Duke Mathematical Journal Articles (Project Euclid)
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The latest articles from Duke Mathematical Journal on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 01 Jun 2011 09:20 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Crofton measures and Minkowski valuations
http://projecteuclid.org/euclid.dmj/1279140505
<strong>Franz E. Schuster</strong><p><strong>Source: </strong>Duke Math. J., Volume 154, Number 1, 1--30.</p><p><strong>Abstract:</strong><br/>
A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations </p>projecteuclid.org/euclid.dmj/1279140505_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTSpectral asymptotics for sub-Riemannian Laplacians, I: Quantum ergodicity and quantum limits in the 3-dimensional contact casehttps://projecteuclid.org/euclid.dmj/1510304420<strong>Yves Colin de Verdière</strong>, <strong>Luc Hillairet</strong>, <strong>Emmanuel Trélat</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 1, 109--174.</p><p><strong>Abstract:</strong><br/> This is the first paper of a series in which we plan to study spectral asymptotics for sub-Riemannian (sR) Laplacians and to extend results that are classical in the Riemannian case concerning Weyl measures, quantum limits, quantum ergodicity, quasimodes, and trace formulae. Even if hypoelliptic operators have been well studied from the point of view of PDEs, global geometrical and dynamical aspects have not been the subject of much attention. As we will see, already in the simplest case, the statements of the results in the sR setting are quite different from those in the Riemannian one. Let us consider an sR metric on a closed $3$ -dimensional manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We establish a quantum ergodicity (QE) theorem for the eigenfunctions of any associated sR Laplacian under the assumption that the Reeb flow is ergodic. The limit measure is given by the normalized Popp measure. This is the first time that such a result is established for a hypoelliptic operator, whereas the usual Shnirelman theorem yields QE for the Laplace–Beltrami operator on a closed Riemannian manifold with ergodic geodesic flow. To prove our theorem, we first establish a microlocal Weyl law, which allows us to identify the limit measure and to prove the microlocal concentration of the eigenfunctions on the characteristic manifold of the sR Laplacian. Then, we derive a Birkhoff normal form along this characteristic manifold, thus showing that, in some sense, all $3$ -dimensional contact structures are microlocally equivalent. The quantum version of this normal form provides a useful microlocal factorization of the sR Laplacian. Using the normal form, the factorization, and the ergodicity assumption, we finally establish a variance estimate, from which QE follows. We also obtain a second result, which is valid without any ergodicity assumption: every quantum limit (QL) can be decomposed in a sum of two mutually singular measures: the first measure is supported on the unit cotangent bundle and is invariant under the sR geodesic flow, and the second measure is supported on the characteristic manifold of the sR Laplacian and is invariant under the lift of the Reeb flow. Moreover, we prove that the first measure is zero for most QLs. </p>projecteuclid.org/euclid.dmj/1510304420_20180104040138Thu, 04 Jan 2018 04:01 ESTOn the marked length spectrum of generic strictly convex billiard tableshttps://projecteuclid.org/euclid.dmj/1512702098<strong>Guan Huang</strong>, <strong>Vadim Kaloshin</strong>, <strong>Alfonso Sorrentino</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 1, 175--209.</p><p><strong>Abstract:</strong><br/>
In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard map from the (maximal) marked length spectrum of the domain.
</p>projecteuclid.org/euclid.dmj/1512702098_20180104040138Thu, 04 Jan 2018 04:01 ESTThe Abelianization of the real Cremona grouphttps://projecteuclid.org/euclid.dmj/1513998141<strong>Susanna Zimmermann</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 2, 211--267.</p><p><strong>Abstract:</strong><br/>
We present the Abelianization of the group of birational transformations of $\mathbb{P}^{2}_{\mathbb{R}}$ .
</p>projecteuclid.org/euclid.dmj/1513998141_20180129040341Mon, 29 Jan 2018 04:03 ESTCurrent fluctuations of the stationary ASEP and six-vertex modelhttps://projecteuclid.org/euclid.dmj/1515812504<strong>Amol Aggarwal</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 2, 269--384.</p><p><strong>Abstract:</strong><br/> Our results in this article are twofold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time $T$ , and show that they are of order $T^{1/3}$ along a characteristic line. Upon scaling by $T^{1/3}$ , we establish that these fluctuations converge to the long-time height fluctuations of the stationary Kardar–Parisi–Zhang (KPZ) equation, that is, to the Baik–Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and Spohn from 2005 (when they established the same result for the TASEP) and the work of Balázs and Seppäläinen from 2010 (when they established the $T^{1/3}$ -scaling for the general ASEP). Second, we introduce a class of translation-invariant Gibbs measures that characterizes a one-parameter family of slopes for an arbitrary ferroelectric, symmetric six-vertex model. This family of slopes corresponds to what is known as the conical singularity (or tricritical point ) in the free-energy profile for the ferroelectric six-vertex model. We consider fluctuations of the height function of this model on a large grid of size $T$ and show that they too are of order $T^{1/3}$ along a certain characteristic line; this confirms a prediction of Bukman and Shore from 1995, stating that the ferroelectric six-vertex model should exhibit KPZ growth at the conical singularity. Upon scaling the height fluctuations by $T^{1/3}$ , we again recover the Baik–Rains distribution in the large $T$ limit. Recasting this statement in terms of the (asymmetric) stochastic six-vertex model confirms a prediction of Gwa and Spohn from 1992. </p>projecteuclid.org/euclid.dmj/1515812504_20180129040341Mon, 29 Jan 2018 04:03 ESTTotaro’s question on zero-cycles on torsorshttps://projecteuclid.org/euclid.dmj/1513998140<strong>R. Gordon-Sarney</strong>, <strong>V. Suresh</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 2, 385--395.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a smooth connected linear algebraic group, and let $X$ be a $G$ -torsor. Totaro asked: If $X$ admits a zero-cycle of degree $d\geq1$ , then does $X$ have a closed étale point of degree dividing $d$ ? While the literature contains affirmative answers in some special cases, we give examples to show that the answer is negative in general.
