Abstract We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archimedean local field of characteristic zero.
摘要证明了例外群$G_2(F)$的局部朗兰兹猜想，其中F是特征为零的非阿基米德局部域。
{"title":"The Local Langlands Conjecture for","authors":"Wee Teck Gan, Gordan Savin","doi":"10.1017/fmp.2023.27","DOIUrl":"https://doi.org/10.1017/fmp.2023.27","url":null,"abstract":"Abstract We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archimedean local field of characteristic zero.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135053491","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 study higher uniformity properties of the Möbius function $mu $ , the von Mangoldt function $Lambda $ , and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{theta +varepsilon } leq H leq X^{1-varepsilon }$ for a fixed constant $0 leq theta < 1$ and any $varepsilon>0$ . More precisely, letting $Lambda ^sharp $ and $d_k^sharp $ be suitable approximants of $Lambda $ and $d_k$ and $mu ^sharp = 0$ , we show for instance that, for any nilsequence $F(g(n)Gamma )$ , we have $$begin{align*}sum_{X < n leq X+H} (f(n)-f^sharp(n)) F(g(n) Gamma) ll H log^{-A} X end{align*}$$ when $theta = 5/8$ and $f in {Lambda , mu , d_k}$ or $theta = 1/3$ and $f = d_2$ . As a consequence, we show that the short interval Gowers norms $|f-f^sharp |_{U^s(X,X+H]}$ are also asymptotically small for any fixed s for these choices of $f,theta $ . As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in $L^2$ . Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type $II$ sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.
摘要研究了Möbius函数$mu $、von Mangoldt函数$Lambda $和短间隔$(X,X+H]$上的除数函数$d_k$的高均匀性，其中$X^{theta +varepsilon } leq H leq X^{1-varepsilon }$对于固定常数$0 leq theta < 1$和任意$varepsilon>0$。更准确地说，让$Lambda ^sharp $和$d_k^sharp $成为$Lambda $、$d_k$和$mu ^sharp = 0$的合适近似值，例如，我们表明，对于任何nilsequence $F(g(n)Gamma )$，当$theta = 5/8$和$f in {Lambda , mu , d_k}$或$theta = 1/3$和$f = d_2$时，我们有$$begin{align*}sum_{X < n leq X+H} (f(n)-f^sharp(n)) F(g(n) Gamma) ll H log^{-A} X end{align*}$$。结果表明，对于任意固定的s，对于$f,theta $的这些选择，短区间Gowers范数$|f-f^sharp |_{U^s(X,X+H]}$也是渐近小的。作为应用，我们证明了短区间内素数线性方程解个数的渐近公式，并证明了短区间内沿素数的多个遍历平均收敛于$L^2$。我们的创新包括使用多参数nilsequence等分布定理来控制$II$型和，以及将双曲线的邻域分解为等差数列来控制$I_2$型和。
{"title":"Higher uniformity of arithmetic functions in short intervals I. All intervals","authors":"Kaisa Matomäki, Xuancheng Shao, Terence Tao, Joni Teräväinen","doi":"10.1017/fmp.2023.28","DOIUrl":"https://doi.org/10.1017/fmp.2023.28","url":null,"abstract":"Abstract We study higher uniformity properties of the Möbius function $mu $ , the von Mangoldt function $Lambda $ , and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{theta +varepsilon } leq H leq X^{1-varepsilon }$ for a fixed constant $0 leq theta < 1$ and any $varepsilon>0$ . More precisely, letting $Lambda ^sharp $ and $d_k^sharp $ be suitable approximants of $Lambda $ and $d_k$ and $mu ^sharp = 0$ , we show for instance that, for any nilsequence $F(g(n)Gamma )$ , we have $$begin{align*}sum_{X < n leq X+H} (f(n)-f^sharp(n)) F(g(n) Gamma) ll H log^{-A} X end{align*}$$ when $theta = 5/8$ and $f in {Lambda , mu , d_k}$ or $theta = 1/3$ and $f = d_2$ . As a consequence, we show that the short interval Gowers norms $|f-f^sharp |_{U^s(X,X+H]}$ are also asymptotically small for any fixed s for these choices of $f,theta $ . As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in $L^2$ . Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type $II$ sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135053473","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 this paper, we study multiple zeta values (abbreviated as MZV’s) over function fields in positive characteristic. Our main result is to prove Thakur’s basis conjecture, which plays the analogue of Hoffman’s basis conjecture for real MZV’s. As a consequence, we derive Todd’s dimension conjecture, which is the analogue of Zagier’s dimension conjecture for classical real MZV’s.
