Forum of Mathematics Pi最新文献

{"title":"Local parameters of supercuspidal representations","authors":"Wee Teck Gan, Michael Harris, Will Sawin, Raphaël Beuzart-Plessis","doi":"10.1017/fmp.2024.10","DOIUrl":"https://doi.org/10.1017/fmp.2024.10","url":null,"abstract":"For a connected reductive group <jats:italic>G</jats:italic> over a nonarchimedean local field <jats:italic>F</jats:italic> of positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000106_inline1.png\"/> <jats:tex-math> ${mathcal {L}}^{ss}(pi )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> to each irreducible representation <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000106_inline2.png\"/> <jats:tex-math> $pi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Our first result shows that the Genestier-Lafforgue parameter of a tempered <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000106_inline3.png\"/> <jats:tex-math> $pi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> can be uniquely refined to a tempered L-parameter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000106_inline4.png\"/> <jats:tex-math> ${mathcal {L}}(pi )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000106_inline5.png\"/> <jats:tex-math> ${mathcal {L}}^{ss}(pi )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for unramified <jats:italic>G</jats:italic> and supercuspidal <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000106_inline6.png\"/> <jats:tex-math> $pi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> constructed by induction from an open compact (modulo center) subgroup. If <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000106_inline7.png\"/> <jats:tex-math> ${mathcal {L}}^{ss}(pi )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is pure in an appropriate sense, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000106_inline8.png\"/> <jats:tex-math> ${mathcal {L}}^{ss}(pi )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is ramified (unless <jats:italic>G</jats:italic> is a torus). If the inducing subgroup is sufficiently small in a precise sense, we ","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196463","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}

{"title":"Strichartz estimates and global well-posedness of the cubic NLS on","authors":"Sebastian Herr, Beomjong Kwak","doi":"10.1017/fmp.2024.11","DOIUrl":"https://doi.org/10.1017/fmp.2024.11","url":null,"abstract":"The optimal <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000118_inline2.png\"/> <jats:tex-math> $L^4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000118_inline3.png\"/> <jats:tex-math> $mathbb {T}^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach yields a stronger <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000118_inline4.png\"/> <jats:tex-math> $L^4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> bound on a logarithmic time scale, which implies global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger equation in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000118_inline5.png\"/> <jats:tex-math> $H^s(mathbb {T}^2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000118_inline6.png\"/> <jats:tex-math> $s&gt;0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and data that are small in the critical norm.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196480","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}

{"title":"The definable content of homological invariants II: Čech cohomology and homotopy classification","authors":"Jeffrey Bergfalk, Martino Lupini, Aristotelis Panagiotopoulos","doi":"10.1017/fmp.2024.7","DOIUrl":"https://doi.org/10.1017/fmp.2024.7","url":null,"abstract":"This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000076_inline1.png\"/&gt; on the category of locally compact separable metric spaces each factor into (i) what we term their &lt;jats:italic&gt;definable version&lt;/jats:italic&gt;, a functor &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000076_inline2.png\"/&gt; taking values in the category &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000076_inline3.png\"/&gt; &lt;jats:tex-math&gt; $mathsf {GPC}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; of &lt;jats:italic&gt;groups with a Polish cover&lt;/jats:italic&gt; (a category first introduced in this work’s predecessor), followed by (ii) a forgetful functor from &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000076_inline4.png\"/&gt; &lt;jats:tex-math&gt; $mathsf {GPC}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of &lt;jats:italic&gt;d&lt;/jats:italic&gt;-spheres or &lt;jats:italic&gt;d&lt;/jats:italic&gt;-tori for any &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000076_inline5.png\"/&gt; &lt;jats:tex-math&gt; $dgeq 1$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000076_inline6.png\"/&gt; to show that a seminal problem in the development of algebraic topology – namely, Borsuk and Eilenberg’s 1936 problem of classifying, up to homotopy, the maps from a solenoid complement &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000076_inline7.png\"/&gt; &lt;jats:tex-math&gt; $S^3backslash Sigma $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; to the &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000076_inline8.png\"/&gt; &lt;jats:tex-math&gt; $2$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-sphere – is essentially hyperfini","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142196461","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}

