOIUrl":"https://doi.org/arxiv-2408.01564","url":null,"abstract":"By slicing the Heegaard diagram for a given $3$-manifold in a particular way,\u0000it is possible to construct $mathcal{A}_{infty}$-bimodules, the tensor\u0000product of which retrieves the Heegaard Floer homology of the original\u00003-manifold. The first step in this is to construct algebras corresponding to\u0000the individual slices. In this paper, we use the graphical calculus for\u0000$mathcal{A}_{infty}$-structures introduced in arXiv:2009.05222v3 to construct\u0000Koszul dual $mathcal{A}_{infty}$ algebras $mathcal{A}$ and $mathcal{B}$ for\u0000a particular star-shaped class of slice. Using $mathcal{A}_{infty}$-bimodules\u0000over $mathcal{A}$ and $mathcal{B}$, we then verify the Koszul duality\u0000relation.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Growth of eigenvalues of Floer Hessians","authors":"Urs Frauenfelder, Joa Weber","doi":"arxiv-2408.00269","DOIUrl":"https://doi.org/arxiv-2408.00269","url":null,"abstract":"In this article we prove that the space of Floer Hessians has infinitely many\u0000connected components.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rabinowitz Floer homology as a Tate vector space","authors":"Kai Cieliebak, Alexandru Oancea","doi":"arxiv-2407.21741","DOIUrl":"https://doi.org/arxiv-2407.21741","url":null,"abstract":"We show that the category of linearly topologized vector spaces over discrete\u0000fields constitutes the correct framework for algebraic structures on Floer\u0000homologies with field coefficients. Our case in point is the Poincar'e duality\u0000theorem for Rabinowitz Floer homology. We prove that Rabinowitz Floer homology\u0000is a locally linearly compact vector space in the sense of Lefschetz, or,\u0000equivalently, a Tate vector space in the sense of Beilinson-Feigin-Mazur.\u0000Poincar'e duality and the graded Frobenius algebra structure on Rabinowitz\u0000Floer homology then hold in the topological sense. Along the way, we develop in\u0000a largely self-contained manner the theory of linearly topologized vector\u0000spaces, with special emphasis on duality and completed tensor products,\u0000complementing results of Beilinson-Drinfeld, Beilinson, Rojas, Positselski, and\u0000Esposito-Penkov.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"124 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Solution to the Periodic Square Peg Problem","authors":"Cole Hugelmeyer","doi":"arxiv-2407.20412","DOIUrl":"https://doi.org/arxiv-2407.20412","url":null,"abstract":"We resolve the periodic square peg problem using a simple Lagrangian Floer\u0000homology argument. Inscribed squares are interpreted as intersections between\u0000two non-displaceable Lagrangian sub-manifolds of a symplectic 4-torus.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"124 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Symplectic structures on the space of space curves","authors":"Martin Bauer, Sadashige Ishida, Peter W. Michor","doi":"arxiv-2407.19908","DOIUrl":"https://doi.org/arxiv-2407.19908","url":null,"abstract":"We present symplectic structures on the shape space of unparameterized space\u0000curves that generalize the classical Marsden-Weinstein structure. Our method\u0000integrates the Liouville 1-form of the Marsden-Weinstein structure with\u0000Riemannian structures that have been introduced in mathematical shape analysis.\u0000We also derive Hamiltonian vector fields for several classical Hamiltonian\u0000functions with respect to these new symplectic structures.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"If a Minkowski billiard is projective, it is the standard billiard","authors":"Alexey Glutsyuk, Vladimir S. Matveev","doi":"arxiv-2407.20159","DOIUrl":"https://doi.org/arxiv-2407.20159","url":null,"abstract":"In the recent paper arXiv:2405.13258, the first author of this note proved\u0000that if a billiard in a convex domain in $mathbb{R}^n$ is simultaneously\u0000projective and Minkowski, then it is the standard Euclidean billiard in an\u0000appropriate Euclidean structure. The proof was quite complicated and required\u0000high smoothness. Here we present a direct simple proof of this result which\u0000works in $C^1$-smoothness. In addition we prove the semi-local and local\u0000versions of the result","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Castelnuovo bound for curves in projective 3-folds","authors":"Zhiyu Liu","doi":"arxiv-2407.20161","DOIUrl":"https://doi.org/arxiv-2407.20161","url":null,"abstract":"The Castelnuovo bound conjecture, which is proposed by physicists, predicts\u0000an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau\u00003-folds of Picard number one. Previously, it is only known for a few cases and\u0000all the proofs rely on the Bogomolov-Gieseker conjecture of Bayer-Macr`i-Toda. In this paper, we prove the Castelnuovo bound conjecture for any Calabi-Yau\u00003-folds of Picard number one, up to a linear term and finitely many degree,\u0000without assuming the conjecture of Bayer-Macr`i-Toda. Furthermore, we prove an\u0000effective vanishing theorem for surface-counting invariants of Calabi-Yau\u00004-folds of Picard number one. We also apply our techniques to study low-degree\u0000curves on some explicit Calabi-Yau 3-folds. Our approach is based on a general iterative method to obtain upper bounds\u0000for the genus of one-dimensional closed subschemes in a fixed 3-fold, which is\u0000a combination of classical techniques and the wall-crossing of weak stability\u0000conditions on derived categories, and works for any projective 3-fold with at\u0000worst isolated singularities over any algebraically closed field.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"110 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Traveling Mailman: Topological Optimization Methods for User-Centric Redistricting","authors":"Nelson A. Colón Vargas","doi":"arxiv-2407.19535","DOIUrl":"https://doi.org/arxiv-2407.19535","url":null,"abstract":"This study introduces a new districting approach using the US Postal Service\u0000network to measure community connectivity. We combine Topological Data Analysis\u0000with Markov Chain Monte Carlo methods to assess district boundaries' impact on\u0000community integrity. Using Iowa as a case study, we generate and refine\u0000districting plans using KMeans clustering and stochastic rebalancing. Our\u0000method produces plans with fewer cut edges and more compact shapes than the\u0000official Iowa plan under relaxed conditions. The low likelihood of finding\u0000plans as disruptive as the official one suggests potential inefficiencies in\u0000existing boundaries. Gaussian Mixture Model analysis reveals three distinct\u0000distributions in the districting landscape. This framework offers a more\u0000accurate reflection of community interactions for fairer political\u0000representation.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
求助内容：
应助结果提醒方式：