In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of Matheus and Moreira [Comment. Math. Helv. 95 (2020), pp. 593–633]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain–Kontorovich [Ann. of Math. (2) 180 (2014), pp. 137–196], Huang [An improvement to Zaremba’s conjecture, ProQuest LLC, Ann Arbor, MI, 2015] and Kan [Mat. Sb. 210 (2019), pp. 75–130]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen [Amer. J. Math. 120 (1998), pp. 691-721]. In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in this paper in a way that is straightforward to implement. These estimates apparently cannot be obtained by other known methods.
在这篇文章中，我们将描述一种简单实用的方法来得到一些一维马尔可夫迭代函数格式的极限集的Hausdorff维的严格界限。一般的问题已经引起了相当大的关注，但我们特别关心的是豪斯多夫维数在解决数学其他领域的猜想和问题中的作用。作为我们的第一个应用，我们证实并经常加强丢芬图分析中关于拉格朗日和马尔可夫谱的差异的猜想，这些猜想出现在Matheus和Moreira的著作中[注释]。数学。《中国科学》(2020)，第593-633页。作为第二个应用，我们(重新)验证和改进了与bourgin - kontorovich [Ann]的工作中使用的数论中的Zaremba猜想有关的估计。的数学。(2) 180(2014)，第137-196页]，Huang[对Zaremba猜想的改进，ProQuest LLC, Ann Arbor, MI, 2015]和Kan [Mat. Sb. 210(2019)，第75-130页]。作为第三个几何应用，我们严格限定了无限面积表面的拉普拉斯谱的底部，如McMullen [Amer]研究的一个例子所示。数学学报(1998)，第691-721页。在所有估计极限集维数的方法中，都存在关于算法效率、所需计算量和边界严谨性的问题。我们使用的方法具有简单和有效的优点，我们在本文中以一种直接实现的方式呈现它。这些估计显然不能用其他已知的方法得到。
{"title":"Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups","authors":"M. Pollicott, P. Vytnova","doi":"10.1090/btran/109","DOIUrl":"https://doi.org/10.1090/btran/109","url":null,"abstract":"In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of Matheus and Moreira [Comment. Math. Helv. 95 (2020), pp. 593–633]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain–Kontorovich [Ann. of Math. (2) 180 (2014), pp. 137–196], Huang [An improvement to Zaremba’s conjecture, ProQuest LLC, Ann Arbor, MI, 2015] and Kan [Mat. Sb. 210 (2019), pp. 75–130]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen [Amer. J. Math. 120 (1998), pp. 691-721].\u0000\u0000In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in this paper in a way that is straightforward to implement.\u0000\u0000These estimates apparently cannot be obtained by other known methods.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126466323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the real cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class on projective space, or the hyperbolic ideal volume on spheres. In rank one, this leads to an isomorphism between the cohomology of the group and of this boundary model. In higher rank, additional classes appear, which we determine completely.
{"title":"The cohomology of semi-simple Lie groups, viewed from infinity","authors":"N. Monod","doi":"10.1090/btran/85","DOIUrl":"https://doi.org/10.1090/btran/85","url":null,"abstract":"We prove that the real cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class on projective space, or the hyperbolic ideal volume on spheres.\u0000\u0000In rank one, this leads to an isomorphism between the cohomology of the group and of this boundary model. In higher rank, additional classes appear, which we determine completely.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125205275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct examples where the spaces are not quasiisometrically equivalent.
{"title":"Hyperbolic distance versus quasihyperbolic distance in plane domains","authors":"D. Herron, Jeff Lindquist","doi":"10.1090/btran/73","DOIUrl":"https://doi.org/10.1090/btran/73","url":null,"abstract":"We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct examples where the spaces are not quasiisometrically equivalent.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124404032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.
{"title":"Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser","authors":"K. Bringmann, B. Kane, Larry Rolen, Z. Tripp","doi":"10.1090/BTRAN/77","DOIUrl":"https://doi.org/10.1090/BTRAN/77","url":null,"abstract":"Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114619770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper studies the Hausdorff dimension of harmonic measures on various boundaries of a relatively hyperbolic group which are associated with random walks driven by a probability measure with finite first moment. With respect to the Floyd metric and the shortcut metric, we prove that the Hausdorff dimension of the harmonic measure equals the ratio of the entropy and the drift of the random walk. If the group is infinitely-ended, the same dimension formula is obtained for the end boundary endowed with a visual metric. In addition, the Hausdorff dimension of the visual metric is identified with the growth rate of the word metric. These results are complemented by a characterization of doubling visual metrics for accessible infinitely-ended groups: the visual metrics on the end boundary is doubling if and only if the group is virtually free. Consequently, there are at least two different bi-Hölder classes (and thus quasi-symmetric classes) of visual metrics on the end boundary.
