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KP governs random growth off a 1-dimensional substrate KP控制着一维基底上的随机生长
IF 2.3 1区 数学
Forum of Mathematics Pi Pub Date : 2019-08-27 DOI: 10.1017/fmp.2021.9
J. Quastel, Daniel Remenik
{"title":"KP governs random growth off a 1-dimensional substrate","authors":"J. Quastel, Daniel Remenik","doi":"10.1017/fmp.2021.9","DOIUrl":"https://doi.org/10.1017/fmp.2021.9","url":null,"abstract":"Abstract The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev–Petviashvili (KP) equation. This is derived algebraically from a Fredholm determinant obtained in [MQR17] for the Kardar–Parisi–Zhang (KPZ) fixed point as the limit of the transition probabilities of TASEP, a special solvable model in the KPZ universality class. The Tracy–Widom distributions appear as special self-similar solutions of the KP and Korteweg–de Vries equations. In addition, it is noted that several known exact solutions of the KPZ equation also solve the KP equation.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43794223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 26
Modules over algebraic cobordism 代数协上的模
IF 2.3 1区 数学
Forum of Mathematics Pi Pub Date : 2019-08-06 DOI: 10.1017/fmp.2020.13
E. Elmanto, Marc Hoyois, Adeel A. Khan, V. Sosnilo, Maria Yakerson
{"title":"Modules over algebraic cobordism","authors":"E. Elmanto, Marc Hoyois, Adeel A. Khan, V. Sosnilo, Maria Yakerson","doi":"10.1017/fmp.2020.13","DOIUrl":"https://doi.org/10.1017/fmp.2020.13","url":null,"abstract":"Abstract We prove that the $infty $-category of $mathrm{MGL} $-modules over any scheme is equivalent to the $infty $-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $mathbf{P} ^1$-loop spaces, we deduce that very effective $mathrm{MGL} $-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $Omega ^infty _{mathbf{P} ^1}mathrm{MGL} $ is the $mathbf{A} ^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$, $Omega ^infty _{mathbf{P} ^1} Sigma ^n_{mathbf{P} ^1} mathrm{MGL} $ is the $mathbf{A} ^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2020.13","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49273910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 41
Simultaneously vanishing higher derived limits 同时消失的更高衍生极限
IF 2.3 1区 数学
Forum of Mathematics Pi Pub Date : 2019-07-26 DOI: 10.1017/fmp.2021.4
J. Bergfalk, C. Lambie-Hanson
{"title":"Simultaneously vanishing higher derived limits","authors":"J. Bergfalk, C. Lambie-Hanson","doi":"10.1017/fmp.2021.4","DOIUrl":"https://doi.org/10.1017/fmp.2021.4","url":null,"abstract":"Abstract In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system \u0000$textbf {A}$\u0000 with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that \u0000$lim ^ntextbf {A}$\u0000 (the nth derived limit of \u0000$textbf {A}$\u0000 ) vanishes for every \u0000$n>0$\u0000 . Since that time, the question of whether it is consistent with the \u0000$mathsf {ZFC}$\u0000 axioms that \u0000$lim ^n textbf {A}=0$\u0000 for every \u0000$n>0$\u0000 has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the \u0000$mathsf {ZFC}$\u0000 axioms that \u0000$lim ^n textbf {A}=0$\u0000 for all \u0000$n>0$\u0000 . We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to \u0000$lim ^ntextbf {A}=0$\u0000 will hold for each \u0000$n>0$\u0000 . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions \u0000$mathbb {N}^2to mathbb {Z}$\u0000 which are indexed in turn by n-tuples of functions \u0000$f:mathbb {N}to mathbb {N}$\u0000 . The triviality and coherence in question here generalise the classical and well-studied case of \u0000$n=1$\u0000 .","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2021.4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42314984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS 用于检查类别、字符槽和表示的内窥镜检查
IF 2.3 1区 数学
Forum of Mathematics Pi Pub Date : 2019-04-02 DOI: 10.1017/fmp.2020.9
G. Lusztig, Zhiwei Yun
{"title":"ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS","authors":"G. Lusztig, Zhiwei Yun","doi":"10.1017/fmp.2020.9","DOIUrl":"https://doi.org/10.1017/fmp.2020.9","url":null,"abstract":"For a reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$, after passing to asymptotic versions. The other is a similar relationship between representations of $G(mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(mathbb{F}_{q})$.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2020.9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47337090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 23
GLOBAL NEARLY-PLANE-SYMMETRIC SOLUTIONS TO THE MEMBRANE EQUATION 膜方程的全局近平面对称解
IF 2.3 1区 数学
Forum of Mathematics Pi Pub Date : 2019-03-08 DOI: 10.1017/fmp.2020.10
L. Abbrescia, W. Wong
{"title":"GLOBAL NEARLY-PLANE-SYMMETRIC SOLUTIONS TO THE MEMBRANE EQUATION","authors":"L. Abbrescia, W. Wong","doi":"10.1017/fmp.2020.10","DOIUrl":"https://doi.org/10.1017/fmp.2020.10","url":null,"abstract":"We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $dgeqslant 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher order peeling in our vector-field-based method. In particular, the higher order energies (in fact, all energies at order $2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments, where only the ‘top’ order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has ‘infinite energy’ and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in $C^{2}$ by carefully analysing the nonlinear interactions and exposing a certain ‘vestigial’ null structure in the equations.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2020.10","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43155667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
ON THE COHOMOLOGY OF TORELLI GROUPS 关于托雷利群的上同调
