We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle algebras.
{"title":"The Representation Theory of the Increasing Monoid","authors":"Sema Gunturkun, A. Snowden","doi":"10.1090/memo/1420","DOIUrl":"https://doi.org/10.1090/memo/1420","url":null,"abstract":"We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle algebras.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46656786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.
{"title":"Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations","authors":"Oscar Dom'inguez, S. Tikhonov","doi":"10.1090/memo/1393","DOIUrl":"https://doi.org/10.1090/memo/1393","url":null,"abstract":"In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47177533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It has been known that twistor spaces associated to self-dual metrics on compact 4-manifolds are source of interesting examples of non-projective Moishezon threefolds. In this paper we investigate the structure of a variety of new Moishezon twistor spaces. The anti-canonical line bundle on any twistor space admits a canonical half, and we analyze the structure of twistor spaces by using the pluri-half-anti-canonical map from the twistor spaces. Specifically, each of the present twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of rational curves, and it is an anti-canonical curve on smooth members of the pencil. These twistor spaces are naturally classified into four types according to the type of singularities of the branch divisor, or equivalently, those of the Del Pezzo surfaces in the pencil. We also show that the quartic hypersurface satisfies a strong constraint and as a result the defining polynomial of the quartic hypersurface has to be of a specific form. Together with our previous result in cite{Hon_{C}re1}, the present result completes a classification of Moishezon twistor spaces whose half-anti-canonical system is a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil have been understood for a long time before. In the opposite direction, no example is known of a Moishezon twistor space whose half-anti-canonical system is smaller than a pencil. Twistor spaces which have a similar structure were studied in cite{Hon_{I}nv} and cite{Hon_{C}re2}, and they are very special examples among the present twistor spaces.
{"title":"Twistors, Quartics, and del Pezzo Fibrations","authors":"N. Honda","doi":"10.1090/memo/1414","DOIUrl":"https://doi.org/10.1090/memo/1414","url":null,"abstract":"It has been known that twistor spaces associated to self-dual metrics on compact 4-manifolds are source of interesting examples of non-projective Moishezon threefolds. In this paper we investigate the structure of a variety of new Moishezon twistor spaces. The anti-canonical line bundle on any twistor space admits a canonical half, and we analyze the structure of twistor spaces by using the pluri-half-anti-canonical map from the twistor spaces.\u0000\u0000Specifically, each of the present twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of rational curves, and it is an anti-canonical curve on smooth members of the pencil.\u0000\u0000These twistor spaces are naturally classified into four types according to the type of singularities of the branch divisor, or equivalently, those of the Del Pezzo surfaces in the pencil. We also show that the quartic hypersurface satisfies a strong constraint and as a result the defining polynomial of the quartic hypersurface has to be of a specific form.\u0000\u0000Together with our previous result in cite{Hon_{C}re1}, the present result completes a classification of Moishezon twistor spaces whose half-anti-canonical system is a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil have been understood for a long time before. In the opposite direction, no example is known of a Moishezon twistor space whose half-anti-canonical system is smaller than a pencil.\u0000\u0000Twistor spaces which have a similar structure were studied in cite{Hon_{I}nv} and cite{Hon_{C}re2}, and they are very special examples among the present twistor spaces.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46420794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that for Bernoulli percolation on Zdmathbb {Z}^d, d≥2dgeq 2, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that pcbond>1/2p_c^{bond} >1/2 for certain families of triangulations for which Benjamini & Schramm conjectured that pcsite≤1/2p_c^{site} leq 1/2.
