We study the degree of an L-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $X_k$ is the connected sum of k copies of $mathbb CP^2$ for $k ge 4$ , then we prove that the maximum degree of an L-Lipschitz self-map of $X_k$ is between $C_1 L^4 (log L)^{-4}$ and $C_2 L^4 (log L)^{-1/2}$ . More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected n-manifolds, the maximal degree is $sim L^n$ . For formal but nonscalable simply connected n-manifolds, the maximal degree grows roughly like $L^n (log L)^{-theta (1)}$ . And for nonformal simply connected n-manifolds, the maximal degree is bounded by
{"title":"Degrees of maps and multiscale geometry","authors":"Aleksandr Berdnikov, Larry Guth, Fedor Manin","doi":"10.1017/fmp.2023.33","DOIUrl":"https://doi.org/10.1017/fmp.2023.33","url":null,"abstract":"We study the degree of an <jats:italic>L</jats:italic>-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000331_inline1.png\" /> <jats:tex-math> $X_k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the connected sum of <jats:italic>k</jats:italic> copies of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000331_inline2.png\" /> <jats:tex-math> $mathbb CP^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000331_inline3.png\" /> <jats:tex-math> $k ge 4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then we prove that the maximum degree of an <jats:italic>L</jats:italic>-Lipschitz self-map of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000331_inline4.png\" /> <jats:tex-math> $X_k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is between <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000331_inline5.png\" /> <jats:tex-math> $C_1 L^4 (log L)^{-4}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000331_inline6.png\" /> <jats:tex-math> $C_2 L^4 (log L)^{-1/2}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected <jats:italic>n</jats:italic>-manifolds, the maximal degree is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000331_inline7.png\" /> <jats:tex-math> $sim L^n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. For formal but nonscalable simply connected <jats:italic>n</jats:italic>-manifolds, the maximal degree grows roughly like <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000331_inline8.png\" /> <jats:tex-math> $L^n (log L)^{-theta (1)}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. And for nonformal simply connected <jats:italic>n</jats:italic>-manifolds, the maximal degree is bounded by <jats:inline-formula> <jats:alternatives> <jats:inline","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139498374","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}
Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $mathbf {Z}/pmathbf {Z}$-extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For $mathbf {Z}/pmathbf {Z}$-extensions of local function fields, we construct a base change homomorphism for the mod p Bernstein center of any reductive group. We then use this to prove existence of local base change for mod p irreducible representation along $mathbf {Z}/pmathbf {Z}$-extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann-Venkatesh.
The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from modular representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod p spherical Hecke algebras, in a joint appendix with Gus Lonergan.
Lafforgue 和 Genestier-Lafforgue 为函数域上的任意还原群构建了全局和（半简化的）局部朗兰兹对应关系。我们为这些对应关系建立了关于循环基变化的函数性的各种性质：对于全局函数域的 $mathbf {Z}/pmathbf {Z}$ 扩展，我们证明了任意还原群上 mod p 自形形式的基底变化的存在性。对于局部函数域的 $mathbf {Z}/pmathbf {Z}$ 扩展，我们为任意还原群的模 p 伯恩斯坦中心构造了一个基变同态。然后，我们用它证明了沿着 $mathbf {Z}/pmathbf {Z}$ 扩展的模 p 不可还原表示的局部基变的存在，以及塔特同调实现了基变下降，验证了特鲁曼-文卡特什一个猜想的函数场版本。证明基于shtukas模空间的等变本地化论证，同时还借鉴了模块表示理论的新工具，包括奇偶性剪和史密斯-特鲁曼理论。特别是，在与古斯-侬纳根（Gus Lonergan）的联合附录中，我们利用这些工具为模 p 球形赫克代数建立了基变同态的分类。
{"title":"Smith theory and cyclic base change functoriality","authors":"Tony Feng","doi":"10.1017/fmp.2023.32","DOIUrl":"https://doi.org/10.1017/fmp.2023.32","url":null,"abstract":"<p>Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112084941107-0818:S205050862300032X:S205050862300032X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {Z}/pmathbf {Z}$</span></span></img></span></span>-extensions of global function fields, we prove the existence of base change for mod <span>p</span> automorphic forms on arbitrary reductive groups. For <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112084941107-0818:S205050862300032X:S205050862300032X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {Z}/pmathbf {Z}$</span></span></img></span></span>-extensions of local function fields, we construct a base change homomorphism for the mod <span>p</span> Bernstein center of any reductive group. We then use this to prove existence of local base change for mod <span>p</span> irreducible representation along <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112084941107-0818:S205050862300032X:S205050862300032X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {Z}/pmathbf {Z}$</span></span></img></span></span>-extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann-Venkatesh.</p><p>The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from modular representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod <span>p</span> spherical Hecke algebras, in a joint appendix with Gus Lonergan.</p>","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139469632","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 show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle $E to B$ if and only if it holds for the base B. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of B, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.
