Let $D 0$ is an absolute positive constant independent of $D$. More generally, let $K$ be a bounded degree number field over $mathbb{Q}$ with the discriminant $D_K$ and the class number $h_K.$ We conjecture that a positive proportion of the ideal classes of $K$ contain a prime ideal with a norm less than $h_Klog(|D_K|)$.
{"title":"The least prime number represented by a binary quadratic form","authors":"Naser Talebizadeh Sardari","doi":"10.4171/jems/1031","DOIUrl":"https://doi.org/10.4171/jems/1031","url":null,"abstract":"Let $D 0$ is an absolute positive constant independent of $D$. More generally, let $K$ be a bounded degree number field over $mathbb{Q}$ with the discriminant $D_K$ and the class number $h_K.$ We conjecture that a positive proportion of the ideal classes of $K$ contain a prime ideal with a norm less than $h_Klog(|D_K|)$.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"76 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83852736","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}
The main result of this paper is that when $M_0$, $M_1$ are two simply connected spin manifolds of the same dimension $d geq 5$ which both admit a metric of positive scalar curvature, the spaces $mathcal{R}^+(M_0)$ and $mathcal{R}^+(M_1)$ of such metrics are homotopy equivalent. This supersedes a previous result of Chernysh and Walsh which gives the same conclusion when $M_0$ and $M_1$ are also spin cobordant. We also prove an analogous result for simply connected manifolds which do not admit a spin structure; we need to assume that $d neq 8$ in that case.
{"title":"On the homotopy type of the space of metrics of positive scalar curvature","authors":"Johannes Ebert, M. Wiemeler","doi":"10.4171/JEMS/1333","DOIUrl":"https://doi.org/10.4171/JEMS/1333","url":null,"abstract":"The main result of this paper is that when $M_0$, $M_1$ are two simply connected spin manifolds of the same dimension $d geq 5$ which both admit a metric of positive scalar curvature, the spaces $mathcal{R}^+(M_0)$ and $mathcal{R}^+(M_1)$ of such metrics are homotopy equivalent. This supersedes a previous result of Chernysh and Walsh which gives the same conclusion when $M_0$ and $M_1$ are also spin cobordant. \u0000We also prove an analogous result for simply connected manifolds which do not admit a spin structure; we need to assume that $d neq 8$ in that case.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"7 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90094742","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}
Let $U_q'(mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $mathcal{C}_{mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $mathcal{C}_{mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Lambda$ and $Lambda^infty$ introduced by the authors. We next define the reflections $mathcal{S}_k$ and $mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp. complete) duality datum, then $mathcal{S}_k(D)$ and $mathcal{S}_k^{-1}(D)$ are also strong (resp. complete ) duality data. We finally introduce the notion of affine cuspidal modules in $mathcal{C}_{mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.
{"title":"PBW theory for quantum affine algebras","authors":"M. Kashiwara, Myungho Kim, Se-jin Oh, E. Park","doi":"10.4171/jems/1323","DOIUrl":"https://doi.org/10.4171/jems/1323","url":null,"abstract":"Let $U_q'(mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $mathcal{C}_{mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $mathcal{C}_{mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Lambda$ and $Lambda^infty$ introduced by the authors. We next define the reflections $mathcal{S}_k$ and $mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp. complete) duality datum, then $mathcal{S}_k(D)$ and $mathcal{S}_k^{-1}(D)$ are also strong (resp. complete ) duality data. We finally introduce the notion of affine cuspidal modules in $mathcal{C}_{mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"31 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89140468","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}
L. Manivel, M. Michałek, Leonid Monin, Tim Seynnaeve, Martin Vodivcka
We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.
{"title":"Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming","authors":"L. Manivel, M. Michałek, Leonid Monin, Tim Seynnaeve, Martin Vodivcka","doi":"10.4171/jems/1330","DOIUrl":"https://doi.org/10.4171/jems/1330","url":null,"abstract":"We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"63 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80232001","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}
Let S be a Gorenstein local ring and suppose that M is a finitely generated Cohen-Macaulay S-module of codimension c. Given a regular sequence f1, . . . , fc in the annihilator of M we set R = S/(f1, . . . , fc) and construct layered S-free and R-free resolutions of M . The construction inductively reduces the problem to the case of a Cohen-Macaulay module of codimension c 1 and leads to the inductive construction of a higher matrix factorization for M . In the case where M is a su ciently high R-syzygy of some module of finite projective dimension over S, the layered resolutions are minimal and coincide with the resolutions defined from higher matrix factorizations we described in [EP].