证明了在Z d mathbb Z{^d, d≥2 d }geq 2上的伯努利渗流，渗流密度是超临界区间参数的解析函数。为此，我们将介绍一些具有进一步含义的技术。特别地，我们证明了对于所有传递的短期或长期模型，在亚临界区间的磁化率是解析的。对于某些三角划分族，Benjamini ＆ Schramm推测p c site≤1/2 p＿c^{site}{}leq 1/2, p c bo on和>1/2 p＿c^bond >1/2。
{"title":"Analyticity Results in Bernoulli Percolation","authors":"Agelos Georgakopoulos, C. Panagiotis","doi":"10.1090/memo/1431","DOIUrl":"https://doi.org/10.1090/memo/1431","url":null,"abstract":"<p>We prove that for Bernoulli percolation on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z Superscript d\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {Z}^d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">dgeq 2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p Subscript c Superscript b o n d Baseline greater-than 1 slash 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>b</mml:mi>\u0000 <mml:mi>o</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msubsup>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">p_c^{bond} >1/2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for certain families of triangulations for which Benjamini & Schramm conjectured that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p Subscript c Superscript s i t e Baseline less-than-or-equal-to 1 slash 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mi>i</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mi>e</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msubsup>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">p_c^{site} leq 1/2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48544614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Ivanisvili, D. Stolyarov, V. Vasyunin, P. Zatitskiy
{"title":"Bellman Function for Extremal Problems in BMO\u0000 II: Evolution","authors":"P. Ivanisvili, D. Stolyarov, V. Vasyunin, P. Zatitskiy","doi":"10.1090/MEMO/1220","DOIUrl":"https://doi.org/10.1090/MEMO/1220","url":null,"abstract":"","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74676899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze the abstract structure of algebraic groups over an algebraically closed field K, using techniques from the theory of groups of
利用群的理论，分析了代数闭域K上代数群的抽象结构
{"title":"Algebraic ℚ-groups as abstract groups","authors":"Olivier Frécon","doi":"10.1090/MEMO/1219","DOIUrl":"https://doi.org/10.1090/MEMO/1219","url":null,"abstract":"We analyze the abstract structure of algebraic groups over an algebraically closed field K, using techniques from the theory of groups of","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86784185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume and bounded cohomology. In order to study the main properties of simplicial volume, Gromov himself initiated the dual theory of bounded cohomology, which then developed into a very active and independent research field. Gromov’s theory of bounded cohomology of topological spaces was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets. The first aim of this paper is to lay the foundation of the theory of multicomplexes. After setting the main definitions, we construct the singular multicomplex K ( X ) mathcal {K}(X) associated to a topological space X X , and we prove that the geometric realization of K ( X ) mathcal {K}(X) is homotopy equivalent to X X for every CW complex X X . Following Gromov, we introduce the notion of completeness, which, roughly speaking, translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes, and we show that K ( X ) mathcal {K}(X) is complete for every CW complex X X . In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. The third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the ℓ 1 ell ^1 -invisibility of closed manifolds in terms of amenable covers. As an application, we give the first detailed proof of the vanishing of the simplicial volume of the product of three open manifolds.
{"title":"Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume","authors":"R. Frigerio, M. Moraschini","doi":"10.1090/memo/1402","DOIUrl":"https://doi.org/10.1090/memo/1402","url":null,"abstract":"The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume and bounded cohomology. In order to study the main properties of simplicial volume, Gromov himself initiated the dual theory of bounded cohomology, which then developed into a very active and independent research field. Gromov’s theory of bounded cohomology of topological spaces was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets.\u0000\u0000The first aim of this paper is to lay the foundation of the theory of multicomplexes. After setting the main definitions, we construct the singular multicomplex \u0000\u0000 \u0000 \u0000 \u0000 K\u0000 \u0000 (\u0000 X\u0000 )\u0000 \u0000 mathcal {K}(X)\u0000 \u0000\u0000 associated to a topological space \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000, and we prove that the geometric realization of \u0000\u0000 \u0000 \u0000 \u0000 K\u0000 \u0000 (\u0000 X\u0000 )\u0000 \u0000 mathcal {K}(X)\u0000 \u0000\u0000 is homotopy equivalent to \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000 for every CW complex \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000. Following Gromov, we introduce the notion of completeness, which, roughly speaking, translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes, and we show that \u0000\u0000 \u0000 \u0000 \u0000 K\u0000 \u0000 (\u0000 X\u0000 )\u0000 \u0000 mathcal {K}(X)\u0000 \u0000\u0000 is complete for every CW complex \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000.\u0000\u0000In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity.\u0000\u0000The third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the \u0000\u0000 \u0000 \u0000 ℓ\u0000 1\u0000 \u0000 ell ^1\u0000 \u0000\u0000-invisibility of closed manifolds in terms of amenable covers. As an application, we give the first detailed proof of the vanishing of the simplicial volume of the product of three open manifolds.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45989417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe higher dimensional generalizations of Ramanujan’s classical differential relations satisfied by the Eisenstein series E 2 E_2 , E 4 E_4 , E 6 E_6 . Such “higher Ramanujan equations” are given geometrically in terms of vector fields living on certain moduli stacks classifying abelian schemes equipped with suitable frames of their first de Rham cohomology. These vector fields are canonically constructed by means of the Gauss-Manin connection and the Kodaira-Spencer isomorphism. Using Mumford’s theory of degenerating families of abelian varieties, we construct remarkable solutions of these differential equations generalizing ( E 2 , E 4 , E 6 ) (E_2,E_4,E_6) , which are also shown to be defined over Z mathbf {Z} . This geometric framework taking account of integrality issues is mainly motivated by questions in Transcendental Number Theory regarding an extension of Nesterenko’s celebrated theorem on the algebraic independence of values of Eisenstein series. In this direction, we discuss the precise relation between periods of abelian varieties and the values of the above referred solutions of the higher Ramanujan equations, thereby linking the study of such differential equations to Grothendieck’s Period Conjecture. Working in the complex analytic category, we prove “functional” transcendence results, such as the Zariski-density of every leaf of the holomorphic foliation induced by the higher Ramanujan equations.