我们证明了格罗莫夫-维滕理论中的维拉索罗猜想（Virasoro conjecture）在环束 $E to B$ 的总空间中成立，前提是且仅当它在基 B 中成立时：(i) 我们建立了一个局部化公式，用环状纤维的零属不变式和 B 的全属不变式来表达 E 的格罗莫夫-维滕不变式，相对于纤维环状作用等变，以及 (ii) 我们利用布朗关于环状束的镜像定理来传递这个公式中的非变极限。
{"title":"Virasoro Constraints for Toric Bundles","authors":"Tom Coates, Alexander Givental, Hsian-Hua Tseng","doi":"10.1017/fmp.2024.2","DOIUrl":"https://doi.org/10.1017/fmp.2024.2","url":null,"abstract":"<p>We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203073427923-0125:S2050508624000027:S2050508624000027_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$E to B$</span></span></img></span></span> if and only if it holds for the base <span>B</span>. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of <span>E</span>, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of <span>B</span>, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.</p>","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139689992","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}
Martin Ravn Christiansen, Christian Hainzl, Phan Thành Nam
We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three dimensions in the mean-field scaling regime. We consider a system of N fermions on a torus, interacting via a two-body repulsive potential proportional to $N^{-frac {1}{3}}$ . In the limit $Nrightarrow infty $ , we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.
我们提出了一种通用方法，用以证明均相费米气体在三维均场缩放机制中的随机相近似。我们考虑了一个环上由 N 个费米子组成的系统，该系统通过与 $N^{-frac {1}{3}$ 成比例的双体斥力势相互作用。在极限 $Nrightarrow infty $ 中，我们推导出了系统的相关能和玻色基本激元的精确前导阶，这与物理学文献中随机相近似的预测是一致的。
{"title":"The Random Phase Approximation for Interacting Fermi Gases in the Mean-Field Regime","authors":"Martin Ravn Christiansen, Christian Hainzl, Phan Thành Nam","doi":"10.1017/fmp.2023.31","DOIUrl":"https://doi.org/10.1017/fmp.2023.31","url":null,"abstract":"We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three dimensions in the mean-field scaling regime. We consider a system of <jats:italic>N</jats:italic> fermions on a torus, interacting via a two-body repulsive potential proportional to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000318_inline1.png\" /> <jats:tex-math> $N^{-frac {1}{3}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In the limit <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000318_inline2.png\" /> <jats:tex-math> $Nrightarrow infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139025552","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}
This article is devoted to a general class of one-dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data are both small and localized. However, except for the completely integrable case, no such results have been known for small but not necessarily localized initial data. In this article, we introduce a new, nonperturbative method to prove global well-posedness and scattering for $L^2$ initial data which are small and nonlocalized. Our main structural assumption is that our nonlinearity is defocusing. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of Interaction Morawetz estimates, developed almost 20 years ago by the I-team. In terms of scattering, we prove that our global solutions satisfy both global $L^6$ Strichartz estimates and bilinear $L^2$ bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS.1 There, by scaling, our result also admits a large data counterpart.
