Forum of Mathematics Sigma最新文献

{"title":"t-Design Curves and Mobile Sampling on the Sphere","authors":"Martin Ehler, Karlheinz Gröchenig","doi":"10.1017/fms.2023.106","DOIUrl":"https://doi.org/10.1017/fms.2023.106","url":null,"abstract":"In analogy to classical spherical <jats:italic>t</jats:italic>-design points, we introduce the concept of <jats:italic>t</jats:italic>-design curves on the sphere. This means that the line integral along a <jats:italic>t</jats:italic>-design curve integrates polynomials of degree <jats:italic>t</jats:italic> exactly. For low degrees, we construct explicit examples. We also derive lower asymptotic bounds on the lengths of <jats:italic>t</jats:italic>-design curves. Our main results prove the existence of asymptotically optimal <jats:italic>t</jats:italic>-design curves in the Euclidean <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001068_inline1.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-sphere and the existence of <jats:italic>t</jats:italic>-design curves in the <jats:italic>d</jats:italic>-sphere.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability conditions for polarised varieties","authors":"Ruadhaí Dervan","doi":"10.1017/fms.2023.104","DOIUrl":"https://doi.org/10.1017/fms.2023.104","url":null,"abstract":"We introduce an analogue of Bridgeland’s stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of <jats:italic>Z</jats:italic>-stability is modelled on the notion of K-stability of polarised varieties. We then introduce an analytic counterpart to stability, through the notion of a <jats:italic>Z</jats:italic>-critical Kähler metric, modelled on the constant scalar curvature Kähler condition. Our main result shows that a polarised variety which is analytically K-semistable and asymptotically <jats:italic>Z</jats:italic>-stable admits <jats:italic>Z</jats:italic>-critical Kähler metrics in the large volume regime. We also prove a local converse and explain how these results can be viewed in terms of local wall crossing. A special case of our framework gives a manifold analogue of the deformed Hermitian Yang–Mills equation.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Length functions in Teichmüller and anti-de Sitter geometry","authors":"Filippo Mazzoli, Gabriele Viaggi","doi":"10.1017/fms.2023.100","DOIUrl":"https://doi.org/10.1017/fms.2023.100","url":null,"abstract":"We establish a link between the behavior of length functions on Teichmüller space and the geometry of certain anti-de Sitter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001007_inline1.png\" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-manifolds. As an application, we give new purely anti-de Sitter proofs of results of Teichmüller theory such as (strict) convexity of length functions along shear paths and geometric bounds on their second variation along earthquakes. Along the way, we provide shear-bend coordinates for GHMC anti-de Sitter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001007_inline2.png\" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-manifolds.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields","authors":"Gebhard Böckle, Ann-Kristin Juschka","doi":"10.1017/fms.2023.82","DOIUrl":"https://doi.org/10.1017/fms.2023.82","url":null,"abstract":"Let <jats:italic>K</jats:italic> be a finite extension of the p-adic field <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000828_inline1.png\" /> <jats:tex-math> ${mathbb {Q}}_p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of degree <jats:italic>d</jats:italic>, let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000828_inline2.png\" /> <jats:tex-math> ${{mathbb {F}},!{}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a finite field of characteristic <jats:italic>p</jats:italic> and let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000828_inline3.png\" /> <jats:tex-math> ${overline {{D}}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be an <jats:italic>n</jats:italic>-dimensional pseudocharacter in the sense of Chenevier of the absolute Galois group of <jats:italic>K</jats:italic> over the field <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000828_inline4.png\" /> <jats:tex-math> ${{mathbb {F}},!{}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. For the universal mod <jats:italic>p</jats:italic> pseudodeformation ring <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000828_inline5.png\" /> <jats:tex-math> ${overline {R}{{phantom {overline {overline m}}}}^{operatorname {univ}}_{{{overline {{D}}}}}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000828_inline6.png\" /> <jats:tex-math> ${overline {{D}}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we prove the following: The ring <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000828_inline7.png\" /> <jats:tex-math> $overline R_{{overline {{D}}}}^{mathrm {ps}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is equidimensional of dimension <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000828_inline8.png\" /> <jats:tex-math> $dn^2+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Its reduced quotient <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000828_inline9.png\" /> <jats:tex-math> ${overline {R}{{phantom {overline {overline m}}}}^{operator","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders","authors":"Philipp Lücke, Sandra Müller","doi":"10.1017/fms.2023.102","DOIUrl":"https://doi.org/10.1017/fms.2023.102","url":null,"abstract":"Abstract Given an uncountable cardinal \u0000$kappa $\u0000 , we consider the question of whether subsets of the power set of \u0000$kappa $\u0000 that are usually constructed with the help of the axiom of choice are definable by \u0000$Sigma _1$\u0000 -formulas that only use the cardinal \u0000$kappa $\u0000 and sets of hereditary cardinality less than \u0000$kappa $\u0000 as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical nondefinability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of \u0000$kappa $\u0000 of length at least \u0000$kappa ^+$\u0000 implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of \u0000$Sigma _1$\u0000 -definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal \u0000$omega _1$\u0000 .","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Double