</p>projecteuclid.org/euclid.dmj/1513998140_20180129040341Mon, 29 Jan 2018 04:03 ESTThe colored HOMFLYPT function is $q$ -holonomichttps://projecteuclid.org/euclid.dmj/1510304421<strong>Stavros Garoufalidis</strong>, <strong>Aaron D. Lauda</strong>, <strong>Thang T. Q. Lê</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 3, 397--447.</p><p><strong>Abstract:</strong><br/>
We prove that the HOMFLYPT polynomial of a link colored by partitions with a fixed number of rows is a $q$ -holonomic function. By specializing to the case of knots colored by a partition with a single row, it proves the existence of an $(a,q)$ superpolynomial of knots in $3$ -space, as was conjectured by string theorists. Our proof uses skew-Howe duality that reduces the evaluation of web diagrams and their ladders to a Poincaré–Birkhoff–Witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram. The result is a concrete and algorithmic web evaluation algorithm that is manifestly $q$ -holonomic.
</p>projecteuclid.org/euclid.dmj/1510304421_20180209220044Fri, 09 Feb 2018 22:00 ESTCanonical growth conditions associated to ample line bundleshttps://projecteuclid.org/euclid.dmj/1515143006<strong>David Witt Nyström</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 3, 449--495.</p><p><strong>Abstract:</strong><br/>
We propose a new construction which associates to any ample (or big) line bundle $L$ on a projective manifold $X$ a canonical growth condition (i.e., a choice of a plurisubharmonic (psh) function well defined up to a bounded term) on the tangent space $T_{p}X$ of any given point $p$ . We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows one to recover all the infinitesimal Okounkov bodies of $L$ at $p$ . The construction is inspired by toric geometry and the theory of Okounkov bodies; in the toric case, the growth condition is “equivalent” to the moment polytope. As in the toric case, the growth condition says a lot about the Kähler geometry of the manifold. We prove a theorem about Kähler embeddings of large balls, which generalizes the well-known connection between Seshadri constants and Gromov width established by McDuff and Polterovich.
</p>projecteuclid.org/euclid.dmj/1515143006_20180209220044Fri, 09 Feb 2018 22:00 ESTCarathéodory’s metrics on Teichmüller spaces and $L$ -shaped pillowcaseshttps://projecteuclid.org/euclid.dmj/1516762971<strong>Vladimir Markovic</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 3, 497--535.</p><p><strong>Abstract:</strong><br/>
One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each $\mathcal{T}_{g}$ , the Teichmüller space of a closed surface of genus $g\ge2$ . The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential $\varphi$ . First, we construct a holomorphic embedding $\mathcal{E}:\mathbb{H}^{k}\to\mathcal{T}_{g,n}$ corresponding to $\varphi$ . The criterion says that the two metrics agree on this disk if and only if a certain function $\mathbf{\Phi}:\mathcal{E}(\mathbb{H}^{k})\to\mathbb{H}$ can be extended to a holomorphic function $\mathbf{\Phi}:\mathcal{T}_{g,n}\to\mathbb{H}$ . We then show by explicit computation that this is not the case for quadratic differentials arising from $L$ -shaped pillowcases.
</p>projecteuclid.org/euclid.dmj/1516762971_20180209220044Fri, 09 Feb 2018 22:00 ESTGroups quasi-isometric to right-angled Artin groupshttps://projecteuclid.org/euclid.dmj/1515467194<strong>Jingyin Huang</strong>, <strong>Bruce Kleiner</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 3, 537--602.</p><p><strong>Abstract:</strong><br/>
We characterize groups quasi-isometric to a right-angled Artin group (RAAG) $G$ with finite outer automorphism group. In particular, all such groups admit a geometric action on a $\operatorname{CAT}(0)$ cube complex that has an equivariant “fibering” over the Davis building of $G$ . This characterization will be used in forthcoming work of the first author to give a commensurability classification of the groups quasi-isometric to certain RAAGs.
</p>projecteuclid.org/euclid.dmj/1515467194_20180209220044Fri, 09 Feb 2018 22:00 ESTThe Breuil–Mézard conjecture when $l\neq p$https://projecteuclid.org/euclid.dmj/1513998139<strong>Jack Shotton</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 4, 603--678.</p><p><strong>Abstract:</strong><br/> Let $l$ and $p$ be primes, let $F/\mathbb{Q}_{p}$ be a finite extension with absolute Galois group $G_{F}$ , let $\mathbb{F}$ be a finite field of characteristic $l$ , and let \[\overline{\rho}:G_{F}\rightarrow \operatorname{GL}_{n}(\mathbb{F})\] be a continuous representation. Let $R^{\square}(\overline{\rho})$ be the universal framed deformation ring for $\overline{\rho}$ . If $l=p$ , then the Breuil–Mézard conjecture (as recently formulated by Emerton and Gee) relates the mod $l$ reduction of certain cycles in $R^{\square}(\overline{\rho})$ to the mod $l$ reduction of certain representations of $\operatorname{GL}_{n}(\mathcal{O}_{F})$ . We state an analogue of the Breuil–Mézard conjecture when $l\neq p$ , and we prove it whenever $l\gt 2$ using automorphy lifting theorems. We give a local proof when $l$ is “quasibanal” for $F$ and $\overline{\rho}$ is tamely ramified. We also analyze the reduction modulo $l$ of the types $\sigma(\tau)$ defined by Schneider and Zink. </p>projecteuclid.org/euclid.dmj/1513998139_20180301040030Thu, 01 Mar 2018 04:00 EST$\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of stratahttps://projecteuclid.org/euclid.dmj/1517281370<strong>Paul Apisa</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 4, 679--742.</p><p><strong>Abstract:</strong><br/>
The object of this article is to study $\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of strata of Abelian differentials. The main result is that all higher-rank affine invariant submanifolds in hyperelliptic components are branched covering constructions; that is, every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of Abelian differentials. This result implies the finiteness of algebraically primitive Teichmüller curves in all hyperelliptic components for genus greater than two. A classification of all $\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided. Our main theorem resolves a pair of conjectures of Mirzakhani in the case of hyperelliptic components of moduli space.