IF 2.3 1区 数学
Forum of Mathematics Pi Pub Date : 2019-01-07 DOI: 10.1017/fmp.2020.5
A. Kupers, O. Randal-Williams
{"title":"ON THE COHOMOLOGY OF TORELLI GROUPS","authors":"A. Kupers, O. Randal-Williams","doi":"10.1017/fmp.2020.5","DOIUrl":"https://doi.org/10.1017/fmp.2020.5","url":null,"abstract":"We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds $#^{g}S^{n}times S^{n}$ relative to a disc in a stable range, for $2ngeqslant 6$. Our calculation is also valid for $2n=2$ assuming that the rational cohomology groups of these Torelli groups are finite-dimensional in a stable range.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2020.5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49642683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
HIGHER GENUS GROMOV–WITTEN THEORY OF $mathsf{Hilb}^{n}(mathbb{C}^{2})$ AND $mathsf{CohFTs}$ ASSOCIATED TO LOCAL CURVES 局部曲线上$mathsf{Hilb}^{n}(mathbb{C}^{2})$和$mathsf{CohFTs}$的高格GROMOV-WITTEN理论
IF 2.3 1区 数学
Forum of Mathematics Pi Pub Date : 2019-01-01 DOI: 10.1017/fmp.2019.4
R. Pandharipande, Hsian-Hua Tseng
{"title":"HIGHER GENUS GROMOV–WITTEN THEORY OF $mathsf{Hilb}^{n}(mathbb{C}^{2})$ AND $mathsf{CohFTs}$ ASSOCIATED TO LOCAL CURVES","authors":"R. Pandharipande, Hsian-Hua Tseng","doi":"10.1017/fmp.2019.4","DOIUrl":"https://doi.org/10.1017/fmp.2019.4","url":null,"abstract":"We study the higher genus equivariant Gromov–Witten theory of the Hilbert scheme of $n$ points of $mathbb{C}^{2}$ . Since the equivariant quantum cohomology, computed by Okounkov and Pandharipande [Invent. Math. 179 (2010), 523–557], is semisimple, the higher genus theory is determined by an $mathsf{R}$ -matrix via the Givental–Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required $mathsf{R}$ -matrix by explicit data in degree $0$ . As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme $mathsf{Hilb}^{n}(mathbb{C}^{2})$ and the Gromov–Witten/Donaldson–Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the analytic continuation of the fundamental solution of the QDE of the Hilbert scheme of points determined by Okounkov and Pandharipande [Transform. Groups 15 (2010), 965–982]. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov–Witten theory of the symmetric product $mathsf{Sym}^{n}(mathbb{C}^{2})$ is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [Bryan and Graber, Algebraic Geometry–Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 23–42; Coates et al., Geom. Topol. 13 (2009), 2675–2744; Coates & Ruan, Ann. Inst. Fourier (Grenoble) 63 (2013), 431–478].","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2019.4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56602190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS 普遍可测同态的连续性
IF 2.3 1区 数学
Forum of Mathematics Pi Pub Date : 2019-01-01 DOI: 10.1017/fmp.2019.5
Christian Rosendal
{"title":"CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS","authors":"Christian Rosendal","doi":"10.1017/fmp.2019.5","DOIUrl":"https://doi.org/10.1017/fmp.2019.5","url":null,"abstract":"Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $text{ZF}+text{DC}$ , the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on ${0,1}^{mathbb{N}}$ has finite chromatic number.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2019.5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56602225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
IGUSA’S CONJECTURE FOR EXPONENTIAL SUMS: OPTIMAL ESTIMATES FOR NONRATIONAL SINGULARITIES 指数和的IGUSA猜想：非有理奇点的最优估计
IF 2.3 1区 数学
Forum of Mathematics Pi Pub Date : 2018-10-26 DOI: 10.1017/fmp.2019.3
R. Cluckers, M. Mustaţă, K. Nguyen
{"title":"IGUSA’S CONJECTURE FOR EXPONENTIAL SUMS: OPTIMAL ESTIMATES FOR NONRATIONAL SINGULARITIES","authors":"R. Cluckers, M. Mustaţă, K. Nguyen","doi":"10.1017/fmp.2019.3","DOIUrl":"https://doi.org/10.1017/fmp.2019.3","url":null,"abstract":"We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa’s conjecture on exponential sums, with the log canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for nonrational singularities by comparing the log canonical threshold with a local notion of the motivic oscillation index.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2018-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2019.3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48680586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
Frobenius splitting of Schubert varieties of semi-infinite flag manifolds 半无限旗流形的Schubert变种的Frobenius分裂
IF 2.3 1区 数学
Forum of Mathematics Pi Pub Date : 2018-10-16 DOI: 10.1017/fmp.2021.5
Syu Kato
{"title":"Frobenius splitting of Schubert varieties of semi-infinite flag manifolds","authors":"Syu Kato","doi":"10.1017/fmp.2021.5","DOIUrl":"https://doi.org/10.1017/fmp.2021.5","url":null,"abstract":"Abstract We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field ${mathbb K}$ of characteristic $neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind-)scheme structure, its projective coordinate ring has a $mathbb {Z}$-model and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. 371 no.2 (2018)]) when $mathsf {char}, {mathbb K} =0$ or $gg 0$, and the higher-cohomology vanishing of their nef line bundles in arbitrary characteristic $neq 2$. Some particular cases of these results play crucial roles in our proof [47] of a conjecture by Lam, Li, Mihalcea and Shimozono [60] that describes an isomorphism between affine and quantum K-groups of a flag manifold.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2018-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2021.5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44530314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 20
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