Schubert polynomials do have saturated Newton polytopes","authors":"Federico Castillo, Yairon Cid-Ruiz, Fatemeh Mohammadi, Jonathan Montaño","doi":"10.1017/fms.2023.101","DOIUrl":"https://doi.org/10.1017/fms.2023.101","url":null,"abstract":"We prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen–Macaulay prime ideal in a nonstandard multigrading, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algebraic relations among Goss’s zeta values on elliptic curves","authors":"Nathan Green, Tuan Ngo Dac","doi":"10.1017/fms.2023.94","DOIUrl":"https://doi.org/10.1017/fms.2023.94","url":null,"abstract":"In 2007 Chang and Yu determined all the algebraic relations among Goss’s zeta values for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000944_inline1.png\" /> <jats:tex-math> $A=mathbb F_q[theta ]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, also known as the Carlitz zeta values. Goss raised the problem of determining all algebraic relations among Goss’s zeta values at positive integers for a general base ring <jats:italic>A</jats:italic>, but very little is known. In this paper, we develop a general method, and we determine all algebraic relations among Goss’s zeta values for the base ring <jats:italic>A</jats:italic> which is the coordinate ring of an elliptic curve defined over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000944_inline2.png\" /> <jats:tex-math> $mathbb F_q$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. To our knowledge, this is the first work tackling Goss’s problem when the base ring has class number strictly greater than 1.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Collapse and diffusion in harmonic activation and transport","authors":"Jacob Calvert, Shirshendu Ganguly, Alan Hammond","doi":"10.1017/fms.2023.81","DOIUrl":"https://doi.org/10.1017/fms.2023.81","url":null,"abstract":"For an <jats:italic>n</jats:italic>-element subset <jats:italic>U</jats:italic> of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000816_inline1.png\" /> <jats:tex-math> $mathbb {Z}^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, select <jats:italic>x</jats:italic> from <jats:italic>U</jats:italic> according to harmonic measure from infinity, remove <jats:italic>x</jats:italic> from <jats:italic>U</jats:italic> and start a random walk from <jats:italic>x</jats:italic>. If the walk leaves from <jats:italic>y</jats:italic> when it first enters the rest of <jats:italic>U</jats:italic>, add <jats:italic>y</jats:italic> to it. Iterating this procedure constitutes the process we call <jats:italic>harmonic activation and transport</jats:italic> (HAT). HAT exhibits a phenomenon we refer to as <jats:italic>collapse</jats:italic>: Informally, the diameter shrinks to its logarithm over a number of steps which is comparable to this logarithm. Collapse implies the existence of the stationary distribution of HAT, where configurations are viewed up to translation, and the exponential tightness of diameter at stationarity. Additionally, collapse produces a renewal structure with which we establish that the center of mass process, properly rescaled, converges in distribution to two-dimensional Brownian motion. To characterize the phenomenon of collapse, we address fundamental questions about the extremal behavior of harmonic measure and escape probabilities. Among <jats:italic>n</jats:italic>-element subsets of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000816_inline2.png\" /> <jats:tex-math> $mathbb {Z}^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, what is the least positive value of harmonic measure? What is the probability of escape from the set to a distance of, say, <jats:italic>d</jats:italic>? Concerning the former, examples abound for which the harmonic measure is exponentially small in <jats:italic>n</jats:italic>. We prove that it can be no smaller than exponential in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000816_inline3.png\" /> <jats:tex-math> $n log n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Regarding the latter, the escape probability is at most the reciprocal of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000816_inline4.png\" /> <jats:tex-math> $log d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, up to a constant factor. We prove it is always at least this much, up to an <jats:italic>n</jats:italic>-dependent factor.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Calabi–Yau structures on (quasi-)bisymplectic algebras","authors":"Tristan Bozec, Damien Calaque, Sarah Scherotzke","doi":"10.1017/fms.2023.88","DOIUrl":"https://doi.org/10.1017/fms.2023.88","url":null,"abstract":"We show that relative Calabi–Yau structures on noncommutative moment maps give rise to (quasi-)bisymplectic structures, as introduced by Crawley-Boevey–Etingof–Ginzburg (in the additive case) and Van den Bergh (in the multiplicative case). We prove along the way that the fusion process (a) corresponds to the composition of Calabi–Yau cospans with ‘pair-of-pants’ ones and (b) preserves the duality between non-degenerate double quasi-Poisson structures and quasi-bisymplectic structures. As an application, we obtain that Van den Bergh’s Poisson structures on the moduli spaces of representations of deformed multiplicative preprojective algebras coincide with the ones induced by the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000889_inline1.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-Calabi–Yau structures on (dg-versions of) these algebras.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability for hyperbolic groups acting on boundary spheres","authors":"Kathryn Mann, Jason Fox Manning","doi":"10.1017/fms.2023.78","DOIUrl":"https://doi.org/10.1017/fms.2023.78","url":null,"abstract":"A hyperbolic group <jats:italic>G</jats:italic> acts by homeomorphisms on its Gromov boundary. We show that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423000786_inline1.png\" /> <jats:tex-math> $partial G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a topological <jats:italic>n</jats:italic>–sphere, the action is <jats:italic>topologically stable</jats:italic> in the dynamical sense: any nearby action is semi-conjugate to the standard boundary action.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}