{"title":"On Thakur’s basis conjecture for multiple zeta values in positive characteristic","authors":"Chieh-Yu Chang, Yen-Tsung Chen, Yoshinori Mishiba","doi":"10.1017/fmp.2023.26","DOIUrl":"https://doi.org/10.1017/fmp.2023.26","url":null,"abstract":"Abstract In this paper, we study multiple zeta values (abbreviated as MZV’s) over function fields in positive characteristic. Our main result is to prove Thakur’s basis conjecture, which plays the analogue of Hoffman’s basis conjecture for real MZV’s. As a consequence, we derive Todd’s dimension conjecture, which is the analogue of Zagier’s dimension conjecture for classical real MZV’s.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136301929","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 Iterating the skew RSK correspondence discovered by Sagan and Stanley in the late 1980s, we define deterministic dynamics on the space of pairs of skew Young tableaux $(P,Q)$ . We find that these skew RSK dynamics display conservation laws which, in the picture of Viennot’s shadow line construction, identify generalizations of Greene invariants. The introduction of a novel realization of $0$ -th Kashiwara operators reveals that the skew RSK dynamics possess symmetries induced by an affine bicrystal structure, which, combined with connectedness properties of Demazure crystals, leads to the linearization of the time evolution. Studying asymptotic evolution of the dynamics started from a pair of skew tableaux $(P,Q)$ , we discover a new bijection $Upsilon : (P,Q) mapsto (V,W; kappa , nu )$ . Here, $(V,W)$ is a pair of vertically strict tableaux, that is, column strict fillings of Young diagrams with no condition on rows, with the shape prescribed by the Greene invariant, $kappa $ is an array of nonnegative weights and $nu $ is a partition. An application of this construction is the first bijective proof of Cauchy and Littlewood identities involving q -Whittaker polynomials. New identities relating sums of q -Whittaker and Schur polynomials are also presented.
{"title":"Skew RSK dynamics: Greene invariants, affine crystals and applications to <i>q</i>-Whittaker polynomials","authors":"Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto","doi":"10.1017/fmp.2023.23","DOIUrl":"https://doi.org/10.1017/fmp.2023.23","url":null,"abstract":"Abstract Iterating the skew RSK correspondence discovered by Sagan and Stanley in the late 1980s, we define deterministic dynamics on the space of pairs of skew Young tableaux $(P,Q)$ . We find that these skew RSK dynamics display conservation laws which, in the picture of Viennot’s shadow line construction, identify generalizations of Greene invariants. The introduction of a novel realization of $0$ -th Kashiwara operators reveals that the skew RSK dynamics possess symmetries induced by an affine bicrystal structure, which, combined with connectedness properties of Demazure crystals, leads to the linearization of the time evolution. Studying asymptotic evolution of the dynamics started from a pair of skew tableaux $(P,Q)$ , we discover a new bijection $Upsilon : (P,Q) mapsto (V,W; kappa , nu )$ . Here, $(V,W)$ is a pair of vertically strict tableaux, that is, column strict fillings of Young diagrams with no condition on rows, with the shape prescribed by the Greene invariant, $kappa $ is an array of nonnegative weights and $nu $ is a partition. An application of this construction is the first bijective proof of Cauchy and Littlewood identities involving q -Whittaker polynomials. New identities relating sums of q -Whittaker and Schur polynomials are also presented.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009011","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}
Jean Bourgain, Mariusz Mirek, Elias M. Stein, James Wright
Abstract We prove convergence in norm and pointwise almost everywhere on $L^p$ , $pin (1,infty )$ , for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.