{"title":"The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups","authors":"Matthieu Dussaule, Wen-yuan Yang","doi":"10.1090/btran/145","DOIUrl":"https://doi.org/10.1090/btran/145","url":null,"abstract":"The paper studies the Hausdorff dimension of harmonic measures on various boundaries of a relatively hyperbolic group which are associated with random walks driven by a probability measure with finite first moment. With respect to the Floyd metric and the shortcut metric, we prove that the Hausdorff dimension of the harmonic measure equals the ratio of the entropy and the drift of the random walk.\u0000\u0000If the group is infinitely-ended, the same dimension formula is obtained for the end boundary endowed with a visual metric. In addition, the Hausdorff dimension of the visual metric is identified with the growth rate of the word metric. These results are complemented by a characterization of doubling visual metrics for accessible infinitely-ended groups: the visual metrics on the end boundary is doubling if and only if the group is virtually free. Consequently, there are at least two different bi-Hölder classes (and thus quasi-symmetric classes) of visual metrics on the end boundary.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123682125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On any closed Riemannian manifold of dimension n ≥ 3 ngeq 3 , we prove that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close, in a quantitative sense, to a minimizing Yamabe metric in the conformal class. Generically, this distance is controlled quadratically by the Yamabe energy deficit. Finally, we produce an example for which this quadratic estimate is false.
{"title":"Quantitative stability for minimizing Yamabe metrics","authors":"","doi":"10.1090/btran/111","DOIUrl":"https://doi.org/10.1090/btran/111","url":null,"abstract":"On any closed Riemannian manifold of dimension \u0000\u0000 \u0000 \u0000 n\u0000 ≥\u0000 3\u0000 \u0000 ngeq 3\u0000 \u0000\u0000, we prove that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close, in a quantitative sense, to a minimizing Yamabe metric in the conformal class. Generically, this distance is controlled quadratically by the Yamabe energy deficit. Finally, we produce an example for which this quadratic estimate is false.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125059364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zhibek Kadyrsizova, J. Kenkel, Janet Page, J. Singh, Karen E. Smith, Adela Vraciu, E. Witt
We prove that if ff is a reduced homogeneous polynomial of degree dd, then its FF-pure threshold at the unique homogeneous maximal ideal is at least 1d−1frac {1}{d-1}. We show, furthermore, that its FF-pure threshold equals 1d−1frac {1}{d-1} if and only if f∈m[q]fin mathfrak m^{[q]}
我们证明了如果f f是d次的简化齐次多项式，那么它在唯一齐次极大理想处的f f纯阈值至少为1 d−1 frac {1}{d-1}。进一步证明，当且仅当F∈m [q] F in mathfrak m^{[q]}且d=q+1 d=q+1时，它的F - F纯阈值等于1 d−1 frac {1}{d-1}，其中q q是p p的幂。直到坐标的线性变化(在一个固定的代数闭域上)，我们对这类“极端奇点”进行了分类，并证明了最多存在一个孤立奇点。最后，我们指出了由这种形式定义的射影超曲面是“极值”的几种方式，例如，就它们可以包含的线的配置而言。
{"title":"Lower bounds on the F-pure threshold and extremal singularities","authors":"Zhibek Kadyrsizova, J. Kenkel, Janet Page, J. Singh, Karen E. Smith, Adela Vraciu, E. Witt","doi":"10.1090/btran/106","DOIUrl":"https://doi.org/10.1090/btran/106","url":null,"abstract":"<p>We prove that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a reduced homogeneous polynomial of degree <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\u0000 <mml:semantics>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, then its <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-pure threshold at the unique homogeneous maximal ideal is at least <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartFraction 1 Over d minus 1 EndFraction\">\u0000 <mml:semantics>\u0000 <mml:mfrac>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:mfrac>\u0000 <mml:annotation encoding=\"application/x-tex\">frac {1}{d-1}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We show, furthermore, that its <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-pure threshold equals <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartFraction 1 Over d minus 1 EndFraction\">\u0000 <mml:semantics>\u0000 <mml:mfrac>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:mfrac>\u0000 <mml:annotation encoding=\"application/x-tex\">frac {1}{d-1}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> if and only if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f element-of German m Superscript left-bracket q right-bracket\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">m</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">fin mathfrak m^{[q]}</mml:annotation>\u0000 </mml:sema","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130661206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by the theory of Cuntz-Krieger algebras we define and study C ∗ C^ast -algebras associated to directed quantum graphs. For classical graphs the C ∗ C^ast -algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to K K KK -equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these C ∗ C^ast -algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of K K KK -theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.
