{"title":"Morse homology and equivariance","authors":"Erkao Bao, Tyler Lawson","doi":"arxiv-2409.04694","DOIUrl":"https://doi.org/arxiv-2409.04694","url":null,"abstract":"In this paper, we develop methods for calculating equivariant homology from\u0000equivariant Morse functions on a closed manifold with the action of a finite\u0000group. We show how to alter $G$-equivariant Morse functions to a stable one,\u0000where the descending manifold from a critical point $p$ has the same stabilizer\u0000group as $p$, giving a better-behaved cell structure on $M$. For an\u0000equivariant, stable Morse function, we show that a generic equivariant metric\u0000satisfies the Morse--Smale condition. In the process, we give a proof that a generic equivariant function is Morse,\u0000and that equivariant, stable Morse functions form a dense subset in the\u0000$C^0$-topology within the space of all equivariant functions. Finally, we give an expository account of equivariant homology and cohomology\u0000theories, as well as their interaction with Morse theory. We show that any\u0000equivariant Morse function gives a filtration of $M$ that induces a Morse\u0000spectral sequence, computing the equivariant homology of $M$ from information\u0000about how the stabilizer group of a critical point acts on its tangent space.\u0000In the case of a stable Morse function, we show that this can be further\u0000reduced to a Morse chain complex.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202200","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":"On Kepler's geometric approach to consonances","authors":"Urs Frauenfelder","doi":"arxiv-2409.04119","DOIUrl":"https://doi.org/arxiv-2409.04119","url":null,"abstract":"Kepler's thinking is highly original and the inspiration for discovering his\u0000famous third law is based on his rather curious geometric approach in his\u0000Harmonices mundi for explaining consonances. In this article we try to use a\u0000modern mathematical approach based on Kepler's ideas how to characterize the\u0000seven consonances with the help of the numbers of edges of polygons\u0000constructible by ruler and compass.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202202","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":"On the exponential type conjecture","authors":"Zihong Chen","doi":"arxiv-2409.03922","DOIUrl":"https://doi.org/arxiv-2409.03922","url":null,"abstract":"We prove that the small quantum t-connection on a closed monotone symplectic\u0000manifold is of exponential type and has quasi-unipotent regularized monodromies\u0000at t=0. This answers a conjecture of Katzarkov-Kontsevich-Pantev and\u0000Galkin-Golyshev-Iritani for those classes of symplectic manifolds. The proof\u0000follows a reduction to positive characteristics argument, and the main tools of\u0000the proof are Katz's local monodromy theorem in differential equations and\u0000quantum Steenrod operations in equivariant Gromov-Witten theory with mod p\u0000coefficients.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202201","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 Moore-Tachikawa conjecture via shifted symplectic geometry","authors":"Peter Crooks, Maxence Mayrand","doi":"arxiv-2409.03532","DOIUrl":"https://doi.org/arxiv-2409.03532","url":null,"abstract":"We use shifted symplectic geometry to construct the Moore-Tachikawa\u0000topological quantum field theories (TQFTs) in a category of Hamiltonian\u0000schemes. Our new and overarching insight is an algebraic explanation for the\u0000existence of these TQFTs, i.e. that their structure comes naturally from three\u0000ingredients: Morita equivalence, as well as multiplication and identity\u0000bisections in abelian symplectic groupoids. Using this insight, we generalize\u0000the Moore-Tachikawa TQFTs in two directions. The first generalization concerns a 1-shifted version of the Weinstein\u0000symplectic category $mathbf{WS}_1$. Each abelianizable quasi-symplectic\u0000groupoid $mathcal{G}$ is shown to determine a canonical 2-dimensional TQFT\u0000$eta_{mathcal{G}}:mathbf{Cob}_2longrightarrowmathbf{WS}_1$. We recover the\u0000open Moore-Tachikawa TQFT and its multiplicative counterpart as special cases. Our second generalization is an affinization process for TQFTs. We first\u0000enlarge Moore and Tachikawa's category $mathbf{MT}$ of holomorphic symplectic\u0000varieties with Hamiltonian actions to $mathbf{AMT}$, a category of affine\u0000Poisson schemes with Hamiltonian actions of affine symplectic groupoids. We\u0000then show that if $mathcal{G} rightrightarrows X$ is an affine symplectic\u0000groupoid that is abelianizable when restricted to an open subset $U subseteq\u0000X$ statisfying Hartogs' theorem, then $mathcal{G}$ determines a TQFT\u0000$eta_{mathcal{G}} : mathbf{Cob}_2 longrightarrow mathbf{AMT}$. In more\u0000detail, we first devise an affinization process sending 1-shifted Lagrangian\u0000correspondences in $mathbf{WS}_1$ to Hamiltonian Poisson schemes in\u0000$mathbf{AMT}$. The TQFT is obtained by composing this affinization process\u0000with the TQFT $eta_{mathcal{G}|_U} : mathbf{Cob}_2 longrightarrow\u0000mathbf{WS}_1$ of the previous paragraph. Our results are also shown to yield\u0000new TQFTs outside of the Moore-Tachikawa setting.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202203","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":"Higher-Categorical Associahedra","authors":"Spencer Backman, Nathaniel Bottman, Daria Poliakova","doi":"arxiv-2409.03633","DOIUrl":"https://doi.org/arxiv-2409.03633","url":null,"abstract":"The second author introduced 2-associahedra as a tool for investigating\u0000functoriality properties of Fukaya categories, and he conjectured that they\u0000could be realized as face posets of convex polytopes. We introduce a family of\u0000posets called categorical $n$-associahedra, which naturally extend the second\u0000author's 2-associahedra and the classical associahedra. Categorical\u0000$n$-associahedra give a combinatorial model for the poset of strata of a\u0000compactified real moduli space of a tree arrangement of affine coordinate\u0000subspaces. We construct a family of complete polyhedral fans, called velocity\u0000fans, whose coordinates encode the relative velocities of pairs of colliding\u0000coordinate subspaces, and whose face posets are the categorical\u0000$n$-associahedra. In