</p>projecteuclid.org/euclid.dmj/1517281370_20180301040030Thu, 01 Mar 2018 04:00 ESTA $p$ -adic Waldspurger formulahttps://projecteuclid.org/euclid.dmj/1518166812<strong>Yifeng Liu</strong>, <strong>Shouwu Zhang</strong>, <strong>Wei Zhang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 4, 743--833.</p><p><strong>Abstract:</strong><br/>
In this article, we study $p$ -adic torus periods for certain $p$ -adic-valued functions on Shimura curves of classical origin. We prove a $p$ -adic Waldspurger formula for these periods as a generalization of recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic $p$ -adic $L$ -function of Rankin–Selberg type. At a character of positive weight, the $p$ -adic $L$ -function interpolates the central critical value of the complex Rankin–Selberg $L$ -function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding $p$ -adic torus period.
</p>projecteuclid.org/euclid.dmj/1518166812_20180301040030Thu, 01 Mar 2018 04:00 ESTPicard–Lefschetz oscillators for the Drinfeld–Lafforgue–Vinberg degeneration for $\operatorname{SL}_{2}$https://projecteuclid.org/euclid.dmj/1520906430<strong>Simon Schieder</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 5, 835--921.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a reductive group, and let $\operatorname{Bun}_{G}$ denote the moduli stack of $G$ -bundles on a smooth projective curve. We begin the study of the singularities of a canonical compactification of $\operatorname{Bun}_{G}$ due to Drinfeld (unpublished), which we refer to as the Drinfeld–Lafforgue–Vinberg compactification $\overline{\operatorname{Bun}}_{G}$ . For $G=\operatorname{GL}_{2}$ and $G=\operatorname{GL}_{n}$ , certain smooth open substacks of this compactification have already appeared in the work of Drinfeld and Lafforgue on the Langlands correspondence for function fields. The stack $\overline{\operatorname{Bun}}_{G}$ is, however, already singular for $G=\operatorname{SL}_{2}$ ; questions about its singularities arise naturally in the geometric Langlands program, and form the topic of the present article. Drinfeld’s definition of $\overline{\operatorname{Bun}}_{G}$ for a general reductive group $G$ relies on the Vinberg semigroup of $G$ (we will study this case in a forthcoming article). In the present article we focus on the case $G=\operatorname{SL}_{2}$ , where the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli space of $\operatorname{SL}_{2}$ -bundles. We study the singularities of this one-parameter degeneration via the weight-monodromy theory of its nearby cycles. We give an explicit description of the nearby cycles sheaf together with its monodromy action in terms of certain novel perverse sheaves which we call Picard–Lefschetz oscillators and which govern the singularities of $\overline{\operatorname{Bun}}_{G}$ . We then use this description to determine the intersection cohomology sheaf of $\overline{\operatorname{Bun}}_{G}$ and other invariants of its singularities. Our proofs rely on the construction of certain local models which themselves form one-parameter families of spaces which are factorizable in the sense of Beilinson and Drinfeld. We also briefly discuss the relationship of our results for $G=\operatorname{SL}_{2}$ with the miraculous duality of Drinfeld and Gaitsgory in the geometric Langlands program, as well as two applications of our results to the classical theory on the level of functions: to Drinfeld’s and Wang’s strange invariant bilinear form on the space of automorphic forms, and to the categorification of the Bernstein asymptotics map studied by Bezrukavnikov and Kazhdan as well as by Sakellaridis and Venkatesh.
</p>projecteuclid.org/euclid.dmj/1520906430_20180330040050Fri, 30 Mar 2018 04:00 EDTA geometric characterization of toric varietieshttps://projecteuclid.org/euclid.dmj/1520046166<strong>Morgan V. Brown</strong>, <strong>James McKernan</strong>, <strong>Roberto Svaldi</strong>, <strong>Hong R. Zong</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 5, 923--968.</p><p><strong>Abstract:</strong><br/>
We prove a conjecture of Shokurov which characterizes toric varieties using log pairs.
