Pub Date : 2022-10-05DOI: 10.1017/S0305004123000178
Bhishan Jacelon
Abstract What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most k extremal traces? What is the expected value of the radius of comparison of a random Villadsen-type AH algebra? What is the probability that such an algebra is $mathcal{Z}$ -stable? What is the probability that a random Cuntz–Krieger algebra is purely infinite and simple, and what can be said about the distribution of its K-theory? By constructing $mathrm{C}^*$ -algebras associated with suitable random (walks on) graphs, we provide context in which these are meaningful questions with computable answers.
{"title":"Random amenable C*-algebras","authors":"Bhishan Jacelon","doi":"10.1017/S0305004123000178","DOIUrl":"https://doi.org/10.1017/S0305004123000178","url":null,"abstract":"Abstract What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most k extremal traces? What is the expected value of the radius of comparison of a random Villadsen-type AH algebra? What is the probability that such an algebra is \u0000$mathcal{Z}$\u0000 -stable? What is the probability that a random Cuntz–Krieger algebra is purely infinite and simple, and what can be said about the distribution of its K-theory? By constructing \u0000$mathrm{C}^*$\u0000 -algebras associated with suitable random (walks on) graphs, we provide context in which these are meaningful questions with computable answers.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86746099","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}
Pub Date : 2022-08-17DOI: 10.1017/s0305004122000329
{"title":"PSP volume 173 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0305004122000329","DOIUrl":"https://doi.org/10.1017/s0305004122000329","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85241319","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}
Pub Date : 2022-08-17DOI: 10.1017/s0305004122000330
{"title":"PSP volume 173 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0305004122000330","DOIUrl":"https://doi.org/10.1017/s0305004122000330","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82585825","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}
Pub Date : 2022-08-09DOI: 10.1017/S0305004123000142
Nadav Yesha
Abstract We study intermediate-scale statistics for the fractional parts of the sequence $left(alpha a_{n}right)_{n=1}^{infty}$ , where $left(a_{n}right)_{n=1}^{infty}$ is a positive, real-valued lacunary sequence, and $alphainmathbb{R}$ . In particular, we consider the number of elements $S_{N}!left(L,alpharight)$ in a random interval of length $L/N$ , where $L=O!left(N^{1-epsilon}right)$ , and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. $alpha$ , which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in $alphainmathbb{R}$ when $L=O!left(N^{1/2-epsilon}right)$ . For slowly growing L, we further prove a central limit theorem for $S_{N}!left(L,alpharight)$ which holds for almost all $alphainmathbb{R}$ .
{"title":"Intermediate-scale statistics for real-valued lacunary sequences","authors":"Nadav Yesha","doi":"10.1017/S0305004123000142","DOIUrl":"https://doi.org/10.1017/S0305004123000142","url":null,"abstract":"Abstract We study intermediate-scale statistics for the fractional parts of the sequence \u0000$left(alpha a_{n}right)_{n=1}^{infty}$\u0000 , where \u0000$left(a_{n}right)_{n=1}^{infty}$\u0000 is a positive, real-valued lacunary sequence, and \u0000$alphainmathbb{R}$\u0000 . In particular, we consider the number of elements \u0000$S_{N}!left(L,alpharight)$\u0000 in a random interval of length \u0000$L/N$\u0000 , where \u0000$L=O!left(N^{1-epsilon}right)$\u0000 , and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. \u0000$alpha$\u0000 , which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in \u0000$alphainmathbb{R}$\u0000 when \u0000$L=O!left(N^{1/2-epsilon}right)$\u0000 . For slowly growing L, we further prove a central limit theorem for \u0000$S_{N}!left(L,alpharight)$\u0000 which holds for almost all \u0000$alphainmathbb{R}$\u0000 .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76243809","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}
Pub Date : 2022-06-13DOI: 10.1017/s0305004122000287
{"title":"PSP volume 173 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s0305004122000287","DOIUrl":"https://doi.org/10.1017/s0305004122000287","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88730043","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}
Pub Date : 2022-06-13DOI: 10.1017/s0305004122000275
{"title":"PSP volume 173 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s0305004122000275","DOIUrl":"https://doi.org/10.1017/s0305004122000275","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78839386","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}
Pub Date : 2022-05-17DOI: 10.1017/S0305004123000075
Alena Ernst, K. Schmidt
Abstract A subset Y of the general linear group $text{GL}(n,q)$ is called t-intersecting if $text{rk}(x-y)le n-t$ for all $x,yin Y$ , or equivalently x and y agree pointwise on a t-dimensional subspace of $mathbb{F}_q^n$ for all $x,yin Y$ . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of $mathbb{F}_q^n$ . In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $text{GL}(n,q)$ that intersect not necessarily pointwise in t-dimensional subspaces of $mathbb{F}_q^n$ and for cross-intersecting subsets of $text{GL}(n,q)$ . These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $text{GL}(n,q)$ .
