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The geometry of dissipation 耗散几何
arXiv - MATH - Symplectic Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.11947
Asier López-Gordón
{"title":"The geometry of dissipation","authors":"Asier López-Gordón","doi":"arxiv-2409.11947","DOIUrl":"https://doi.org/arxiv-2409.11947","url":null,"abstract":"Dissipative phenomena manifest in multiple mechanical systems. In this\u0000dissertation, different geometric frameworks for modelling non-conservative\u0000dynamics are considered. The objective is to generalize several results from\u0000conservative systems to dissipative systems, specially those concerning the\u0000symmetries and integrability of these systems. More specifically, three classes\u0000of geometric frameworks modelling dissipative systems are considered: systems\u0000with external forces, contact systems and systems with impacts. The first two\u0000allow modelling a continuous dissipation of energy over time, while the latter\u0000also permits considering abrupt changes of energy in the instants of the\u0000impacts.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256833","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}
引用次数: 0
On four-dimensional Dehn twists and Milnor fibrations 关于四维 Dehn 扭转和 Milnor 纤度
arXiv - MATH - Symplectic Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.11961
Hokuto Konno, Jianfeng Lin, Anubhav Mukherjee, Juan Muñoz-Echániz
{"title":"On four-dimensional Dehn twists and Milnor fibrations","authors":"Hokuto Konno, Jianfeng Lin, Anubhav Mukherjee, Juan Muñoz-Echániz","doi":"arxiv-2409.11961","DOIUrl":"https://doi.org/arxiv-2409.11961","url":null,"abstract":"We study the monodromy diffeomorphism of Milnor fibrations of isolated\u0000complex surface singularities, by computing the family Seiberg--Witten\u0000invariant of Seifert-fibered Dehn twists using recent advances in monopole\u0000Floer homology. More precisely, we establish infinite order non-triviality\u0000results for boundary Dehn twists on indefinite symplectic fillings of links of\u0000minimally elliptic surface singularities. Using this, we exhibit a wide variety\u0000of new phenomena in dimension four: (1) smoothings of isolated complex surface\u0000singularities whose Milnor fibration has monodromy with infinite order as a\u0000diffeomorphism but with finite order as a homeomorphism, (2) robust Torelli\u0000symplectomorphisms that do not factor as products of Dehn--Seidel twists, (3)\u0000compactly supported exotic diffeomorphisms of exotic $mathbb{R}^4$'s and\u0000contractible manifolds.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256827","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}
引用次数: 0
Bohr-Sommerfeld profile surgeries and Disk Potentials 玻尔-索默菲尔德剖面手术和磁盘电位
arXiv - MATH - Symplectic Geometry Pub Date : 2024-09-17 DOI: arxiv-2409.11603
Soham Chanda
{"title":"Bohr-Sommerfeld profile surgeries and Disk Potentials","authors":"Soham Chanda","doi":"arxiv-2409.11603","DOIUrl":"https://doi.org/arxiv-2409.11603","url":null,"abstract":"We construct a new surgery type operation by switching between two exact\u0000fillings of Legendrians which we call a BSP surgery. In certain cases, this\u0000surgery can preserve monotonicity of Lagrangians. We prove a wall-crossing type\u0000formula for the change of the disk-potential under surgery with Bohr-Sommerfeld\u0000profiles. As an application, we show that Biran's circle-bundle lifts admit a\u0000Bohr-Sommerfeld type surgery. We use the wall-crossing theorem about\u0000disk-potentials to construct exotic monotone Lagrangian tori in $bP^n$.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256825","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}
引用次数: 0
Computable, obstructed Morse homology for clean intersections 干净交叉点的可计算、受阻莫尔斯同源性
arXiv - MATH - Symplectic Geometry Pub Date : 2024-09-17 DOI: arxiv-2409.11565
Erkao Bao, Ke Zhu