在Cuntz-Krieger代数理论的激励下，我们定义并研究了与有向量子图相关的C * C^ast -代数。对于经典图，用这种方法得到的C * C^ast -代数可以看作是Cuntz-Krieger代数的自由类似物，而不必是核的。我们详细研究了两类特殊的量子图，即平凡量子图和完全量子图。对于单个矩阵块上的平凡量子图，我们证明了相关的量子Cuntz-Krieger代数既不是一元的，也不是核的，也不是简单的，并且不依赖于矩阵块的大小，直到KK KK -等价。在完全量子图的情况下，我们使用量子对称性来证明，在某些情况下，相应的量子康茨-克里格代数与康茨代数是同构的。这些同构，从定义上看似乎并不明显，特别暗示这些C∗C^ast -代数对于不同维数的完全量子图都是成对非同构的，即使在KK KK -理论的水平上也是如此。我们解释了量子信息论中统一误差基的概念如何有助于阐明这种情况。我们也一般讨论了量子Cuntz-Krieger代数的量子对称性。
{"title":"Quantum Cuntz-Krieger algebras","authors":"Mike Brannan, Kari Eifler, C. Voigt, Moritz Weber","doi":"10.1090/btran/88","DOIUrl":"https://doi.org/10.1090/btran/88","url":null,"abstract":"Motivated by the theory of Cuntz-Krieger algebras we define and study \u0000\u0000 \u0000 \u0000 C\u0000 ∗\u0000 \u0000 C^ast\u0000 \u0000\u0000-algebras associated to directed quantum graphs. For classical graphs the \u0000\u0000 \u0000 \u0000 C\u0000 ∗\u0000 \u0000 C^ast\u0000 \u0000\u0000-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear.\u0000\u0000We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to \u0000\u0000 \u0000 \u0000 K\u0000 K\u0000 \u0000 KK\u0000 \u0000\u0000-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these \u0000\u0000 \u0000 \u0000 C\u0000 ∗\u0000 \u0000 C^ast\u0000 \u0000\u0000-algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of \u0000\u0000 \u0000 \u0000 K\u0000 K\u0000 \u0000 KK\u0000 \u0000\u0000-theory.\u0000\u0000We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation.\u0000\u0000We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115826842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3 3 -fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3 3 -fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3 3 -fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3 3 -folds by building degenerations to K-polystable toric Fano 3 3 -folds.
{"title":"On toric geometry and K-stability of Fano varieties","authors":"Anne-Sophie Kaloghiros, Andrea Petracci","doi":"10.1090/btran/82","DOIUrl":"https://doi.org/10.1090/btran/82","url":null,"abstract":"We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000-fold, while in the other they are non-reduced near the closed point associated to the toric Fano \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000-folds by building degenerations to K-polystable toric Fano \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000-folds.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125622386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on R mathbb {R} and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for ∞ infty . We extend aspects of this theory in the setting of rational functions with poles on R ¯ = R ∪ { ∞ } overline {mathbb {R}} = mathbb {R} cup {infty } , obtaining a formulation which allows multiple poles and proving an invariance with respect to R ¯ overline {mathbb {R}} -preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.
{"title":"Orthogonal rational functions with real poles, root asymptotics, and GMP matrices","authors":"B. Eichinger, Milivoje Luki'c, Giorgio Young","doi":"10.1090/btran/117","DOIUrl":"https://doi.org/10.1090/btran/117","url":null,"abstract":"There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on \u0000\u0000 \u0000 \u0000 R\u0000 \u0000 mathbb {R}\u0000 \u0000\u0000 and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for \u0000\u0000 \u0000 ∞\u0000 infty\u0000 \u0000\u0000. We extend aspects of this theory in the setting of rational functions with poles on \u0000\u0000 \u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 ¯\u0000 \u0000 =\u0000 \u0000 R\u0000 \u0000 ∪\u0000 {\u0000 ∞\u0000 }\u0000 \u0000 overline {mathbb {R}} = mathbb {R} cup {infty }\u0000 \u0000\u0000, obtaining a formulation which allows multiple poles and proving an invariance with respect to \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 ¯\u0000 \u0000 overline {mathbb {R}}\u0000 \u0000\u0000-preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116216988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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