{"title":"Layered resolutions of Cohen–Macaulay modules","authors":"D. Eisenbud, I. Peeva","doi":"10.4171/jems/1024","DOIUrl":"https://doi.org/10.4171/jems/1024","url":null,"abstract":"Let S be a Gorenstein local ring and suppose that M is a finitely generated Cohen-Macaulay S-module of codimension c. Given a regular sequence f1, . . . , fc in the annihilator of M we set R = S/(f1, . . . , fc) and construct layered S-free and R-free resolutions of M . The construction inductively reduces the problem to the case of a Cohen-Macaulay module of codimension c 1 and leads to the inductive construction of a higher matrix factorization for M . In the case where M is a su ciently high R-syzygy of some module of finite projective dimension over S, the layered resolutions are minimal and coincide with the resolutions defined from higher matrix factorizations we described in [EP].","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"89 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91256786","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}
In this paper we continue our investigation of the potential theory of Markov processes with jump kernels decaying at the boundary. To be more precise, we consider processes in ${mathbb R}^d_+$ with jump kernels of the form ${mathcal B}(x,y) |x-y|^{-d-alpha}$ and killing potentials $kappa(x)=cx_d^{-alpha}$, $0
{"title":"Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary","authors":"P. Kim, R. Song, Z. Vondravcek","doi":"10.4171/jems/1322","DOIUrl":"https://doi.org/10.4171/jems/1322","url":null,"abstract":"In this paper we continue our investigation of the potential theory of Markov processes with jump kernels decaying at the boundary. To be more precise, we consider processes in ${mathbb R}^d_+$ with jump kernels of the form ${mathcal B}(x,y) |x-y|^{-d-alpha}$ and killing potentials $kappa(x)=cx_d^{-alpha}$, $0<alpha<2$. The boundary part ${mathcal B}(x,y)$ is comparable to the product of three terms with parameters $beta_1, beta_2$ and $beta_3$ appearing as exponents in these terms. The constant $c$ in the killing term can be written as a function of $alpha$, ${mathcal B}$ and a parameter $pin ((alpha-1)_+, alpha+beta_1)$, which is strictly increasing in $p$, decreasing to $0$ as $pdownarrow (alpha-1)_+$ and increasing to $infty$ as $puparrowalpha+beta_1$. We establish sharp two-sided estimates on the Green functions of these processes for all $pin ((alpha-1)_+, alpha+beta_1)$ and all admissible values of $beta_1, beta_2$ and $beta_3$. Depending on the regions where $beta_1$, $beta_2$ and $p$ belong, the estimates on the Green functions are different. In fact, the estimates have three different forms depending on the regions the parameters belong to. As applications, we prove that the boundary Harnack principle holds in certain region of the parameters and fails in some other region of the parameters. Combined with the main results of cite{KSV},we completely determine the region of the parameters where the boundary Harnack principle holds.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"64 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90902962","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}
We describe a time-dependent functional involving the relative entropy and the $dot{H}^1$ seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functionial sheds light on the competition between the dissipation and the nonlinearity for this equation. It enables to obtain new results concerning regularity/blowup issues for the Landau equation with Coulomb potential.
{"title":"A new monotonicity formula for the spatially homogeneous Landau equation with Coulomb potential and its applications","authors":"L. Desvillettes, Lingbing He, Jin-Cheng Jiang","doi":"10.4171/jems/1313","DOIUrl":"https://doi.org/10.4171/jems/1313","url":null,"abstract":"We describe a time-dependent functional involving the relative entropy and the $dot{H}^1$ seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functionial sheds light on the competition between the dissipation and the nonlinearity for this equation. It enables to obtain new results concerning regularity/blowup issues for the Landau equation with Coulomb potential.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"94 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80355364","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}
In this paper we study the dynamics of an incompressible viscous fluid evolving in an open-top container in two dimensions. The fluid mechanics are dictated by the Navier-Stokes equations. The upper boundary of the fluid is free and evolves within the container. The fluid is acted upon by a uniform gravitational field, and capillary forces are accounted for along the free boundary. The triple-phase interfaces where the fluid, air above the vessel, and solid vessel wall come in contact are called contact points, and the angles formed at the contact point are called contact angles. The model that we consider integrates boundary conditions that allow for full motion of the contact points and angles. Equilibrium configurations consist of quiescent fluid within a domain whose upper boundary is given as the graph of a function minimizing a gravity-capillary energy functional, subject to a fixed mass constraint. The equilibrium contact angles can take on any values between $0$ and $pi$ depending on the choice of capillary parameters. The main thrust of the paper is the development of a scheme of a priori estimates that show that solutions emanating from data sufficiently close to the equilibrium exist globally in time and decay to equilibrium at an exponential rate.
