Pub Date : 2021-10-01DOI: 10.1215/00127094-2021-0059
S. Popa
We prove that any separable II1 factor M admits a coarse decomposition over the hyperfinite II1 factor R—that is, there exists an embedding R↪M such that L2M⊖L2R is a multiple of the coarse Hilbert R-bimodule L2R⊗‾L2Rop. Equivalently, the von Neumann algebra generated by left and right multiplication by R on L2M⊖L2R is isomorphic to R⊗‾Rop. Moreover, if Q⊂M is an infinite-index irreducible subfactor, then R↪M can be constructed to be coarse with respect to Q as well. This implies the existence of maximal abelian ∗-subalgebras that are mixing, strongly malnormal, and with infinite multiplicity, in any given separable II1 factor.
{"title":"Coarse decomposition of II1 factors","authors":"S. Popa","doi":"10.1215/00127094-2021-0059","DOIUrl":"https://doi.org/10.1215/00127094-2021-0059","url":null,"abstract":"We prove that any separable II1 factor M admits a coarse decomposition over the hyperfinite II1 factor R—that is, there exists an embedding R↪M such that L2M⊖L2R is a multiple of the coarse Hilbert R-bimodule L2R⊗‾L2Rop. Equivalently, the von Neumann algebra generated by left and right multiplication by R on L2M⊖L2R is isomorphic to R⊗‾Rop. Moreover, if Q⊂M is an infinite-index irreducible subfactor, then R↪M can be constructed to be coarse with respect to Q as well. This implies the existence of maximal abelian ∗-subalgebras that are mixing, strongly malnormal, and with infinite multiplicity, in any given separable II1 factor.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45505254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-06DOI: 10.1215/00127094-2022-0075
Arvind Ayyer, S. Chhita, K. Johansson
The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter $Delta$. When $Delta = 0$, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all $Delta$, there has been very little progress in understanding its statistics in the scaling limit for other values. In this work, we focus on the six-vertex model with domain wall boundary conditions at $Delta = 1/2$, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We show that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy--Widom distribution after appropriate rescaling. A key ingredient in our proof is Zeilberger's proof of the ASM conjecture. As far as we know, this is the first edge fluctuation result away from the tangency points for the domain-wall six-vertex model when we are not in the free fermion case.
{"title":"GOE fluctuations for the maximum of the top path in alternating sign matrices","authors":"Arvind Ayyer, S. Chhita, K. Johansson","doi":"10.1215/00127094-2022-0075","DOIUrl":"https://doi.org/10.1215/00127094-2022-0075","url":null,"abstract":"The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter $Delta$. When $Delta = 0$, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all $Delta$, there has been very little progress in understanding its statistics in the scaling limit for other values. In this work, we focus on the six-vertex model with domain wall boundary conditions at $Delta = 1/2$, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We show that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy--Widom distribution after appropriate rescaling. A key ingredient in our proof is Zeilberger's proof of the ASM conjecture. As far as we know, this is the first edge fluctuation result away from the tangency points for the domain-wall six-vertex model when we are not in the free fermion case.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44587447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-15DOI: 10.1215/00127094-2020-0079
T. Dokchitser
Let C be a smooth projective curve over a discretely valued field K, defined by an affine equation f(x,y)=0. We construct a model of C over the ring of integers of K using a toroidal embedding associated to the Newton polygon of f. We show that under “generic” conditions it is regular with normal crossings, and we determine when it is minimal, the global sections of its relative dualizing sheaf, and the tame part of the first etale cohomology of C.
{"title":"Models of curves over discrete valuation rings","authors":"T. Dokchitser","doi":"10.1215/00127094-2020-0079","DOIUrl":"https://doi.org/10.1215/00127094-2020-0079","url":null,"abstract":"Let C be a smooth projective curve over a discretely valued field K, defined by an affine equation f(x,y)=0. We construct a model of C over the ring of integers of K using a toroidal embedding associated to the Newton polygon of f. We show that under “generic” conditions it is regular with normal crossings, and we determine when it is minimal, the global sections of its relative dualizing sheaf, and the tame part of the first etale cohomology of C.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42468897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-30DOI: 10.1215/00127094-2022-0034
R. Shankar, Yu Yuan
We show that every general semiconvex entire solution to the sigma-2 equation is a quadratic polynomial. A decade ago, this result was shown for almost convex solutions.