{"title":"Higher Ramanujan Equations and Periods of Abelian Varieties","authors":"T. Fonseca","doi":"10.1090/memo/1391","DOIUrl":"https://doi.org/10.1090/memo/1391","url":null,"abstract":"We describe higher dimensional generalizations of Ramanujan’s classical differential relations satisfied by the Eisenstein series \u0000\u0000 \u0000 \u0000 E\u0000 2\u0000 \u0000 E_2\u0000 \u0000\u0000, \u0000\u0000 \u0000 \u0000 E\u0000 4\u0000 \u0000 E_4\u0000 \u0000\u0000, \u0000\u0000 \u0000 \u0000 E\u0000 6\u0000 \u0000 E_6\u0000 \u0000\u0000. Such “higher Ramanujan equations” are given geometrically in terms of vector fields living on certain moduli stacks classifying abelian schemes equipped with suitable frames of their first de Rham cohomology. These vector fields are canonically constructed by means of the Gauss-Manin connection and the Kodaira-Spencer isomorphism. Using Mumford’s theory of degenerating families of abelian varieties, we construct remarkable solutions of these differential equations generalizing \u0000\u0000 \u0000 \u0000 (\u0000 \u0000 E\u0000 2\u0000 \u0000 ,\u0000 \u0000 E\u0000 4\u0000 \u0000 ,\u0000 \u0000 E\u0000 6\u0000 \u0000 )\u0000 \u0000 (E_2,E_4,E_6)\u0000 \u0000\u0000, which are also shown to be defined over \u0000\u0000 \u0000 \u0000 Z\u0000 \u0000 mathbf {Z}\u0000 \u0000\u0000.\u0000\u0000This geometric framework taking account of integrality issues is mainly motivated by questions in Transcendental Number Theory regarding an extension of Nesterenko’s celebrated theorem on the algebraic independence of values of Eisenstein series. In this direction, we discuss the precise relation between periods of abelian varieties and the values of the above referred solutions of the higher Ramanujan equations, thereby linking the study of such differential equations to Grothendieck’s Period Conjecture. Working in the complex analytic category, we prove “functional” transcendence results, such as the Zariski-density of every leaf of the holomorphic foliation induced by the higher Ramanujan equations.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49627915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the mating-of-trees framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to 8 / 3 sqrt {8/3} -LQG and SLE 6 _6 . In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of 8 / 3 sqrt {8/3} -LQG and SLE 6 _6 . For instance, we show that the exploration tree of the percolation converges to a branching SLE 6 _6 , and that the collection of percolation cycles converges to the conformal loop ensemble CLE 6 _6 . We also prove convergence of counting measure on the pivotal points of the percolation. Our results play an essential role in several other works, including a program for showing convergence of the conformal structure of uniform triangulations and works which study the behavior of random walk on the uniform infinite planar triangulation.
{"title":"Percolation on Triangulations: A Bijective Path to Liouville Quantum Gravity","authors":"O. Bernardi, N. Holden, Xin Sun","doi":"10.1090/memo/1440","DOIUrl":"https://doi.org/10.1090/memo/1440","url":null,"abstract":"We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the mating-of-trees framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to \u0000\u0000 \u0000 \u0000 8\u0000 \u0000 /\u0000 \u0000 3\u0000 \u0000 sqrt {8/3}\u0000 \u0000\u0000-LQG and SLE\u0000\u0000 \u0000 \u0000 \u0000 6\u0000 \u0000 _6\u0000 \u0000\u0000. In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of \u0000\u0000 \u0000 \u0000 8\u0000 \u0000 /\u0000 \u0000 3\u0000 \u0000 sqrt {8/3}\u0000 \u0000\u0000-LQG and SLE\u0000\u0000 \u0000 \u0000 \u0000 6\u0000 \u0000 _6\u0000 \u0000\u0000. For instance, we show that the exploration tree of the percolation converges to a branching SLE\u0000\u0000 \u0000 \u0000 \u0000 6\u0000 \u0000 _6\u0000 \u0000\u0000, and that the collection of percolation cycles converges to the conformal loop ensemble CLE\u0000\u0000 \u0000 \u0000 \u0000 6\u0000 \u0000 _6\u0000 \u0000\u0000. We also prove convergence of counting measure on the pivotal points of the percolation. Our results play an essential role in several other works, including a program for showing convergence of the conformal structure of uniform triangulations and works which study the behavior of random walk on the uniform infinite planar triangulation.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46614292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Croydon, Tsuyoshi Kato, M. Sasada, S. Tsujimoto