</p>projecteuclid.org/euclid.dmj/1520046166_20180330040050Fri, 30 Mar 2018 04:00 EDTRegularization under diffusion and anticoncentration of the information contenthttps://projecteuclid.org/euclid.dmj/1515747886<strong>Ronen Eldan</strong>, <strong>James R. Lee</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 5, 969--993.</p><p><strong>Abstract:</strong><br/>
Under the Ornstein–Uhlenbeck semigroup $\{U_{t}\}$ , any nonnegative measurable $f:\mathbb{R}^{n}\to\mathbb{R}_{+}$ exhibits a uniform tail bound better than that implied by Markov’s inequality and conservation of mass. For every $\alpha\geq e^{3}$ , and $t\gt 0$ ,
\[\gamma_{n}(\{x\in\mathbb{R}^{n}:U_{t}f(x)\gt \alpha\int f\,d\gamma_{n}\})\leq C(t)\frac{1}{\alpha}\sqrt{\frac{\log\log\alpha}{\log\alpha}},\] where $\gamma_{n}$ is the $n$ -dimensional Gaussian measure and $C(t)$ is a constant depending only on $t$ . This confirms positively the Gaussian limiting case of Talagrand’s convolution conjecture (1989). This is shown to follow from a more general phenomenon. Suppose that $f:\mathbb{R}^{n}\to\mathbb{R}_{+}$ is semi-log-convex in the sense that for some $\beta\gt 0$ , for all $x\in\mathbb{R}^{n}$ , the eigenvalues of $\nabla^{2}\log f(x)$ are at least $-\beta$ . Then $f$ satisfies a tail bound asymptotically better than that implied by Markov’s inequality.
</p>projecteuclid.org/euclid.dmj/1515747886_20180330040050Fri, 30 Mar 2018 04:00 EDTOdd degree number fields with odd class numberhttps://projecteuclid.org/euclid.dmj/1520046167<strong>Wei Ho</strong>, <strong>Arul Shankar</strong>, <strong>Ila Varma</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 5, 995--1047.</p><p><strong>Abstract:</strong><br/>
For every odd integer $n\geq3$ , we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_{n}$ whose class number is odd. To do so, we study the class groups of families of number fields of degree $n$ whose rings of integers arise as the coordinate rings of the subschemes of ${\mathbb{P}}^{1}$ cut out by integral binary $n$ -ic forms. By obtaining upper bounds on the mean number of $2$ -torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to $1$ as $n$ tends to $\infty$ ) of such fields have trivial $2$ -torsion subgroups in their class groups and narrow class groups. Conditional on a tail estimate, we also prove the corresponding lower bounds and obtain the exact values of these averages, which are consistent with the heuristics of Cohen and Lenstra, Cohen and Martinet, Malle, and Dummit and Voight. Additionally, for any order ${\mathcal{O}}_{f}$ of degree $n$ arising from an integral binary $n$ -ic form $f$ , we compare the sizes of $\operatorname{Cl}_{2}({\mathcal{O}}_{f})$ , the $2$ -torsion subgroup of ideal classes in ${\mathcal{O}}_{f}$ , and of ${\mathcal{I}}_{2}({\mathcal{O}}_{f})$ , the $2$ -torsion subgroup of ideals in ${\mathcal{O}}_{f}$ . For the family of orders arising from integral binary $n$ -ic forms and contained in fields with fixed signature $(r_{1},r_{2})$ , we prove that the mean value of the difference $\vert \operatorname{Cl}_{2}({\mathcal{O}}_{f})\vert -{2^{1-r_{1}-r_{2}}}\vert {\mathcal{I}}_{2}({\mathcal{O}}_{f})\vert $ is equal to $1$ , generalizing a result of Bhargava and the third-named author for cubic fields. Conditional on certain tail estimates, we also prove that the mean value of $\vert \operatorname{Cl}_{2}({\mathcal{O}}_{f})\vert -{2^{1-r_{1}-r_{2}}}\vert {\mathcal{I}}_{2}({\mathcal{O}}_{f})\vert $ remains $1$ for certain families obtained by imposing local splitting and maximality conditions.
</p>projecteuclid.org/euclid.dmj/1520046167_20180330040050Fri, 30 Mar 2018 04:00 EDTGroup cubizationhttps://projecteuclid.org/euclid.dmj/1520499610<strong>Damian Osajda</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 6, 1049--1055.</p><p><strong>Abstract:</strong><br/>
We present a procedure of group cubization: it results in a group whose features resemble some of those of a given group and which acts without fixed points on a $\operatorname{CAT}(0)$ cubical complex. As a main application, we establish the lack of Kazhdan’s property (T) for Burnside groups.
</p>projecteuclid.org/euclid.dmj/1520499610_20180413040030Fri, 13 Apr 2018 04:00 EDTGalois and Cartan cohomology of real groupshttps://projecteuclid.org/euclid.dmj/1520928011<strong>Jeffrey Adams</strong>, <strong>Olivier Taïbi</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 6, 1057--1097.</p><p><strong>Abstract:</strong><br/>
Suppose that $G$ is a complex, reductive algebraic group. A real form of $G$ is an antiholomorphic involutive automorphism $\sigma$ , so $G(\mathbb{R})=G(\mathbb{C})^{\sigma}$ is a real Lie group. Write $H^{1}(\sigma,G)$ for the Galois cohomology (pointed) set $H^{1}(\operatorname{Gal}(\mathbb{C}/\mathbb{R}),G)$ . A Cartan involution for $\sigma$ is an involutive holomorphic automorphism $\theta$ of $G$ , commuting with $\sigma$ , so that $\theta\sigma$ is a compact real form of $G$ . Let $H^{1}(\theta,G)$ be the set $H^{1}(\mathbb{Z}_{2},G)$ , where the action of the nontrivial element of $\mathbb{Z}_{2}$ is by $\theta$ . By analogy with the Galois group, we refer to $H^{1}(\theta,G)$ as the Cartan cohomology of $G$ with respect to $\theta$ . Cartan’s classification of real forms of a connected group, in terms of their maximal compact subgroups, amounts to an isomorphism $H^{1}(\sigma,G_{\mathrm{ad}})\simeq H^{1}(\theta,G_{\mathrm{ad}})$ , where $G_{\mathrm{ad}}$ is the adjoint group. Our main result is a generalization of this: there is a canonical isomorphism $H^{1}(\sigma,G)\simeq H^{1}(\theta,G)$ .