{"title":"Computable, obstructed Morse homology for clean intersections","authors":"Erkao Bao, Ke Zhu","doi":"arxiv-2409.11565","DOIUrl":"https://doi.org/arxiv-2409.11565","url":null,"abstract":"In this paper, we develop a method to compute the Morse homology of a\u0000manifold when descending manifolds and ascending manifolds intersect cleanly,\u0000but not necessarily transversely. While obstruction bundle gluing defined by Hutchings and Taubes is a\u0000computable tool to handle non-transverse intersections, it has only been\u0000developed for specific cases. In contrast, most virtual techniques apply to\u0000general cases but lack computational efficiency. To address this, we construct\u0000minimal semi-global Kuranishi structures for the moduli spaces of Morse\u0000trajectories, which generalize obstruction bundle gluing while maintaining its\u0000computability feature. Through this construction, we obtain iterated gluing\u0000equals simultaneous gluing.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256826","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}
引用次数: 0
Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory 重新审视莫尔斯-波特理论中的科恩-琼斯-西格尔结构
arXiv - MATH - Symplectic Geometry Pub Date : 2024-09-17 DOI: arxiv-2409.11278
Ciprian Mircea Bonciocat
{"title":"Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory","authors":"Ciprian Mircea Bonciocat","doi":"arxiv-2409.11278","DOIUrl":"https://doi.org/arxiv-2409.11278","url":null,"abstract":"In 1995, Cohen, Jones and Segal proposed a method of upgrading any given\u0000Floer homology to a stable homotopy-valued invariant. For a generic\u0000pseudo-gradient Morse-Bott flow on a closed smooth manifold $M$, we rigorously\u0000construct the conjectural stable normal framings, which are an essential\u0000ingredient in their construction, and give a rigorous proof that the resulting\u0000stable homotopy type recovers $Sigma^infty_+ M$. We further show that one can\u0000recover Thom spectra $M^E$ for all reduced $KO$-theory classes $E$ on $M$, by\u0000using slightly modified stable normal framings. Our paper also includes a\u0000construction of the smooth corner structure on compactified moduli spaces of\u0000broken flow lines with free endpoint, a formal construction of\u0000Piunikhin-Salamon-Schwarz type continuation maps, and a way to relax the stable\u0000normal framing condition to orientability in orthogonal spectra.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256828","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}
引用次数: 0
Kähler compactification of $mathbb{C}^n$ and Reeb dynamics $mathbb{C}^n$的凯勒紧凑化与里布动力学
arXiv - MATH - Symplectic Geometry Pub Date : 2024-09-16 DOI: arxiv-2409.10275
Chi Li, Zhengyi Zhou
{"title":"Kähler compactification of $mathbb{C}^n$ and Reeb dynamics","authors":"Chi Li, Zhengyi Zhou","doi":"arxiv-2409.10275","DOIUrl":"https://doi.org/arxiv-2409.10275","url":null,"abstract":"Let $X$ be a smooth complex manifold. Assume that $Ysubset X$ is a\u0000K\"{a}hler submanifold such that $Xsetminus Y$ is biholomorphic to\u0000$mathbb{C}^n$. We prove that $(X, Y)$ is biholomorphic to the standard example\u0000$(mathbb{P}^n, mathbb{P}^{n-1})$. We then study certain K\"{a}hler orbifold\u0000compactifications of $mathbb{C}^n$ and prove that on $mathbb{C}^3$ the flat\u0000metric is the only asymptotically conical Ricci-flat K\"{a}hler metric whose\u0000metric cone at infinity has a smooth link. As a key technical ingredient, a new\u0000formula for minimal discrepancy of isolated Fano cone singularities in terms of\u0000generalized Conley-Zehnder indices in symplectic geometry is derived.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256770","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}
引用次数: 0
Counting in Calabi--Yau categories, with applications to Hall algebras and knot polynomials 卡拉比--尤范畴中的计数，以及在霍尔代数和结多项式中的应用
arXiv - MATH - Symplectic Geometry Pub Date : 2024-09-16 DOI: arxiv-2409.10154
Mikhail Gorsky, Fabian Haiden
{"title":"Counting in Calabi--Yau categories, with applications to Hall algebras and knot polynomials","authors":"Mikhail Gorsky, Fabian Haiden","doi":"arxiv-2409.10154","DOIUrl":"https://doi.org/arxiv-2409.10154","url":null,"abstract":"We show that homotopy cardinality -- a priori ill-defined for many\u0000dg-categories, including all periodic ones -- has a reasonable definition for\u0000even-dimensional Calabi--Yau (evenCY) categories and their relative\u0000generalizations (under appropriate finiteness conditions). As a first application we solve the problem of defining an intrinsic Hall\u0000algebra for degreewise finite pre-triangulated dg-categories in the case