{"title":"A categorical Torelli theorem for hypersurfaces","authors":"Dmitrii Pirozhkov","doi":"10.1112/blms.13117","DOIUrl":"10.1112/blms.13117","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>⊂</mo>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$X subset mathbb {P}^{n+1}$</annotation>\u0000 </semantics></math> be a smooth Fano hypersurface of dimension <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> and degree <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math>. The derived category of coherent sheaves on <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> contains an interesting subcategory called the Kuznetsov component <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mi>X</mi>\u0000 </msub>\u0000 <annotation>$mathcal {A}_X$</annotation>\u0000 </semantics></math>. We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> uniquely if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>></mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$d &gt; 3$</annotation>\u0000 </semantics></math> or if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>=</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$d = 3$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>></mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$n &gt; 3$</annotation>\u0000 </semantics></math>. This generalizes a result by Huybrechts and Rennemo, who proved the same statement under the additional assumption that <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math> divides <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$n+2$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 10","pages":"3075-3089"},"PeriodicalIF":0.8,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141829666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On finite groups with the Magnus Property","authors":"Martino Garonzi, Claude Marion","doi":"10.1112/blms.13119","DOIUrl":"https://doi.org/10.1112/blms.13119","url":null,"abstract":"<p>We investigate finite groups with the Magnus Property (MP), where a group is said to have the MP if whenever two elements have the same normal closure, then they are conjugate or inverse conjugate. In particular, we observe that a finite MP group is solvable, determine the finite primitive MP groups, and determine all the possible orders of the chief factors of a finite MP group. We also determine the MP finite direct products of finite primitive groups, as well as the MP crown-based powers of a finite monolithic primitive group.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 10","pages":"3114-3128"},"PeriodicalIF":0.8,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142435035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Towards Turán's polynomial conjecture","authors":"Pradipto Banerjee, Amit Kundu","doi":"10.1112/blms.13123","DOIUrl":"10.1112/blms.13123","url":null,"abstract":"<p>We revisit an old problem posed by P. Turán asking whether there exists an absolute constant <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>></mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$C&gt;0$</annotation>\u0000 </semantics></math> such that if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 <mo>∈</mo>\u0000 <mi>Z</mi>\u0000 <mo>[</mo>\u0000 <mi>x</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>$f(x)in mathbb {Z}[x]$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>deg</mo>\u0000 <mi>f</mi>\u0000 <mo>=</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$deg f = d$</annotation>\u0000 </semantics></math>, then there is a polynomial <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>w</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 <mo>∈</mo>\u0000 <mi>Z</mi>\u0000 <mo>[</mo>\u0000 <mi>x</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>$w(x)in mathbb {Z}[x]$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>deg</mo>\u0000 <mi>w</mi>\u0000 <mo>⩽</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$deg wleqslant d$</annotation>\u0000 </semantics></math> and the sum of the absolute values of the coefficients of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>w</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$w(x)$</annotation>\u0000 </semantics></math> is less than or equal to <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$C$</annotation>\u0000 </semantics></math> such that the sum <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 <mo>+</mo>\u0000 <mi>w</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$f(x)+w(x)$</annotation>\u0000 </semantics></math> is irreducible over <span></span><math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>$mathbb {Q}$</annotation>\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 10","pages":"3164-3173"},"PeriodicalIF":0.8,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141672466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Graph convex hull bounds as generalized Jensen inequalities","authors":"Ilja Klebanov","doi":"10.1112/blms.13116","DOIUrl":"https://doi.org/10.1112/blms.13116","url":null,"abstract":"<p>Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>:</mo>\u0000 <mi>K</mi>\u0000 <mo>→</mo>\u0000 <mi>R</mi>\u0000 </mrow>\u0000 <annotation>$fcolon K rightarrow mathbb {R}$</annotation>\u0000 </semantics></math> defined on a convex domain <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>K</mi>\u0000 <mo>⊆</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$K subseteq mathbb {R}^{d}$</annotation>\u0000 </semantics></math> and any random variable <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> taking values in <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>E</mi>\u0000 <mo>[</mo>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 <mo>]</mo>\u0000 <mo>⩾</mo>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>E</mi>\u0000 <mo>[</mo>\u0000 <mi>X</mi>\u0000 <mo>]</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathbb {E}[f(X)] geqslant f(mathbb {E}[X])$</annotation>\u0000 </semantics></math>. In this paper, sharp upper and lower bounds on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>E</mi>\u0000 <mo>[</mo>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>$mathbb {E}[f(X)]$</annotation>\u0000 </semantics></math>, termed ‘graph convex hull bounds’, are derived for arbitrary functions <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> on arbitrary domains <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>, thereby extensively generalizing Jensen's inequality. The ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 10","pages":"3061-3074"},"PeriodicalIF":0.8,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13116","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142435033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rectifiable paths with polynomial log-signature are straight lines","authors":"Peter K. Friz, Terry Lyons, Anna Seigal","doi":"10.1112/blms.13110","DOIUrl":"https://doi.org/10.1112/blms.13110","url":null,"abstract":"<p>The signature of a rectifiable path is a tensor series in the tensor algebra whose coefficients are definite iterated integrals of the path. The signature characterizes the path up to a generalized form of reparameterization. It is a classical result of Chen that the log-signature (the logarithm of the signature) is a Lie series. A Lie series is polynomial if it has finite degree. We show that the log-signature is polynomial if and only if the path is a straight line up to reparameterization. Consequently, the log-signature of a rectifiable path either has degree one or infinite support. Though our result pertains to rectifiable paths, the proof uses rough path theory, in particular that the signature characterizes a rough path up to reparameterization.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 9","pages":"2922-2934"},"PeriodicalIF":0.8,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142165465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Carolyn Abbott, Hoang Thanh Nguyen, Alexander J. Rasmussen
{"title":"Largest hyperbolic action of 3-manifold groups","authors":"Carolyn Abbott, Hoang Thanh Nguyen, Alexander J. Rasmussen","doi":"10.1112/blms.13118","DOIUrl":"10.1112/blms.13118","url":null,"abstract":"<p>The set of equivalence classes of cobounded actions of a group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> on different hyperbolic metric spaces carries a natural partial order. Following Abbott–Balasubramanya–Osin, the group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$mathcal {H}$</annotation>\u0000 </semantics></math>-<i>accessible</i> if the resulting poset has a largest element. In this paper, we prove that every nongeometric 3-manifold has a finite cover with <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$mathcal {H}$</annotation>\u0000 </semantics></math>-inaccessible fundamental group and give conditions under which the fundamental group of the original manifold is <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$mathcal {H}$</annotation>\u0000 </semantics></math>-inaccessible. We also prove that every Croke–Kleiner admissible group (a class of graphs of groups that generalizes fundamental groups of three-dimensional graph manifolds) has a finite index subgroup that is <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$mathcal {H}$</annotation>\u0000 </semantics></math>-inaccessible.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 10","pages":"3090-3113"},"PeriodicalIF":0.8,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141678544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quasi-conical domains with embedded eigenvalues","authors":"David Krejčiřík, Vladimir Lotoreichik","doi":"10.1112/blms.13113","DOIUrl":"https://doi.org/10.1112/blms.13113","url":null,"abstract":"<p>The spectrum of the Dirichlet Laplacian on any quasi-conical open set coincides with the non-negative semi-axis. We show that there is a connected quasi-conical open set such that the respective Dirichlet Laplacian has a positive (embedded) eigenvalue. This open set is constructed as the tower of cubes of growing size connected by windows of vanishing size. Moreover, we show that the sizes of the windows in this construction can be chosen so that the absolutely continuous spectrum of the Dirichlet Laplacian is empty.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 9","pages":"2969-2981"},"PeriodicalIF":0.8,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142165464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Separability in Morse local-to-global groups","authors":"Lawk Mineh, Davide Spriano","doi":"10.1112/blms.13121","DOIUrl":"https://doi.org/10.1112/blms.13121","url":null,"abstract":"<p>We show that in a Morse local-to-global group where stable subgroups are separable, the product of any stable subgroups is separable. As an application, we show that the product of stable subgroups in virtually special groups is separable.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 10","pages":"3134-3144"},"PeriodicalIF":0.8,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13121","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142435167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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