我们证明了σ -2方程的所有一般半凸全解都是二次多项式。十年前，这个结果在几乎凸解中得到了证明。
{"title":"Rigidity for general semiconvex entire solutions to the sigma-2 equation","authors":"R. Shankar, Yu Yuan","doi":"10.1215/00127094-2022-0034","DOIUrl":"https://doi.org/10.1215/00127094-2022-0034","url":null,"abstract":"We show that every general semiconvex entire solution to the sigma-2 equation is a quadratic polynomial. A decade ago, this result was shown for almost convex solutions.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46535529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-15DOI: 10.1215/00127094-2022-0061
Dimitrios Ntalampekos, Matthew Romney
We prove that any length metric space homeomorphic to a 2-manifold with boundary, also called a length surface, is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. As an application, using the classical uniformization theorem for Riemann surfaces and a limiting argument, we establish a general"one-sided"quasiconformal uniformization theorem for length surfaces with locally finite Hausdorff 2-measure. Our approach yields a new proof of the Bonk-Kleiner theorem characterizing Ahlfors 2-regular quasispheres.
{"title":"Polyhedral approximation of metric surfaces and applications to uniformization","authors":"Dimitrios Ntalampekos, Matthew Romney","doi":"10.1215/00127094-2022-0061","DOIUrl":"https://doi.org/10.1215/00127094-2022-0061","url":null,"abstract":"We prove that any length metric space homeomorphic to a 2-manifold with boundary, also called a length surface, is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. As an application, using the classical uniformization theorem for Riemann surfaces and a limiting argument, we establish a general\"one-sided\"quasiconformal uniformization theorem for length surfaces with locally finite Hausdorff 2-measure. Our approach yields a new proof of the Bonk-Kleiner theorem characterizing Ahlfors 2-regular quasispheres.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43653453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-02DOI: 10.1215/00127094-2022-0016
Andrew Putman, A. Snowden
We prove that the Steinberg representation of a connected reductive group over an infinite field is irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis.
{"title":"The Steinberg representation is irreducible","authors":"Andrew Putman, A. Snowden","doi":"10.1215/00127094-2022-0016","DOIUrl":"https://doi.org/10.1215/00127094-2022-0016","url":null,"abstract":"We prove that the Steinberg representation of a connected reductive group over an infinite field is irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44515393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-10DOI: 10.1215/00127094-2022-0060
Asaf Ferber, Matthew Kwan, A. Sah, Mehtaab Sawhney
Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of"low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $kge 3$ and $lambda>0$, an ErdH os--R'enyi random graph $Gsimmathbb{G}(n,lambda/n)$ with $n$ vertices and edge probability $lambda/n$ typically has the property that its $k$-core (its largest subgraph with minimum degree at least $k$) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for"extremely sparse'' random matrices with density $O(1/n)$. A key aspect of our proof is a technique to extract high-degree vertices and use them to"boost'' the rank, starting from approximate rank bounds obtainable from (non-quantitative) spectral convergence machinery due to Bordenave, Lelarge and Salez.
{"title":"Singularity of the k-core of a random graph","authors":"Asaf Ferber, Matthew Kwan, A. Sah, Mehtaab Sawhney","doi":"10.1215/00127094-2022-0060","DOIUrl":"https://doi.org/10.1215/00127094-2022-0060","url":null,"abstract":"Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of\"low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $kge 3$ and $lambda>0$, an ErdH os--R'enyi random graph $Gsimmathbb{G}(n,lambda/n)$ with $n$ vertices and edge probability $lambda/n$ typically has the property that its $k$-core (its largest subgraph with minimum degree at least $k$) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for\"extremely sparse'' random matrices with density $O(1/n)$. A key aspect of our proof is a technique to extract high-degree vertices and use them to\"boost'' the rank, starting from approximate rank bounds obtainable from (non-quantitative) spectral convergence machinery due to Bordenave, Lelarge and Salez.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45933639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-26DOI: 10.1215/00127094-2021-0023
S. Yamagishi
Let F ∈ Z[x1, . . . , xn] be a homogeneous form of degree d ≥ 2, and let V ∗ F denote the singular locus of the affine variety V (F ) = {z ∈ C : F (z) = 0}. In this paper, we prove the existence of integer solutions with prime coordinates to the equation F (x1, . . . , xn) = 0 provided F satisfies suitable local conditions and n − dimV ∗ F ≥ 235d(2d− 1)4. Our result improves on what was known previously due to Cook and Magyar (B. Cook and Á. Magyar, ‘Diophantine equations in the primes’. Invent. Math. 198 (2014), 701-737), which required n− dimV ∗ F to be an exponential tower in d.
