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.
{"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}
E. Elmanto, Marc Hoyois, Adeel A. Khan, V. Sosnilo, Maria Yakerson
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 $-n$.
{"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}
Abstract In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $textbf {A}$ 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 $lim ^ntextbf {A}$ (the nth derived limit of $textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $mathsf {ZFC}$ axioms that $lim ^n textbf {A}=0$ for every $n>0$ 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 $mathsf {ZFC}$ axioms that $lim ^n textbf {A}=0$ for all $n>0$ . 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 $lim ^ntextbf {A}=0$ will hold for each $n>0$ . 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 $mathbb {N}^2to mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:mathbb {N}to mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .
{"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}
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 $H(mathbb{F}_{q})$.
{"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}
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.
{"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}
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.
{"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}
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].
研究$mathbb{C}^{2}$ n$点的Hilbert格式的高格等变Gromov-Witten理论。自从等变量子上同调，由Okounkov和Pandharipande[发明]计算。数学，179(2010)，523-557]，是半简单的，高属理论是由一个$mathsf{R}$ -矩阵通过上同调场论(CohFTs)的Givental-Teleman分类确定的。我们唯一指定所需的$mathsf{R}$ -矩阵的显式数据在度$0$。因此，我们将Hilbert方案$mathsf{Hilb}^{n}(mathbb{C}^{2})$的等变量子上同调的基本等价三角形和局部曲线三重理论的Gromov-Witten / Donaldson-Thomas对应提升到所有高属的等价三角形。证明使用了由Okounkov和Pandharipande [Transform]确定的Hilbert格式的QDE的基本解的解析延拓。第15组(2010)，965-982]。高格三角形的GW/DT边涉及稳定曲线模空间中通过改变3重局部曲线定义的新cohft。也证明了对称积$mathsf{Sym}^{n}(mathbb{C}^{2})$的等变轨道Gromov-Witten理论在所有属中都等价于三角形的理论。结果建立了一个完整的蠕变分解猜想[Bryan and Graber, algeaic Geometry-Seattle 2005, Part 1, symposium Proceedings in Pure Mathematics, 80] (American Mathematical Society, Providence, RI, 2009)， 23-42;科茨等人，Geom。植物学报，2009 (3)，2675-2744;科茨和阮，安。傅立叶研究所(格勒诺布尔)63(2013)，431-478]。
{"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}
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.
回答一个长期存在的问题，起源于克里斯滕森对哈尔零集的开创性工作[数学]。科学，28 (1971)，124-128;以色列。数学。13 (1972)，255-260;拓扑学和Borel结构。描述拓扑和集合论及其在泛函分析和测度理论中的应用，北荷数学研究，10 (noas de matatica, No. 51)。北荷兰出版公司，阿姆斯特丹-伦敦;美国Elsevier出版公司，Inc.， New York, 1974)， iii+133 pp]，我们证明了波兰群体之间普遍可测量的同态是自动连续的。利用我们对群同态连续性的一般分析，这个结果被用来校准波兰群之间不连续同态存在的强度。特别地，证明了在模$text{ZF}+text{DC}$时，波兰群间的不连续同态的存在性意味着${0,1}^{mathbb{N}}$上的Hamming图具有有限的色数。
{"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}
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.
{"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}
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.
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：