of\u0000oddCY categories. We compare this definition with To\"en's derived Hall\u0000algebras (in case they are well-defined) and with other approaches based on\u0000extended Hall algebras and central reduction, including a construction of Hall\u0000algebras associated with Calabi--Yau triples of triangulated categories. For a\u0000category equivalent to the root category of a 1CY abelian category $mathcal\u0000A$, the algebra is shown to be isomorphic to the Drinfeld double of the twisted\u0000Ringel--Hall algebra of $mathcal A$, thus resolving in the Calabi--Yau case\u0000the long-standing problem of realizing the latter as a Hall algebra\u0000intrinsically defined for such a triangulated category. Our second application is the proof of a conjecture of\u0000Ng--Rutherford--Shende--Sivek, which provides an intrinsic formula for the\u0000ruling polynomial of a Legendrian knot $L$, and its generalization to\u0000Legendrian tangles, in terms of the augmentation category of $L$.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256769","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}
引用次数: 0
Contact non-squeezing in lens spaces 隐形眼镜在镜片空间内不会产生挤压
arXiv - MATH - Symplectic Geometry Pub Date : 2024-09-16 DOI: arxiv-2409.10334
Pierre-Alexandre Arlove
{"title":"Contact non-squeezing in lens spaces","authors":"Pierre-Alexandre Arlove","doi":"arxiv-2409.10334","DOIUrl":"https://doi.org/arxiv-2409.10334","url":null,"abstract":"We describe some contact non-squeezing phenomena in lens spaces by defining\u0000and computing a contact capacity. This contact capacity comes from the spectral\u0000selectors constructed by Allais, Sandon and the author by the means of\u0000generating functions and Givental's non-linear Maslov index. We also discuss a\u0000potential generalization of these non-squeezing phenomena for orderable closed\u0000prequantizations.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256767","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}
引用次数: 0
The focus-focus addition graph is immersed 焦点-焦点加法图沉浸在
arXiv - MATH - Symplectic Geometry Pub Date : 2024-09-16 DOI: arxiv-2409.10377
Mohammed Abouzaid, Nathaniel Bottman, Yunpeng Niu
{"title":"The focus-focus addition graph is immersed","authors":"Mohammed Abouzaid, Nathaniel Bottman, Yunpeng Niu","doi":"arxiv-2409.10377","DOIUrl":"https://doi.org/arxiv-2409.10377","url":null,"abstract":"For a symplectic 4-manifold $M$ equipped with a singular Lagrangian fibration\u0000with a section, the natural fiberwise addition given by the local Hamiltonian\u0000flow is well-defined on the regular points. We prove, in the case that the\u0000singularities are of focus-focus type, that the closure of the corresponding\u0000addition graph is the image of a Lagrangian immersion in $(M times M)^- times\u0000M$, and we study its geometry. Our main motivation for this result is the\u0000construction of a symmetric monoidal structure on the Fukaya category of such a\u0000manifold.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256768","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}
引用次数: 0
Geometry of bi-Lagrangian Grassmannian 双拉格朗日格拉斯曼几何学
arXiv - MATH - Symplectic Geometry Pub Date : 2024-09-15 DOI: arxiv-2409.09855
I. K. Kozlov
{"title":"Geometry of bi-Lagrangian Grassmannian","authors":"I. K. Kozlov","doi":"arxiv-2409.09855","DOIUrl":"https://doi.org/arxiv-2409.09855","url":null,"abstract":"This paper explores the structure of bi-Lagrangian Grassmanians for pencils\u0000of $2$-forms on real or complex vector spaces. We reduce the analysis to the\u0000pencils whose Jordan-Kronecker Canonical Form consists of Jordan blocks with\u0000the same eigenvalue. We demonstrate that this is equivalent to studying\u0000Lagrangian subspaces invariant under a nilpotent self-adjoint operator. We\u0000calculate the dimension of bi-Lagrangian Grassmanians and describe their open\u0000orbit under the automorphism group. We completely describe the automorphism\u0000orbits in the following three cases: for one Jordan block, for sums of equal\u0000Jordan blocks and for a sum of two distinct Jordan blocks.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256772","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}
引用次数: 0
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