particular, this gives the first fan realization of\u00002-associahedra. In the case of the classical associahedron, the velocity fan\u0000specializes to the normal fan of Loday's realization of the associahedron. For proving that the velocity fan is a fan, we first construct a cone complex\u0000of metric $n$-bracketings and then exhibit a piecewise-linear isomorphism from\u0000this complex to the velocity fan. We demonstrate that the velocity fan, which\u0000is not simplicial, admits a canonical smooth flag triangulation on the same set\u0000of rays, and we describe a second, finer triangulation which provides a new\u0000extension of the braid arrangement. We describe piecewise-unimodular maps on\u0000the velocity fan such that the image of each cone is a union of cones in the\u0000braid arrangement, and we highlight a connection to the theory of building sets\u0000and nestohedra. We explore the local iterated fiber product structure of\u0000categorical $n$-associahedra and the extent to which this structure is realized\u0000by the velocity fan. For the class of concentrated $n$-associahedra we exhibit\u0000generalized permutahedra having velocity fans as their normal fans.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202094","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":"Legendrian Hopf links in L(p,1)","authors":"Rima Chatterjee, Hansjörg Geiges, Sinem Onaran","doi":"arxiv-2409.02582","DOIUrl":"https://doi.org/arxiv-2409.02582","url":null,"abstract":"We classify Legendrian realisations, up to coarse equivalence, of the Hopf\u0000link in the lens spaces L(p,1) with any contact structure.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202204","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":"Multiplicity free covering of a graded manifold","authors":"Elizaveta Vishnyakova","doi":"arxiv-2409.02211","DOIUrl":"https://doi.org/arxiv-2409.02211","url":null,"abstract":"We define and study a multiplicity free covering of a graded manifold. As an\u0000application of our research we give a new conceptual proof of the theorem about\u0000equivalence of categories of graded manifolds and symmetric $n$-fold vector\u0000bundles.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202205","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 survey of equivariant operations on quantum cohomology for symplectic manifolds","authors":"Nicholas Wilkins","doi":"arxiv-2409.01743","DOIUrl":"https://doi.org/arxiv-2409.01743","url":null,"abstract":"In this survey paper, we will collate various different ideas and thoughts\u0000regarding equivariant operations on quantum cohomology (and some in more\u0000general Floer theory) for a symplectic manifold. We will discuss a general\u0000notion of equivariant quantum operations associated to finite groups, in\u0000addition to their properties, examples, and calculations. We will provide a\u0000brief connection to Floer theoretic invariants. We then provide abridged\u0000descriptions (as per the author's understanding) of work by other authors in\u0000the field, along with their major results. Finally we discuss the first step to\u0000compact groups, specifically $S^1$-equivariant operations. Contained within\u0000this survey are also a sketch of the idea of mod-$p$ pseudocycles, and an\u0000in-depth appendix detailing the author's understanding of when one can define\u0000these equivariant operations in an additive way.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202206","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}
Naiara V. de Paulo, Seongchan Kim, Pedro A. S. Salomão, Alexsandro Schneider
{"title":"Transverse foliations for two-degree-of-freedom mechanical systems","authors":"Naiara V. de Paulo, Seongchan Kim, Pedro A. S. Salomão, Alexsandro Schneider","doi":"arxiv-2409.00445","DOIUrl":"https://doi.org/arxiv-2409.00445","url":null,"abstract":"We investigate the dynamics of a two-degree-of-freedom mechanical system for\u0000energies slightly above a critical value. The critical set of the potential\u0000function is assumed to contain a finite number of saddle points. As the energy\u0000increases across the critical value, a disk-like component of the Hill region\u0000gets connected to other components precisely at the saddles. Under certain\u0000convexity assumptions on the critical set, we show the existence of a weakly\u0000convex foliation in the region of the energy surface where the interesting\u0000dynamics takes place. The binding of the foliation is formed by the index-$2$\u0000Lyapunov orbits in the neck region about the rest points and a particular\u0000index-$3$ orbit. Among other dynamical implications, the transverse foliation\u0000forces the existence of periodic orbits, homoclinics, and heteroclinics to the\u0000Lyapunov orbits. We apply the results to the H'enon-Heiles potential for\u0000energies slightly above $1/6$. We also discuss the existence of transverse\u0000foliations for decoupled mechanical systems, including the frozen Hill's lunar\u0000problem with centrifugal force, the Stark problem, the Euler problem of two\u0000centers, and the potential of a chemical reaction.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202208","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":"Instability of Legendrian knottedness, and non-regular Lagrangian concordances of knots","authors":"Georgios Dimitroglou Rizell, Roman Golovko","doi":"arxiv-2409.00290","DOIUrl":"https://doi.org/arxiv-2409.00290","url":null,"abstract":"We show that the family of smoothly non-isotopic Legendrian pretzel knots\u0000from the work of Cornwell-Ng-Sivek that all have the same Legendrian invariants\u0000as the standard unknot have front-spuns that are Legendrian isotopic to the\u0000front-spun of the unknot. Besides that, we construct the first examples of\u0000Lagrangian concordances between Legendrian knots that are not regular, and\u0000hence not decomposable. Finally, we show that the relation of Lagrangian\u0000concordance between Legendrian knots is not anti-symmetric, and hence does not\u0000define a partial order. The latter two results are based upon a new type of\u0000flexibility for Lagrangian concordances with stabilised Legendrian ends.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202207","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}
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