{"title":"Diophantine equations in primes: Density of prime points on affine hypersurfaces","authors":"S. Yamagishi","doi":"10.1215/00127094-2021-0023","DOIUrl":"https://doi.org/10.1215/00127094-2021-0023","url":null,"abstract":"Let F ∈ Z[x1, . . . , xn] be a homogeneous form of degree d ≥ 2, and let V ∗ F denote the singular locus of the affine variety V (F ) = {z ∈ C : F (z) = 0}. In this paper, we prove the existence of integer solutions with prime coordinates to the equation F (x1, . . . , xn) = 0 provided F satisfies suitable local conditions and n − dimV ∗ F ≥ 235d(2d− 1)4. Our result improves on what was known previously due to Cook and Magyar (B. Cook and Á. Magyar, ‘Diophantine equations in the primes’. Invent. Math. 198 (2014), 701-737), which required n− dimV ∗ F to be an exponential tower in d.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47582522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-04DOI: 10.1215/00127094-2022-0074
M. Mustaţă, S. Olano, M. Popa, J. Witaszek
We study the Du Bois complex $underline{Omega}_Z^bullet$ of a hypersurface $Z$ in a smooth complex algebraic variety in terms its minimal exponent $widetilde{alpha}(Z)$. The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein-Sato polynomial of $Z$, and refining the log canonical threshold. We show that if $widetilde{alpha}(Z)geq p+1$, then the canonical morphism $Omega_Z^pto underline{Omega}_Z^p$ is an isomorphism, where $underline{Omega}_Z^p$ is the $p$-th associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, if $Z$ is singular and $widetilde{alpha}(Z)>pgeq 2$, we obtain non-vanishing results for some of the higher cohomologies of $underline{Omega}_Z^{n-p}$.
{"title":"The Du Bois complex of a hypersurface and the minimal exponent","authors":"M. Mustaţă, S. Olano, M. Popa, J. Witaszek","doi":"10.1215/00127094-2022-0074","DOIUrl":"https://doi.org/10.1215/00127094-2022-0074","url":null,"abstract":"We study the Du Bois complex $underline{Omega}_Z^bullet$ of a hypersurface $Z$ in a smooth complex algebraic variety in terms its minimal exponent $widetilde{alpha}(Z)$. The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein-Sato polynomial of $Z$, and refining the log canonical threshold. We show that if $widetilde{alpha}(Z)geq p+1$, then the canonical morphism $Omega_Z^pto underline{Omega}_Z^p$ is an isomorphism, where $underline{Omega}_Z^p$ is the $p$-th associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, if $Z$ is singular and $widetilde{alpha}(Z)>pgeq 2$, we obtain non-vanishing results for some of the higher cohomologies of $underline{Omega}_Z^{n-p}$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46285535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-30DOI: 10.1215/00127094-2022-0079
B. Yu
The purpose of this paper is to classify Anosov flows on the 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by M(r) (r is a ratioanl number), which contains the first class of hyperbolic 3-manifolds admitting Anosov flows in history, discovered by Goodman. Combining with the classification of Anosov flows on the sol-manifold M(0) due to Plante, we have: 1. if r is an integer, up to topological equivalence, M(r) exactly carries a unique Anosov flow, which is constructed by Goodman by doing a Dehn-Fried-Goodman surgery on a suspension Anosov flow; 2. if r is not an integer, M(r) does not carry any Anosov flow. As a consequence of the second result, we get infinitely many closed orientable hyperbolic 3-manifolds which carry taut foliations but does not carry any Anosov flow. The fundamental tool in the proofs is the set of branched surfaces built by Schwider, which is used to carry essential laminations on M(r).
{"title":"Anosov flows on Dehn surgeries on the figure-eight knot","authors":"B. Yu","doi":"10.1215/00127094-2022-0079","DOIUrl":"https://doi.org/10.1215/00127094-2022-0079","url":null,"abstract":"The purpose of this paper is to classify Anosov flows on the 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by M(r) (r is a ratioanl number), which contains the first class of hyperbolic 3-manifolds admitting Anosov flows in history, discovered by Goodman. Combining with the classification of Anosov flows on the sol-manifold M(0) due to Plante, we have: 1. if r is an integer, up to topological equivalence, M(r) exactly carries a unique Anosov flow, which is constructed by Goodman by doing a Dehn-Fried-Goodman surgery on a suspension Anosov flow; 2. if r is not an integer, M(r) does not carry any Anosov flow. As a consequence of the second result, we get infinitely many closed orientable hyperbolic 3-manifolds which carry taut foliations but does not carry any Anosov flow. The fundamental tool in the proofs is the set of branched surfaces built by Schwider, which is used to carry essential laminations on M(r).","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42019788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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