We apply this result to give simple proofs of some well-known structural results: the Kostant–Sekiguchi correspondence of nilpotent orbits; Matsuki duality of orbits on the flag variety; conjugacy classes of Cartan subgroups; and structure of the Weyl group. We also use it to compute $H^{1}(\sigma,G)$ for all simple, simply connected groups and to give a cohomological interpretation of strong real forms. For the applications it is important that we do not assume that $G$ is connected.
</p>projecteuclid.org/euclid.dmj/1520928011_20180413040030Fri, 13 Apr 2018 04:00 EDTAlmost sure multifractal spectrum of Schramm–Loewner evolutionhttps://projecteuclid.org/euclid.dmj/1522224103<strong>Ewain Gwynne</strong>, <strong>Jason Miller</strong>, <strong>Xin Sun</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 6, 1099--1237.</p><p><strong>Abstract:</strong><br/>
Suppose that $\eta$ is a Schramm–Loewner evolution ( $\operatorname{SLE}_{\kappa}$ ) in a smoothly bounded simply connected domain $D\subset{\mathbf{C}}$ and that $\phi$ is a conformal map from $\mathbf{D}$ to a connected component of $D\setminus\eta([0,t])$ for some $t\gt 0$ . The multifractal spectrum of $\eta$ is the function $(-1,1)\to[0,\infty)$ which, for each $s\in(-1,1)$ , gives the Hausdorff dimension of the set of points $x\in\partial\mathbf{D}$ such that $|\phi'((1-\epsilon)x)|=\epsilon^{-s+o(1)}$ as $\epsilon\to0$ . We rigorously compute the almost sure multifractal spectrum of $\operatorname{SLE}$ , confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the almost sure bulk integral means spectrum of $\operatorname{SLE}$ , we obtain the optimal Hölder exponent for a conformal map which uniformizes the complement of an $\operatorname{SLE}$ curve, and we obtain a new derivation of the almost sure Hausdorff dimension of the $\operatorname{SLE}$ curve for $\kappa\leq4$ . Our results also hold for the $\operatorname{SLE}_{\kappa}(\underline{\rho})$ processes with general vectors of weight $\underline{\rho}$ .
</p>projecteuclid.org/euclid.dmj/1522224103_20180413040030Fri, 13 Apr 2018 04:00 EDTIntegration of oscillatory and subanalytic functionshttps://projecteuclid.org/euclid.dmj/1521014410<strong>Raf Cluckers</strong>, <strong>Georges Comte</strong>, <strong>Daniel J. Miller</strong>, <strong>Jean-Philippe Rolin</strong>, <strong>Tamara Servi</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 7, 1239--1309.</p><p><strong>Abstract:</strong><br/>
We prove the stability under integration and under Fourier transform of a concrete class of functions containing all globally subanalytic functions and their complex exponentials. This article extends the investigation started by Lion and Rolin and Cluckers and Miller to an enriched framework including oscillatory functions. It provides a new example of fruitful interaction between analysis and singularity theory.
</p>projecteuclid.org/euclid.dmj/1521014410_20180507220144Mon, 07 May 2018 22:01 EDTThe critical height is a moduli heighthttps://projecteuclid.org/euclid.dmj/1520046165<strong>Patrick Ingram</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 7, 1311--1346.</p><p><strong>Abstract:</strong><br/>
Silverman defined the critical height of a rational function $f(z)$ of degree $d\geq2$ in terms of the asymptotic rate of growth of the Weil height along the critical orbits of $f$ . He also conjectured that this quantity was commensurate to an ample Weil height on the moduli space of rational functions degree $d$ . We prove that conjecture.
</p>projecteuclid.org/euclid.dmj/1520046165_20180507220144Mon, 07 May 2018 22:01 EDTMonodromy dependence and connection formulae for isomonodromic tau functionshttps://projecteuclid.org/euclid.dmj/1520586158<strong>A. R. Its</strong>, <strong>O. Lisovyy</strong>, <strong>A. Prokhorov</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 7, 1347--1432.</p><p><strong>Abstract:</strong><br/>
We discuss an extension of the Jimbo–Miwa–Ueno differential $1$ -form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola, generalizing a previous construction by B. Malgrange. We show how this $1$ -form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for the generic Painlevé VI tau function. The result proves the conjectural formula for this constant proposed by Iorgov, Lisovyy, and Tykhyy. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of the Painlevé II tau function.
</p>projecteuclid.org/euclid.dmj/1520586158_20180507220144Mon, 07 May 2018 22:01 EDTThe Sard conjecture on Martinet surfaceshttps://projecteuclid.org/euclid.dmj/1520046164<strong>André Belotto da Silva</strong>, <strong>Ludovic Rifford</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 8, 1433--1471.</p><p><strong>Abstract:</strong><br/>
Given a totally nonholonomic distribution of rank $2$ on a $3$ -dimensional manifold, we investigate the size of the set of points that can be reached by singular horizontal paths starting from the same point. In this setting, by the Sard conjecture, that set should be a subset of the so-called Martinet surface of $2$ -dimensional Hausdorff measure zero . We prove that the conjecture holds in the case where the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces, and we show that the result holds true under an assumption of nontransversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and some techniques of resolution of singularities.