The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al. recently showed the invariance in distribution of Bernoulli product measures with density strictly less than 1 2 frac {1}{2} , and gave a soliton decomposition for invariant measures more generally. We study the BBS dynamics using the transformation of a nearest neighbour path encoding of the particle configuration given by ‘reflection in the past maximum’, which was famously shown by Pitman to connect Brownian motion and a three-dimensional Bessel process. We use this to characterise the set of configurations for which the dynamics are well-defined and reversible (i.e. can be inverted) for all times. The techniques developed to understand the deterministic dynamics are subsequently applied to study the evolution of the BBS from a random initial configuration. Specifically, we give simple sufficient conditions for random initial conditions to be invariant in distribution under the BBS dynamics, which we check in several natural examples, and also investigate the ergodicity of the relevant transformation. Furthermore, we analyse various probabilistic properties of the BBS that are commonly studied for interacting particle systems, such as the asymptotic behavior of the integrated current of particles and of a tagged particle. Finally, for Bernoulli product measures with parameter p ↑ 1 2 puparrow frac 12 (which may be considered the ‘high density’ regime), the path encoding we consider has a natural scaling limit, which motivates the introduction of a new continuous version of the BBS that we believe will be of independent interest as a dynamical system.
盒式球系统(BBS)由Takahashi和Satsuma于1990年提出，是一种具有孤子行为的元胞自动机。在本文中，我们从一个随机的双面无限粒子结构开始研究BBS。对于这样的模型，Ferrari等人最近证明了密度严格小于12 frac{1}{2}的伯努利积测度分布的不变量，并给出了更一般的不变量测度的孤子分解。我们使用由“过去极大值反射”给出的粒子构型的最近邻路径编码的变换来研究BBS动力学，这是Pitman著名的将布朗运动和三维贝塞尔过程联系起来的方法。我们用它来描述一组配置，其中动力学在任何时候都是定义良好的和可逆的(即可以反转)。为理解确定性动力学而开发的技术随后被应用于研究BBS从随机初始配置的演变。具体地说，我们给出了在BBS动力学下随机初始条件在分布上不变的简单充分条件，并在几个自然例子中进行了验证，同时研究了相关变换的遍历性。此外，我们还分析了通常用于相互作用粒子系统的BBS的各种概率性质，例如粒子和标记粒子的积分电流的渐近行为。最后，对于参数p ^ 1 ^ 2 p uprow frac 12的伯努利积测度(可以被认为是“高密度”状态)，我们考虑的路径编码有一个自然的尺度限制，这促使我们引入新的连续版本的BBS，我们相信它将作为一个动态系统具有独立的兴趣。
{"title":"Dynamics of the Box-Ball System with Random Initial Conditions via Pitman’s Transformation","authors":"D. Croydon, Tsuyoshi Kato, M. Sasada, S. Tsujimoto","doi":"10.1090/memo/1398","DOIUrl":"https://doi.org/10.1090/memo/1398","url":null,"abstract":"The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al. recently showed the invariance in distribution of Bernoulli product measures with density strictly less than \u0000\u0000 \u0000 \u0000 1\u0000 2\u0000 \u0000 frac {1}{2}\u0000 \u0000\u0000, and gave a soliton decomposition for invariant measures more generally. We study the BBS dynamics using the transformation of a nearest neighbour path encoding of the particle configuration given by ‘reflection in the past maximum’, which was famously shown by Pitman to connect Brownian motion and a three-dimensional Bessel process. We use this to characterise the set of configurations for which the dynamics are well-defined and reversible (i.e. can be inverted) for all times. The techniques developed to understand the deterministic dynamics are subsequently applied to study the evolution of the BBS from a random initial configuration. Specifically, we give simple sufficient conditions for random initial conditions to be invariant in distribution under the BBS dynamics, which we check in several natural examples, and also investigate the ergodicity of the relevant transformation. Furthermore, we analyse various probabilistic properties of the BBS that are commonly studied for interacting particle systems, such as the asymptotic behavior of the integrated current of particles and of a tagged particle. Finally, for Bernoulli product measures with parameter \u0000\u0000 \u0000 \u0000 p\u0000 ↑\u0000 \u0000 1\u0000 2\u0000 \u0000 \u0000 puparrow frac 12\u0000 \u0000\u0000 (which may be considered the ‘high density’ regime), the path encoding we consider has a natural scaling limit, which motivates the introduction of a new continuous version of the BBS that we believe will be of independent interest as a dynamical system.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47334698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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