{"title":"Theta functions, fourth moments of eigenforms and the sup-norm problem II","authors":"Ilya Khayutin, Paul D. Nelson, Raphael S. Steiner","doi":"10.1017/fmp.2024.9","DOIUrl":"https://doi.org/10.1017/fmp.2024.9","url":null,"abstract":"Let <jats:italic>f</jats:italic> be an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862400009X_inline1.png\"/> <jats:tex-math> $L^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-normalized holomorphic newform of weight <jats:italic>k</jats:italic> on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862400009X_inline2.png\"/> <jats:tex-math> $Gamma _0(N) backslash mathbb {H}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:italic>N</jats:italic> squarefree or, more generally, on any hyperbolic surface <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862400009X_inline3.png\"/> <jats:tex-math> $Gamma backslash mathbb {H}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> attached to an Eichler order of squarefree level in an indefinite quaternion algebra over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862400009X_inline4.png\"/> <jats:tex-math> $mathbb {Q}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Denote by <jats:italic>V</jats:italic> the hyperbolic volume of said surface. We prove the sup-norm estimate <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862400009X_eqnu1.png\"/> <jats:tex-math> $$begin{align*}| Im(cdot)^{frac{k}{2}} f |_{infty} ll_{varepsilon} (k V)^{frac{1}{4}+varepsilon} end{align*}$$ </jats:tex-math> </jats:alternatives> </jats:disp-formula> with absolute implied constant. For a cuspidal Maaß newform <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862400009X_inline5.png\"/> <jats:tex-math> $varphi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of eigenvalue <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862400009X_inline6.png\"/> <jats:tex-math> $lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> on such a surface, we prove that <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862400009X_eqnu2.png\"/> <jats:tex-math> $$begin{align*}|varphi |_{infty} ll_{lambda,varepsilon} V^{frac{1}{4}+varepsilon}. end{align*}$$ </jats:tex-math> </jats:alternatives> </jats:disp-formula> We establish analogous estimates in the setting of definite quaternion algebras.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141196326","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}

{"title":"ON HOMOMORPHISM GRAPHS","authors":"Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, Zoltán Vidnyánszky","doi":"10.1017/fmp.2024.8","DOIUrl":"https://doi.org/10.1017/fmp.2024.8","url":null,"abstract":"We introduce new types of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks [Mar16]. The motivation for the construction comes from the adaptation of this method to the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000088_inline1.png\"/> <jats:tex-math> $mathsf {LOCAL}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> model of distributed computing [BCG<jats:sup>+</jats:sup>21]. Our approach unifies the previous results in the area, as well as produces new ones. In particular, strengthening the main result of [TV21], we show that for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000088_inline2.png\"/> <jats:tex-math> $Delta&gt;2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, it is impossible to give a simple characterization of acyclic <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000088_inline3.png\"/> <jats:tex-math> $Delta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-regular Borel graphs with Borel chromatic number at most <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000088_inline4.png\"/> <jats:tex-math> $Delta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>: such graphs form a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508624000088_inline5.png\"/> <jats:tex-math> $mathbf {Sigma }^1_2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete set. This implies a strong failure of Brooks-like theorems in the Borel context.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140928280","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}

{"title":"Logarithmic Donaldson–Thomas theory","authors":"Davesh Maulik, Dhruv Ranganathan","doi":"10.1017/fmp.2024.1","DOIUrl":"https://doi.org/10.1017/fmp.2024.1","url":null,"abstract":"<p>Let X be a smooth and projective threefold with a simple normal crossings divisor D. We construct the Donaldson–Thomas theory of the pair <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417062350610-0851:S2050508624000015:S2050508624000015_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(X|D)$</span></span></img></span></span> enumerating ideal sheaves on X relative to D. These moduli spaces are compactified by studying subschemes in expansions of the target geometry, and the moduli space carries a virtual fundamental class leading to numerical invariants with expected properties. We formulate punctual evaluation, rationality and wall-crossing conjectures, in parallel with the standard theory. Our formalism specializes to the Li–Wu theory of relative ideal sheaves when the divisor is smooth and is parallel to recent work on logarithmic Gromov–Witten theory with expansions.</p>","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140615687","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}

{"title":"The Chromatic Fourier Transform","authors":"Tobias Barthel, Shachar Carmeli, Tomer M. Schlank, Lior Yanovski","doi":"10.1017/fmp.2024.5","DOIUrl":"https://doi.org/10.1017/fmp.2024.5","url":null,"abstract":"<p>We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405115914080-0098:S2050508624000052:S2050508624000052_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$n=0$</span></span></img></span></span>, as well as a certain duality for the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405115914080-0098:S2050508624000052:S2050508624000052_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$E_n$</span></span></img></span></span>-(co)homology of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405115914080-0098:S2050508624000052:S2050508624000052_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$pi $</span></span></img></span></span>-finite spectra, established by Hopkins and Lurie, at heights <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405115914080-0098:S2050508624000052:S2050508624000052_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$nge 1$</span></span></img></span></span>. We use this theory to generalize said duality in three different directions. First, we extend it from <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405115914080-0098:S2050508624000052:S2050508624000052_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {Z}$</span></span></img></span></span>-module spectra to all (suitably finite) spectra and use it to compute the discrepancy spectrum of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405115914080-0098:S2050508624000052:S2050508624000052_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$E_n$</span></span></img></span></span>. Second, we lift it to the telescopic setting by replacing <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405115914080-0098:S2050508624000052:S2050508624000052_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$E_n$</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405115914080-0098:S2050508624000052:S2050508624000052_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$T(n)$</span></span></img></span></span>-local higher cyclotomic extensions, from which we deduce various results on affineness, Eilenberg–Moore formulas and Galois extensions in the telescopic setting.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570687","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}