{"title":"On a multi-parameter variant of the Bellow–Furstenberg problem","authors":"Jean Bourgain, Mariusz Mirek, Elias M. Stein, James Wright","doi":"10.1017/fmp.2023.21","DOIUrl":"https://doi.org/10.1017/fmp.2023.21","url":null,"abstract":"Abstract We prove convergence in norm and pointwise almost everywhere on $L^p$ , $pin (1,infty )$ , for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135549839","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 use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety.
{"title":"Stellahedral geometry of matroids","authors":"Christopher Eur, June Huh, Matt Larson","doi":"10.1017/fmp.2023.24","DOIUrl":"https://doi.org/10.1017/fmp.2023.24","url":null,"abstract":"Abstract We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136202134","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 this paper, we determine the number of general points through which a Brill–Noether curve of fixed degree and genus in any projective space can be passed.
摘要本文确定了任意射影空间中固定次属Brill-Noether曲线可通过的一般点的个数。
{"title":"Interpolation for Brill–Noether curves","authors":"Eric Larson, Isabel Vogt","doi":"10.1017/fmp.2023.22","DOIUrl":"https://doi.org/10.1017/fmp.2023.22","url":null,"abstract":"Abstract In this paper, we determine the number of general points through which a Brill–Noether curve of fixed degree and genus in any projective space can be passed.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136258869","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":"Tits Alternative for -dimensional complexes","authors":"Damian Osajda, Piotr Przytycki","doi":"10.1017/fmp.2022.20","DOIUrl":"https://doi.org/10.1017/fmp.2022.20","url":null,"abstract":"We prove the Tits Alternative for groups acting on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000208_inline3.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-dimensional <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050508622000208_inline4.png\" /> <jats:tex-math> $mathrm {CAT}(0)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> complexes with a bound on the order of the cell stabilisers.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519689","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}
M. Moreira, A. Oblomkov, A. Okounkov, R. Pandharipande
Abstract Using new explicit formulas for the stationary Gromov–Witten/Pandharipande–Thomas ( $mathrm {GW}/{mathrm {PT}}$ ) descendent correspondence for nonsingular projective toric threefolds, we show that the correspondence intertwines the Virasoro constraints in Gromov–Witten theory for stable maps with the Virasoro constraints for stable pairs proposed in [18]. Since the Virasoro constraints in Gromov–Witten theory are known to hold in the toric case, we establish the stationary Virasoro constraints for the theory of stable pairs on toric threefolds. As a consequence, new Virasoro constraints for tautological integrals over Hilbert schemes of points on surfaces are also obtained.
{"title":"Virasoro constraints for stable pairs on toric threefolds","authors":"M. Moreira, A. Oblomkov, A. Okounkov, R. Pandharipande","doi":"10.1017/fmp.2022.4","DOIUrl":"https://doi.org/10.1017/fmp.2022.4","url":null,"abstract":"Abstract Using new explicit formulas for the stationary Gromov–Witten/Pandharipande–Thomas ( \u0000$mathrm {GW}/{mathrm {PT}}$\u0000 ) descendent correspondence for nonsingular projective toric threefolds, we show that the correspondence intertwines the Virasoro constraints in Gromov–Witten theory for stable maps with the Virasoro constraints for stable pairs proposed in [18]. Since the Virasoro constraints in Gromov–Witten theory are known to hold in the toric case, we establish the stationary Virasoro constraints for the theory of stable pairs on toric threefolds. As a consequence, new Virasoro constraints for tautological integrals over Hilbert schemes of points on surfaces are also obtained.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2022-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42577154","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 show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $mathbf {R}^4$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$ -close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in $mathbf {R}^4$ . The new proof is more closely related to techniques from the study of strictly positive scalar curvature.
{"title":"Stable anisotropic minimal hypersurfaces in \u0000$mathbf {R}^{4}$","authors":"Otis Chodosh, C. Li","doi":"10.1017/fmp.2023.1","DOIUrl":"https://doi.org/10.1017/fmp.2023.1","url":null,"abstract":"Abstract We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in \u0000$mathbf {R}^4$\u0000 has intrinsic cubic volume growth, provided the parametric elliptic integral is \u0000$C^2$\u0000 -close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in \u0000$mathbf {R}^4$\u0000 . The new proof is more closely related to techniques from the study of strictly positive scalar curvature.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2022-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41930090","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}
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