Abstract We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(log N)^{3/4}(log log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c big ( frac {log k}{log log k} big )^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$.
摘要证明了$[N]$的红-蓝着色性，它没有蓝色的3项等差数列，也没有长度为$e^{C(log N)^{3/4}(log log N)^{1/4}}$的红色等差数列。因此，双色范德华登数$w(3,k)$以$k^{b(k)}$为界，其中$b(k) = c big ( frac {log k}{log log k} big )^{1/3}$。在此之前，有数据支持的推测是$w(3,k) = O(k^2)$。
{"title":"New lower bounds for van der Waerden numbers","authors":"B. Green","doi":"10.1017/fmp.2022.12","DOIUrl":"https://doi.org/10.1017/fmp.2022.12","url":null,"abstract":"Abstract We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(log N)^{3/4}(log log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c big ( frac {log k}{log log k} big )^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42165298","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 consider symmetry-protected topological phases with on-site finite group G symmetry $beta $ for two-dimensional quantum spin systems. We show that they have $H^{3}(G,{mathbb T})$-valued invariant.
{"title":"An $H^{3}(G,{mathbb T})$-valued index of symmetry-protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems","authors":"Y. Ogata","doi":"10.1017/fmp.2021.17","DOIUrl":"https://doi.org/10.1017/fmp.2021.17","url":null,"abstract":"Abstract We consider symmetry-protected topological phases with on-site finite group G symmetry $beta $ for two-dimensional quantum spin systems. We show that they have $H^{3}(G,{mathbb T})$-valued invariant.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47272439","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 This paper studies the critical and near-critical regimes of the planar random-cluster model on $mathbb Z^2$ with cluster-weight $qin [1,4]$ using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents $beta $ , $gamma $ , $delta $ , $eta $ , $nu $ , $zeta $ as well as $alpha $ (when $alpha ge 0$ ). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of the influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalisation of Kesten’s classical scaling relation for Bernoulli percolation involving the ‘mixing rate’ critical exponent $iota $ replacing the four-arm event exponent $xi _4$ .
摘要本文利用新的耦合技术研究了簇权重为$qin[1,4]$的$mathbb Z^2上平面随机簇模型的临界和近临界状态。更准确地说，我们导出了临界指数$beta$、$gamma$、$delta$、$eta$、$snu$、$ zeta$以及$alpha$（当$alpha ge为0时）之间的比例关系。作为一个关键输入，我们使用对混合速率方面的边缘影响概念的新解释，展示了近临界状态下交叉概率的稳定性。作为副产品，我们导出了伯努利渗流的Kesten经典标度关系的推广，涉及“混合速率”临界指数$iota$代替四臂事件指数$neneneba xi _4$。
{"title":"Planar random-cluster model: scaling relations","authors":"H. Duminil-Copin, I. Manolescu","doi":"10.1017/fmp.2022.16","DOIUrl":"https://doi.org/10.1017/fmp.2022.16","url":null,"abstract":"Abstract This paper studies the critical and near-critical regimes of the planar random-cluster model on \u0000$mathbb Z^2$\u0000 with cluster-weight \u0000$qin [1,4]$\u0000 using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents \u0000$beta $\u0000 , \u0000$gamma $\u0000 , \u0000$delta $\u0000 , \u0000$eta $\u0000 , \u0000$nu $\u0000 , \u0000$zeta $\u0000 as well as \u0000$alpha $\u0000 (when \u0000$alpha ge 0$\u0000 ). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of the influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalisation of Kesten’s classical scaling relation for Bernoulli percolation involving the ‘mixing rate’ critical exponent \u0000$iota $\u0000 replacing the four-arm event exponent \u0000$xi _4$\u0000 .","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47039848","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 define, for each quasisyntomic ring R (in the sense of Bhatt et al., Publ. Math. IHES 129 (2019), 199–310), a category $mathrm {DM}^{mathrm {adm}}(R)$ of admissible prismatic Dieudonné crystals over R and a functor from p-divisible groups over R to $mathrm {DM}^{mathrm {adm}}(R)$ . We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.
{"title":"Prismatic Dieudonné Theory","authors":"Johannes Anschütz, Arthur-César Le Bras","doi":"10.1017/fmp.2022.22","DOIUrl":"https://doi.org/10.1017/fmp.2022.22","url":null,"abstract":"Abstract We define, for each quasisyntomic ring R (in the sense of Bhatt et al., Publ. Math. IHES 129 (2019), 199–310), a category \u0000$mathrm {DM}^{mathrm {adm}}(R)$\u0000 of admissible prismatic Dieudonné crystals over R and a functor from p-divisible groups over R to \u0000$mathrm {DM}^{mathrm {adm}}(R)$\u0000 . We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42793226","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 theta correspondence to study the equivalence between Godement–Jacquet and Jacquet–Langlands L-functions for ${mathrm {GL}}(2)$ . We show that the resulting comparison is in fact an expression of an exotic symmetric monoidal structure on the category of ${mathrm {GL}}(2)$ -modules. Moreover, this enables us to construct an abelian category of abstractly automorphic representations, whose irreducible objects are the usual automorphic representations. We speculate that this category is a natural setting for the study of automorphic phenomena for ${mathrm {GL}}(2)$ , and demonstrate its basic properties. This paper is a part of the author’s thesis [4].