{"title":"Global solutions for 1D cubic defocusing dispersive equations: Part I","authors":"Mihaela Ifrim, Daniel Tataru","doi":"10.1017/fmp.2023.30","DOIUrl":"https://doi.org/10.1017/fmp.2023.30","url":null,"abstract":"This article is devoted to a general class of one-dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data are both <jats:italic>small</jats:italic> and <jats:italic>localized</jats:italic>. However, except for the completely integrable case, no such results have been known for small but not necessarily localized initial data. In this article, we introduce a new, nonperturbative method to prove global well-posedness and scattering for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000306_inline1.png\" /> <jats:tex-math> $L^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> initial data which are <jats:italic>small</jats:italic> and <jats:italic>nonlocalized</jats:italic>. Our main structural assumption is that our nonlinearity is <jats:italic>defocusing</jats:italic>. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of Interaction Morawetz estimates, developed almost 20 years ago by the I-team. In terms of scattering, we prove that our global solutions satisfy both global <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000306_inline2.png\" /> <jats:tex-math> $L^6$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> Strichartz estimates and bilinear <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508623000306_inline3.png\" /> <jats:tex-math> $L^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS.<jats:sup>1</jats:sup> There, by scaling, our result also admits a large data counterpart.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519683","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 show that framed deformation rings of mod p representations of the absolute Galois group of a p-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application, we prove density results of loci with prescribed p-adic Hodge theoretic properties.
{"title":"On local Galois deformation rings","authors":"Gebhard Böckle, Ashwin Iyengar, Vytautas Paškūnas","doi":"10.1017/fmp.2023.25","DOIUrl":"https://doi.org/10.1017/fmp.2023.25","url":null,"abstract":"We show that framed deformation rings of mod <jats:italic>p</jats:italic> representations of the absolute Galois group of a <jats:italic>p</jats:italic>-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application, we prove density results of loci with prescribed <jats:italic>p</jats:italic>-adic Hodge theoretic properties.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519682","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}
Matthew Kwan, A. Sah, Lisa Sauermann, Mehtaab Sawhney
Abstract An n-vertex graph is called C-Ramsey if it has no clique or independent set of size $Clog _2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables. The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes.
{"title":"Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture","authors":"Matthew Kwan, A. Sah, Lisa Sauermann, Mehtaab Sawhney","doi":"10.1017/fmp.2023.17","DOIUrl":"https://doi.org/10.1017/fmp.2023.17","url":null,"abstract":"Abstract An n-vertex graph is called C-Ramsey if it has no clique or independent set of size \u0000$Clog _2 n$\u0000 (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables. The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47815580","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 Let $M_{langle mathbf {u},mathbf {v},mathbf {w}rangle }in mathbb C^{mathbf {u}mathbf {v}}{mathord { otimes } } mathbb C^{mathbf {v}mathbf {w}}{mathord { otimes } } mathbb C^{mathbf {w}mathbf {u}}$ denote the matrix multiplication tensor (and write $M_{langle mathbf {n} rangle }=M_{langle mathbf {n},mathbf {n},mathbf {n}rangle }$ ), and let $operatorname {det}_3in (mathbb C^9)^{{mathord { otimes } } 3}$ denote the determinant polynomial considered as a tensor. For a tensor T, let $underline {mathbf {R}}(T)$ denote its border rank. We (i) give the first hand-checkable algebraic proof that $underline {mathbf {R}}(M_{langle 2rangle })=7$ , (ii) prove $underline {mathbf {R}}(M_{langle 223rangle })=10$ and $underline {mathbf {R}}(M_{langle 233rangle })=14$ , where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was $M_{langle 2rangle }$ , (iii) prove $underline {mathbf {R}}( M_{langle 3rangle })geq 17$ , (iv) prove $underline {mathbf {R}}(operatorname {det}_3)=17$ , improving the previous lower bound of $12$ , (v) prove $underline {mathbf {R}}(M_{langle 2mathbf {n}mathbf {n}rangle })geq mathbf {n}^2+1.32mathbf {n}$ for all $mathbf {n}geq 25$ , where previously only $underline {mathbf {R}}(M_{langle 2mathbf {n}mathbf {n}rangle })geq mathbf {n}^2+1$ was known, as well as lower bounds for $4leq mathbf {n}leq 25$ , and (vi) prove $underline {mathbf {R}}(M_{langle 3mathbf {n}mathbf {n}rangle })geq mathbf {n}^2+1.6mathbf {n}$ for all $mathbf {n} ge 18$ , where previously only $underline {mathbf {R}}(M_{langle 3mathbf {n}mathbf {n}rangle })geq mathbf {n}^2+2$ was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors. The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, called border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensor T and an integer r, in a finite number of steps, either outputs that there is no border rank r decomposition for T or produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable when T has a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory.