{"title":"Stability of contact lines in fluids: 2D Navier–Stokes flow","authors":"Yan Guo, Ian Tice","doi":"10.4171/jems/1312","DOIUrl":"https://doi.org/10.4171/jems/1312","url":null,"abstract":"In this paper we study the dynamics of an incompressible viscous fluid evolving in an open-top container in two dimensions. The fluid mechanics are dictated by the Navier-Stokes equations. The upper boundary of the fluid is free and evolves within the container. The fluid is acted upon by a uniform gravitational field, and capillary forces are accounted for along the free boundary. The triple-phase interfaces where the fluid, air above the vessel, and solid vessel wall come in contact are called contact points, and the angles formed at the contact point are called contact angles. The model that we consider integrates boundary conditions that allow for full motion of the contact points and angles. Equilibrium configurations consist of quiescent fluid within a domain whose upper boundary is given as the graph of a function minimizing a gravity-capillary energy functional, subject to a fixed mass constraint. The equilibrium contact angles can take on any values between $0$ and $pi$ depending on the choice of capillary parameters. The main thrust of the paper is the development of a scheme of a priori estimates that show that solutions emanating from data sufficiently close to the equilibrium exist globally in time and decay to equilibrium at an exponential rate.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"17 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81654907","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}
We study a generalization of the Harish-Chandra-Itzykson-Zuber integral to tensors and its expansion over trace-invariants of the two external tensors. This gives rise to natural generalizations of monotone double Hurwitz numbers, which count certain families of constellations. We find an expression of these numbers in terms of monotone Hurwitz numbers, thereby also providing expressions for monotone double Hurwitz numbers of arbitrary genus in terms of the single ones. We give an interpretation of the different combinatorial quantities at play in terms of enumeration of nodal surfaces.
{"title":"The tensor Harish-Chandra–Itzykson–Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers","authors":"B. Collins, R. Gurau, L. Lionni","doi":"10.4171/JEMS/1315","DOIUrl":"https://doi.org/10.4171/JEMS/1315","url":null,"abstract":"We study a generalization of the Harish-Chandra-Itzykson-Zuber integral to tensors and its expansion over trace-invariants of the two external tensors. This gives rise to natural generalizations of monotone double Hurwitz numbers, which count certain families of constellations. We find an expression of these numbers in terms of monotone Hurwitz numbers, thereby also providing expressions for monotone double Hurwitz numbers of arbitrary genus in terms of the single ones. We give an interpretation of the different combinatorial quantities at play in terms of enumeration of nodal surfaces.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"17 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84180697","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}
We develop techniques to construct explicit symplectic Lefschetz fibrations over the 2-sphere with any prescribed signature and any spin type when the signature is divisible by 16. This solves a long-standing conjecture on the existence of such fibrations with positive signature. As applications, we produce symplectic 4-manifolds that are homeomorphic but not diffeomorphic to connected sums of S^2 x S^2, with the smallest topology known to date, as well as larger examples as symplectic Lefschetz fibrations.
我们开发了在2球上构造具有任意指定签名和任意自旋类型的显辛Lefschetz纤振的技术，当签名可被16整除时。这就解决了一个长期存在的关于这种具有正特征的振动存在的猜想。作为应用，我们得到了与S^2 x S^2的连通和同胚但不微分同态的辛4流形，具有迄今为止已知的最小拓扑，以及辛Lefschetz纤振等更大的例子。
{"title":"Lefschetz fibrations with arbitrary signature","authors":"R. Baykur, Noriyuki Hamada","doi":"10.4171/jems/1326","DOIUrl":"https://doi.org/10.4171/jems/1326","url":null,"abstract":"We develop techniques to construct explicit symplectic Lefschetz fibrations over the 2-sphere with any prescribed signature and any spin type when the signature is divisible by 16. This solves a long-standing conjecture on the existence of such fibrations with positive signature. As applications, we produce symplectic 4-manifolds that are homeomorphic but not diffeomorphic to connected sums of S^2 x S^2, with the smallest topology known to date, as well as larger examples as symplectic Lefschetz fibrations.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"3 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78853361","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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