</p>projecteuclid.org/euclid.dmj/1520046164_20180528220342Mon, 28 May 2018 22:03 EDTUniform rectifiability from Carleson measure estimates and $\mathbf{\varepsilon}$ -approximability of bounded harmonic functionshttps://projecteuclid.org/euclid.dmj/1525313238<strong>John Garnett</strong>, <strong>Mihalis Mourgoglou</strong>, <strong>Xavier Tolsa</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 8, 1473--1524.</p><p><strong>Abstract:</strong><br/>
Let $\Omega\subset{\mathbb{R}}^{n+1}$ , $n\geq1$ , be a corkscrew domain with Ahlfors–David regular boundary. In this article we prove that $\partial\Omega$ is uniformly $n$ -rectifiable if every bounded harmonic function on $\Omega$ is $\varepsilon$ -approximable or if every bounded harmonic function on $\Omega$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $\Omega={\mathbb{R}}^{n+1}\setminusE$ and $E$ is Ahlfors–David regular. Our results establish a conjecture posed by Hofmann, Martell, and Mayboroda, in which they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability, one in terms of the so-called $S\lt N$ estimates and another in terms of a suitable corona decomposition involving harmonic measure.
</p>projecteuclid.org/euclid.dmj/1525313238_20180528220342Mon, 28 May 2018 22:03 EDTOn the conservativity of the functor assigning to a motivic spectrum its motivehttps://projecteuclid.org/euclid.dmj/1522224100<strong>Tom Bachmann</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 8, 1525--1571.</p><p><strong>Abstract:</strong><br/>
Given a $0$ -connective motivic spectrum $E\in\mathbf{SH}(k)$ over a perfect field $k$ , we determine $\underline{h}_{0}$ of the associated motive $ME\in\mathbf{DM}(k)$ in terms of $\underline{\pi}_{0}(E)$ . Using this, we show that if $k$ has finite $2$ -étale cohomological dimension, then the functor $M:\mathbf{SH}(k)\to\mathbf{DM}(k)$ is conservative when restricted to the subcategory of compact spectra and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual $2$ -étale cohomological dimension by considering what we call real motives .
</p>projecteuclid.org/euclid.dmj/1522224100_20180528220342Mon, 28 May 2018 22:03 EDTHodge theory of classifying stackshttps://projecteuclid.org/euclid.dmj/1521684060<strong>Burt Totaro</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 8, 1573--1621.</p><p><strong>Abstract:</strong><br/>
We compute the Hodge and the de Rham cohomology of the classifying space $\mathit{BG}$ (defined as étale cohomology on the algebraic stack $\mathit{BG}$ ) for reductive groups $G$ over many fields, including fields of small characteristic. These calculations have a direct relation with representation theory, yielding new results there. The calculations are closely analogous to, but not always the same as, the cohomology of classifying spaces in topology.
</p>projecteuclid.org/euclid.dmj/1521684060_20180528220342Mon, 28 May 2018 22:03 EDTBohr sets and multiplicative Diophantine approximationhttps://projecteuclid.org/euclid.dmj/1521792023<strong>Sam Chow</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 9, 1623--1642.</p><p><strong>Abstract:</strong><br/>
In two dimensions, Gallagher’s theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fiber version of Gallagher’s theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes, and Velani. The idea is to find large generalized arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin–Schaeffer theorem for the problem at hand, via the geometry of numbers.
</p>projecteuclid.org/euclid.dmj/1521792023_20180612040040Tue, 12 Jun 2018 04:00 EDTInteger homology $3$ -spheres admit irreducible representations in $\operatorname{SL}(2,\mathbb{C})$https://projecteuclid.org/euclid.dmj/1525140014<strong>Raphael Zentner</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 9, 1643--1712.</p><p><strong>Abstract:</strong><br/>
We prove that the fundamental group of any integer homology $3$ -sphere different from the $3$ -sphere admits irreducible representations of its fundamental group in $\operatorname{SL}(2,\mathbb{C})$ . For hyperbolic integer homology spheres, this comes with the definition; for Seifert-fibered integer homology spheres, this is well known. We prove that the splicing of any two nontrivial knots in $S^{3}$ admits an irreducible $\operatorname{SU}(2)$ -representation. Using a result of Kuperberg, we get the corollary that the problem of $3$ -sphere recognition is in the complexity class $\mathsf{coNP}$ , provided the generalized Riemann hypothesis holds. To prove our result, we establish a topological fact about the image of the $\operatorname{SU}(2)$ -representation variety of a nontrivial knot complement into the representation variety of its boundary torus, a pillowcase, using holonomy perturbations of the Chern–Simons function in an exhaustive way—showing that any area-preserving self-map of the pillowcase fixing the four singular points, and which is isotopic to the identity, can be $C^{0}$ -approximated by maps realized geometrically through holonomy perturbations of the flatness equation in a thickened torus. We conclude with a stretching argument in instanton gauge theory and a nonvanishing result of Kronheimer and Mrowka for Donaldson’s invariants of a $4$ -manifold which contains the $0$ -surgery of a knot as a splitting hypersurface.