{"title":"New lower bounds for matrix multiplication and \u0000$operatorname {det}_3$","authors":"Austin Conner, Alicia Harper, J. Landsberg","doi":"10.1017/fmp.2023.14","DOIUrl":"https://doi.org/10.1017/fmp.2023.14","url":null,"abstract":"Abstract Let \u0000$M_{langle mathbf {u},mathbf {v},mathbf {w}rangle }in mathbb C^{mathbf {u}mathbf {v}}{mathord { otimes } } mathbb C^{mathbf {v}mathbf {w}}{mathord { otimes } } mathbb C^{mathbf {w}mathbf {u}}$\u0000 denote the matrix multiplication tensor (and write \u0000$M_{langle mathbf {n} rangle }=M_{langle mathbf {n},mathbf {n},mathbf {n}rangle }$\u0000 ), and let \u0000$operatorname {det}_3in (mathbb C^9)^{{mathord { otimes } } 3}$\u0000 denote the determinant polynomial considered as a tensor. For a tensor T, let \u0000$underline {mathbf {R}}(T)$\u0000 denote its border rank. We (i) give the first hand-checkable algebraic proof that \u0000$underline {mathbf {R}}(M_{langle 2rangle })=7$\u0000 , (ii) prove \u0000$underline {mathbf {R}}(M_{langle 223rangle })=10$\u0000 and \u0000$underline {mathbf {R}}(M_{langle 233rangle })=14$\u0000 , where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was \u0000$M_{langle 2rangle }$\u0000 , (iii) prove \u0000$underline {mathbf {R}}( M_{langle 3rangle })geq 17$\u0000 , (iv) prove \u0000$underline {mathbf {R}}(operatorname {det}_3)=17$\u0000 , improving the previous lower bound of \u0000$12$\u0000 , (v) prove \u0000$underline {mathbf {R}}(M_{langle 2mathbf {n}mathbf {n}rangle })geq mathbf {n}^2+1.32mathbf {n}$\u0000 for all \u0000$mathbf {n}geq 25$\u0000 , where previously only \u0000$underline {mathbf {R}}(M_{langle 2mathbf {n}mathbf {n}rangle })geq mathbf {n}^2+1$\u0000 was known, as well as lower bounds for \u0000$4leq mathbf {n}leq 25$\u0000 , and (vi) prove \u0000$underline {mathbf {R}}(M_{langle 3mathbf {n}mathbf {n}rangle })geq mathbf {n}^2+1.6mathbf {n}$\u0000 for all \u0000$mathbf {n} ge 18$\u0000 , where previously only \u0000$underline {mathbf {R}}(M_{langle 3mathbf {n}mathbf {n}rangle })geq mathbf {n}^2+2$\u0000 was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors. The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, called border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensor T and an integer r, in a finite number of steps, either outputs that there is no border rank r decomposition for T or produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable when T has a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47449455","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":"A Proof of the Extended Delta Conjecture – Corrigendum","authors":"","doi":"10.1017/fmp.2023.8","DOIUrl":"https://doi.org/10.1017/fmp.2023.8","url":null,"abstract":"","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48230736","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":"A Shuffle Theorem for Paths Under Any Line – Corrigendum","authors":"","doi":"10.1017/fmp.2023.9","DOIUrl":"https://doi.org/10.1017/fmp.2023.9","url":null,"abstract":"","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44990212","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}
$mathbf {Z}/pmathbf {Z}$-extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For 
$mathbf {Z}/pmathbf {Z}$-extensions of local function fields, we construct a base change homomorphism for the mod p Bernstein center of any reductive group. We then use this to prove existence of local base change for mod p irreducible representation along 
$mathbf {Z}/pmathbf {Z}$-extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann-Venkatesh.
$E to B$ if and only if it holds for the base B. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of B, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.
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