摘要一般线性群$text{GL}(n,q)$的子集Y称为t相交，如果$text{rk}(x- Y) n-t$对于所有$x, Y in Y$，或者等价地，x和Y在$mathbb{F}_q^n$的t维子空间上对所有$x, Y in Y$点方向一致。我们证明，如果n相对于t足够大，则每一个这样的t相交集的大小不超过$mathbb{F}_q^n$的t维子空间的基的稳定子的大小。在相等的情况下，Y的特征向量是这些稳定器的协集的特征向量的线性组合。对于$text{GL}(n,q)$在$mathbb{F}_q^n$的t维子空间中不一定点向相交的子集，以及$text{GL}(n,q)$的交叉子集，我们也给出了类似的结果。这些结果可以看作是极端集合理论中经典Erdős-Ko-Rado定理的变体，并且是对称群中已知的相应结果的q-类似。我们的方法基于特征值技术来估计图中最大独立集的大小，并且关键地涉及到$text{GL}(n,q)$的表示理论。
{"title":"Intersection theorems for finite general linear groups","authors":"Alena Ernst, K. Schmidt","doi":"10.1017/S0305004123000075","DOIUrl":"https://doi.org/10.1017/S0305004123000075","url":null,"abstract":"Abstract A subset Y of the general linear group \u0000$text{GL}(n,q)$\u0000 is called t-intersecting if \u0000$text{rk}(x-y)le n-t$\u0000 for all \u0000$x,yin Y$\u0000 , or equivalently x and y agree pointwise on a t-dimensional subspace of \u0000$mathbb{F}_q^n$\u0000 for all \u0000$x,yin Y$\u0000 . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of \u0000$mathbb{F}_q^n$\u0000 . In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of \u0000$text{GL}(n,q)$\u0000 that intersect not necessarily pointwise in t-dimensional subspaces of \u0000$mathbb{F}_q^n$\u0000 and for cross-intersecting subsets of \u0000$text{GL}(n,q)$\u0000 . These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of \u0000$text{GL}(n,q)$\u0000 .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76892137","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}
Pub Date : 2022-05-01DOI: 10.1017/s0305004122000184
{"title":"PSP volume 172 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s0305004122000184","DOIUrl":"https://doi.org/10.1017/s0305004122000184","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73337961","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}
Pub Date : 2022-05-01DOI: 10.1017/s0305004122000196
{"title":"PSP volume 172 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s0305004122000196","DOIUrl":"https://doi.org/10.1017/s0305004122000196","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89847358","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}
Pub Date : 2022-04-28DOI: 10.1017/S0305004123000233
Humberto A. Diaz
Abstract We obtain examples of smooth projective varieties over ${mathbb C}$ that violate the integral Hodge conjecture and for which the total Chow group is of finite rank. Moreover, we show that there exist such examples defined over number fields.
{"title":"On the integral Hodge conjecture for varieties with trivial Chow group","authors":"Humberto A. Diaz","doi":"10.1017/S0305004123000233","DOIUrl":"https://doi.org/10.1017/S0305004123000233","url":null,"abstract":"Abstract We obtain examples of smooth projective varieties over \u0000${mathbb C}$\u0000 that violate the integral Hodge conjecture and for which the total Chow group is of finite rank. Moreover, we show that there exist such examples defined over number fields.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89378461","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}
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