</p>projecteuclid.org/euclid.dmj/1525140014_20180612040040Tue, 12 Jun 2018 04:00 EDTOn finiteness properties of the Johnson filtrationshttps://projecteuclid.org/euclid.dmj/1525313239<strong>Mikhail Ershov</strong>, <strong>Sue He</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 9, 1713--1759.</p><p><strong>Abstract:</strong><br/>
Let $\Gamma$ be either the automorphism group of the free group of rank $n\geq4$ or the mapping class group of an orientable surface of genus $n\geq12$ with at most $1$ boundary component, and let $G$ be either the subgroup of $\mathrm{IA}$ -automorphisms or the Torelli subgroup of $\Gamma$ . For $N\in\mathbb{N}$ denote by $\gamma_{N}G$ the $N$ th term of the lower central series of $G$ . We prove that
(i) any subgroup of $G$ containing $\gamma_{2}G=[G,G]$ (in particular, the Johnson kernel in the mapping class group case) is finitely generated;
(ii) if $N=2$ or $n\geq8N-4$ and $K$ is any subgroup of $G$ containing $\gamma_{N}G$ (for instance, $K$ can be the $N$ th term of the Johnson filtration of $G$ ), then $G/[K,K]$ is nilpotent and hence the Abelianization of $K$ is finitely generated;
(iii) if $H$ is any finite-index subgroup of $\Gamma$ containing $\gamma_{N}G$ , with $N$ as in (ii), then $H$ has finite Abelianization.
</p>projecteuclid.org/euclid.dmj/1525313239_20180612040040Tue, 12 Jun 2018 04:00 EDTUniversal dynamics for the defocusing logarithmic Schrödinger equationhttps://projecteuclid.org/euclid.dmj/1526436027<strong>Rémi Carles</strong>, <strong>Isabelle Gallagher</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 9, 1761--1801.</p><p><strong>Abstract:</strong><br/>
We consider the Schrödinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time, and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data and remain when an extra power-like nonlinearity is present in the equation. One of the key steps of the proof consists in working in hydrodynamical variables to reduce the equation to a variant of the isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker–Planck operator.
</p>projecteuclid.org/euclid.dmj/1526436027_20180612040040Tue, 12 Jun 2018 04:00 EDTIndependence of $\ell$ for the supports in the decomposition theoremhttps://projecteuclid.org/euclid.dmj/1528855662<strong>Shenghao Sun</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 10, 1803--1823.</p><p><strong>Abstract:</strong><br/>
In this article, we prove a result on the independence of $\ell$ for the supports of irreducible perverse sheaves occurring in the decomposition theorem, as well as for the family of local systems on each support. It generalizes Gabber’s result on the independence of $\ell$ of intersection cohomology to the relative case.
</p>projecteuclid.org/euclid.dmj/1528855662_20180713040712Fri, 13 Jul 2018 04:07 EDTA minimization problem with free boundary related to a cooperative systemhttps://projecteuclid.org/euclid.dmj/1528185619<strong>Luis A. Caffarelli</strong>, <strong>Henrik Shahgholian</strong>, <strong>Karen Yeressian</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 10, 1825--1882.</p><p><strong>Abstract:</strong><br/>
We study the minimum problem for the functional \begin{equation*}\int_{\Omega}(\vert\nabla\mathbf{u}\vert^{2}+Q^{2}\chi_{\{\vert\mathbf{u}\vert\gt 0\}})\,dx\end{equation*} with the constraint $u_{i}\geq0$ for $i=1,\ldots,m$ , where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain and $\mathbf{u}=(u_{1},\ldots,u_{m})\in H^{1}(\Omega;\mathbb{R}^{m})$ . First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set $\{\vert u\vert\gt 0\}$ is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary $\Omega\cap\partial\{\vert u\vert\gt 0\}$ .
</p>projecteuclid.org/euclid.dmj/1528185619_20180713040712Fri, 13 Jul 2018 04:07 EDTAnalytic torsion and R-torsion of Witt representations on manifolds with cuspshttps://projecteuclid.org/euclid.dmj/1528358417<strong>Pierre Albin</strong>, <strong>Frédéric Rochon</strong>, <strong>David Sher</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 10, 1883--1950.</p><p><strong>Abstract:</strong><br/>
We establish a Cheeger–Müller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all noncompact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion, and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a closed manifold undergoing degeneration to a manifold with fibered cusps.
</p>projecteuclid.org/euclid.dmj/1528358417_20180713040712Fri, 13 Jul 2018 04:07 EDTThe $p$ -curvature conjecture and monodromy around simple closed loopshttps://projecteuclid.org/euclid.dmj/1529978488<strong>Ananth N. Shankar</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 10, 1951--1980.</p><p><strong>Abstract:</strong><br/>
The Grothendieck–Katz $p$ -curvature conjecture is an analogue of the Hasse principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$ -curvature vanishes modulo $p$ , for almost all primes $p$ . We prove that if the variety is a generic curve, then every simple closed loop on the curve has finite monodromy.
</p>projecteuclid.org/euclid.dmj/1529978488_20180713040712Fri, 13 Jul 2018 04:07 EDTQuadratic Chabauty and rational points, I: $p$ -adic heightshttps://projecteuclid.org/euclid.dmj/1532073621<strong>Jennifer S. Balakrishnan</strong>, <strong>Netan Dogra</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 11, 1981--2038.</p><p><strong>Abstract:</strong><br/>
We give the first explicit examples beyond the Chabauty–Coleman method where Kim’s nonabelian Chabauty program determines the set of rational points of a curve defined over $\mathbb{Q}$ or a quadratic number field. We accomplish this by studying the role of $p$ -adic heights in explicit non-Abelian Chabauty.
</p>projecteuclid.org/euclid.dmj/1532073621_20180808220157Wed, 08 Aug 2018 22:01 EDTExceptional isogenies between reductions of pairs of elliptic curveshttps://projecteuclid.org/euclid.dmj/1530000176<strong>François Charles</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 11, 2039--2072.</p><p><strong>Abstract:</strong><br/>
Let $E$ and $E'$ be two elliptic curves over a number field. We prove that the reductions of $E$ and $E'$ at a finite place $\mathfrak{p}$ are geometrically isogenous for infinitely many $\mathfrak{p}$ , and we draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether–Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.