{"title":"On the ergodic theory of the real Rel foliation","authors":"Jon Chaika, Barak Weiss","doi":"10.1017/fmp.2024.6","DOIUrl":"https://doi.org/10.1017/fmp.2024.6","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329055111250-0254:S2050508624000064:S2050508624000064_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${{mathcal {H}}}$</span></span></img></span></span> be a stratum of translation surfaces with at least two singularities, let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329055111250-0254:S2050508624000064:S2050508624000064_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$m_{{{mathcal {H}}}}$</span></span></img></span></span> denote the Masur-Veech measure on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329055111250-0254:S2050508624000064:S2050508624000064_inline3.png\"><span data-mathjax-type=\"texmath\"><span>${{mathcal {H}}}$</span></span></img></span></span>, and let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329055111250-0254:S2050508624000064:S2050508624000064_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$Z_0$</span></span></img></span></span> be a flow on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329055111250-0254:S2050508624000064:S2050508624000064_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$({{mathcal {H}}}, m_{{{mathcal {H}}}})$</span></span></img></span></span> obtained by integrating a Rel vector field. We prove that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329055111250-0254:S2050508624000064:S2050508624000064_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$Z_0$</span></span></img></span></span> is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329055111250-0254:S2050508624000064:S2050508624000064_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$({mathcal L}, m_{{mathcal L}})$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329055111250-0254:S2050508624000064:S2050508624000064_inline8.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal L} subset {{mathcal {H}}}$</span></span></img></span></span> is an orbit-closure for the action of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329055111250-0254:S2050508624000064:S2050508624000064_i","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570871","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}

{"title":"Sharp well-posedness for the cubic NLS and mKdV in","authors":"Benjamin Harrop-Griffiths, Rowan Killip, Monica Vişan","doi":"10.1017/fmp.2024.4","DOIUrl":"https://doi.org/10.1017/fmp.2024.4","url":null,"abstract":"<p>We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328063032726-0395:S2050508624000040:S2050508624000040_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$H^s({{mathbb {R}}})$</span></span></img></span></span> for any regularity <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328063032726-0395:S2050508624000040:S2050508624000040_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$s&gt;-frac 12$</span></span></img></span></span>. Well-posedness has long been known for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328063032726-0395:S2050508624000040:S2050508624000040_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$sgeq 0$</span></span></img></span></span>, see [55], but not previously for any <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328063032726-0395:S2050508624000040:S2050508624000040_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$s&lt;0$</span></span></img></span></span>. The scaling-critical value <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328063032726-0395:S2050508624000040:S2050508624000040_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$s=-frac 12$</span></span></img></span></span> is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48].</p><p>We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg–de Vries equations in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328063032726-0395:S2050508624000040:S2050508624000040_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$H^s({{mathbb {R}}})$</span></span></img></span></span> for any <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328063032726-0395:S2050508624000040:S2050508624000040_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$s&gt;-frac 12$</span></span></img></span></span>. The best regularity achieved previously was <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328063032726-0395:S2050508624000040:S2050508624000040_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$sgeq tfrac 14$</span></span></img></span></span> (see [15, 24, 33, 39]).</p><p>To overcome the failure of uniform continuity of the data-to-solution map, we employ the metho","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570782","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}

{"title":"The least singular value of a random symmetric matrix","authors":"Marcelo Campos, Matthew Jenssen, Marcus Michelen, Julian Sahasrabudhe","doi":"10.1017/fmp.2023.29","DOIUrl":"https://doi.org/10.1017/fmp.2023.29","url":null,"abstract":"Let <jats:italic>A</jats:italic> be an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862300029X_inline1.png\" /> <jats:tex-math> $n times n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> symmetric matrix with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862300029X_inline2.png\" /> <jats:tex-math> $(A_{i,j})_{ileqslant j}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> independent and identically distributed according to a subgaussian distribution. We show that <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862300029X_eqnu1.png\" /> <jats:tex-math> $$ begin{align*}mathbb{P}(sigma_{min}(A) leqslant varepsilon n^{-1/2} ) leqslant C varepsilon + e^{-cn},end{align*} $$ </jats:tex-math> </jats:alternatives> </jats:disp-formula> where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862300029X_inline3.png\" /> <jats:tex-math> $sigma _{min }(A)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the least singular value of <jats:italic>A</jats:italic> and the constants <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862300029X_inline4.png\" /> <jats:tex-math> $C,c&gt;0 $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> depend only on the distribution of the entries of <jats:italic>A</jats:italic>. This result confirms the folklore conjecture on the lower tail of the least singular value of such matrices and is best possible up to the dependence of the constants on the distribution of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862300029X_inline5.png\" /> <jats:tex-math> $A_{i,j}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Along the way, we prove that the probability that <jats:italic>A</jats:italic> has a repeated eigenvalue is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050862300029X_inline6.png\" /> <jats:tex-math> $e^{-Omega (n)}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, thus confirming a conjecture of Nguyen, Tao and Vu [<jats:italic>Probab. Theory Relat. Fields</jats:italic> 167 (2017), 777–816].","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139561148","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}