{"title":"Exotic Monoidal Structures and Abstractly Automorphic Representations for \u0000$mathrm {GL}(2)$","authors":"Gal Dor","doi":"10.1017/fmp.2023.18","DOIUrl":"https://doi.org/10.1017/fmp.2023.18","url":null,"abstract":"Abstract We use the theta correspondence to study the equivalence between Godement–Jacquet and Jacquet–Langlands L-functions for \u0000${mathrm {GL}}(2)$\u0000 . We show that the resulting comparison is in fact an expression of an exotic symmetric monoidal structure on the category of \u0000${mathrm {GL}}(2)$\u0000 -modules. Moreover, this enables us to construct an abelian category of abstractly automorphic representations, whose irreducible objects are the usual automorphic representations. We speculate that this category is a natural setting for the study of automorphic phenomena for \u0000${mathrm {GL}}(2)$\u0000 , and demonstrate its basic properties. This paper is a part of the author’s thesis [4].","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44336521","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 This work studies the average complexity of solving structured polynomial systems that are characterised by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that computes, with high probability, an approximate zero of a polynomial system given only as black-box evaluation program. Secondly, we introduce a universal model of random polynomial systems with prescribed evaluation complexity L. Combining both, we show that we can compute an approximate zero of a random structured polynomial system with n equations of degree at most ${D}$ in n variables with only $operatorname {poly}(n, {D}) L$ operations with high probability. This exceeds the expectations implicit in Smale’s 17th problem.
{"title":"Rigid continuation paths II. structured polynomial systems","authors":"Peter Bürgisser, F. Cucker, Pierre Lairez","doi":"10.1017/fmp.2023.7","DOIUrl":"https://doi.org/10.1017/fmp.2023.7","url":null,"abstract":"Abstract This work studies the average complexity of solving structured polynomial systems that are characterised by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that computes, with high probability, an approximate zero of a polynomial system given only as black-box evaluation program. Secondly, we introduce a universal model of random polynomial systems with prescribed evaluation complexity L. Combining both, we show that we can compute an approximate zero of a random structured polynomial system with n equations of degree at most \u0000${D}$\u0000 in n variables with only \u0000$operatorname {poly}(n, {D}) L$\u0000 operations with high probability. This exceeds the expectations implicit in Smale’s 17th problem.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45839622","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 the triply graded Khovanov–Rozansky homology of knots and links over a field of positive odd characteristic p descends to an invariant in the homotopy category finite-dimensional p-complexes. A p-extended differential on the triply graded homology discovered by Cautis is compatible with the p-DG structure. As a consequence, we get a categorification of the Jones polynomial evaluated at a $2p$th root of unity.
{"title":"On some p-differential graded link homologies","authors":"You Qi, Joshua Sussan","doi":"10.1017/fmp.2022.19","DOIUrl":"https://doi.org/10.1017/fmp.2022.19","url":null,"abstract":"Abstract We show that the triply graded Khovanov–Rozansky homology of knots and links over a field of positive odd characteristic p descends to an invariant in the homotopy category finite-dimensional p-complexes. A p-extended differential on the triply graded homology discovered by Cautis is compatible with the p-DG structure. As a consequence, we get a categorification of the Jones polynomial evaluated at a $2p$th root of unity.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41706758","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}
K. De Commer, S. Neshveyev, L. Tuset, M. Yamashita
Abstract We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingof cyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno–Drinfeld type theorem on type $mathrm {B}$ braid group representations defined by the monodromy of KZ-equations and by the Balagović–Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions.
{"title":"Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik–Zamolodchikov equations and Letzter–Kolb coideals","authors":"K. De Commer, S. Neshveyev, L. Tuset, M. Yamashita","doi":"10.1017/fmp.2023.11","DOIUrl":"https://doi.org/10.1017/fmp.2023.11","url":null,"abstract":"Abstract We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingof cyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno–Drinfeld type theorem on type \u0000$mathrm {B}$\u0000 braid group representations defined by the monodromy of KZ-equations and by the Balagović–Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43664950","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 The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.
{"title":"Grothendieck–Serre in the quasi-split unramified case","authors":"Kęstutis Česnavičius","doi":"10.1017/fmp.2022.5","DOIUrl":"https://doi.org/10.1017/fmp.2022.5","url":null,"abstract":"Abstract The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47832809","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 the validity of two special cases of the four-dimensional minimal model program (MMP) in characteristic $p>5$ : for contractions to ${mathbb {Q}}$ -factorial fourfolds and in families over curves (‘semistable MMP’). We also provide their mixed characteristic analogues. As a corollary, we show that liftability of positive characteristic threefolds is stable under the MMP and that liftability of three-dimensional Calabi–Yau varieties is a birational invariant. Our results are partially contingent upon the existence of log resolutions.
{"title":"On the relative minimal model program for fourfolds in positive and mixed characteristic","authors":"C. Hacon, J. Witaszek","doi":"10.1017/fmp.2023.6","DOIUrl":"https://doi.org/10.1017/fmp.2023.6","url":null,"abstract":"Abstract We show the validity of two special cases of the four-dimensional minimal model program (MMP) in characteristic \u0000$p>5$\u0000 : for contractions to \u0000${mathbb {Q}}$\u0000 -factorial fourfolds and in families over curves (‘semistable MMP’). We also provide their mixed characteristic analogues. As a corollary, we show that liftability of positive characteristic threefolds is stable under the MMP and that liftability of three-dimensional Calabi–Yau varieties is a birational invariant. Our results are partially contingent upon the existence of log resolutions.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42362296","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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