</p>projecteuclid.org/euclid.dmj/1530000176_20180808220157Wed, 08 Aug 2018 22:01 EDTEnergy quantization of Willmore surfaces at the boundary of the moduli spacehttps://projecteuclid.org/euclid.dmj/1528358418<strong>Paul Laurain</strong>, <strong>Tristan Rivière</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 11, 2073--2124.</p><p><strong>Abstract:</strong><br/>
We establish an energy quantization result for sequences of Willmore surfaces when the underlying sequence of Riemann surfaces is degenerating in the moduli space. We notably exhibit a new residue which quantifies the possible loss of energy in collar regions. Thanks to this residue, we also establish the compactness (modulo the action of the Möbius group of conformal transformations of ${\mathbb{R}}^{3}\cup\{\infty\}$ ) of the space of Willmore immersions of any arbitrary closed $2$ -dimensional oriented manifold into ${\mathbb{R}}^{3}$ with uniformly bounded conformal class and energy below $12\pi$ .
</p>projecteuclid.org/euclid.dmj/1528358418_20180808220157Wed, 08 Aug 2018 22:01 EDTRigidity of critical circle mapshttps://projecteuclid.org/euclid.dmj/1531900818<strong>Pablo Guarino</strong>, <strong>Marco Martens</strong>, <strong>Welington de Melo</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 11, 2125--2188.</p><p><strong>Abstract:</strong><br/>
We prove that any two $C^{4}$ critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a $C^{1}$ circle diffeomorphism. The conjugacy is $C^{1+\alpha}$ for a full Lebesgue measure set of rotation numbers.
</p>projecteuclid.org/euclid.dmj/1531900818_20180808220157Wed, 08 Aug 2018 22:01 EDTRecognizing a relatively hyperbolic group by its Dehn fillingshttps://projecteuclid.org/euclid.dmj/1532073620<strong>François Dahmani</strong>, <strong>Vincent Guirardel</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 12, 2189--2241.</p><p><strong>Abstract:</strong><br/>
Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a noncompact hyperbolic $3$ -manifold such as hyperbolic knot complements. We prove a rigidity result saying that if two nonelementary relatively hyperbolic groups without certain splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of nonelementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.
</p>projecteuclid.org/euclid.dmj/1532073620_20180828220249Tue, 28 Aug 2018 22:02 EDTOn the maximum of the C $\beta$ E fieldhttps://projecteuclid.org/euclid.dmj/1533866573<strong>Reda Chhaibi</strong>, <strong>Thomas Madaule</strong>, <strong>Joseph Najnudel</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 12, 2243--2345.</p><p><strong>Abstract:</strong><br/>
In this article, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the circular beta ensemble (C $\beta$ E). More precisely, assuming that $X_{n}$ is this characteristic polynomial and ${\mathbb{U}}$ is the unit circle, we prove that
\[\sup_{z\in{\mathbb{U}}}\Re\log X_{n}(z)=\sqrt{\frac{2}{\beta}}(\log n-\frac{3}{4}\log\log n+\mathcal{O}(1)),\] as well as an analogous statement for the imaginary part. The notation ${\mathcal{O}}(1)$ means that the corresponding family of random variables, indexed by $n$ , is tight. This answers a conjecture of Fyodorov, Hiary, and Keating, originally formulated for the $\beta=2$ case, which corresponds to the circular unitary ensemble (CUE) field.
</p>projecteuclid.org/euclid.dmj/1533866573_20180828220249Tue, 28 Aug 2018 22:02 EDTCompactification of strata of Abelian differentialshttps://projecteuclid.org/euclid.dmj/1533866574<strong>Matt Bainbridge</strong>, <strong>Dawei Chen</strong>, <strong>Quentin Gendron</strong>, <strong>Samuel Grushevsky</strong>, <strong>Martin Möller</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 12, 2347--2416.</p><p><strong>Abstract:</strong><br/>
We describe the closure of the strata of Abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne–Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.
</p>projecteuclid.org/euclid.dmj/1533866574_20180828220249Tue, 28 Aug 2018 22:02 EDTThe Gaussian core model in high dimensionshttps://projecteuclid.org/euclid.dmj/1534233620<strong>Henry Cohn</strong>, <strong>Matthew de Courcy-Ireland</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 13, 2417--2455.</p><p><strong>Abstract:</strong><br/>
We prove lower bounds for energy in the Gaussian core model, in which point particles interact via a Gaussian potential. Under the potential function $t\mapsto e^{-\alpha t^{2}}$ with $0\lt \alpha \lt 4\pi/e$ , we show that no point configuration in $\mathbb{R}^{n}$ of density $\rho$ can have energy less than $(\rho+o(1))(\pi/\alpha)^{n/2}$ as $n\to \infty$ with $\alpha$ and $\rho$ fixed. This lower bound asymptotically matches the upper bound of $\rho(\pi/\alpha)^{n/2}$ obtained as the expectation in the Siegel mean value theorem, and it is attained by random lattices. The proof is based on the linear programming bound, and it uses an interpolation construction analogous to those used for the Beurling–Selberg extremal problem in analytic number theory. In the other direction, we prove that the upper bound of $\rho (\pi/\alpha)^{n/2}$ is no longer asymptotically sharp when $\alpha \gt \pi e$ . As a consequence of our results, we obtain bounds in $\mathbb{R}^{n}$ for the minimal energy under inverse power laws $t\mapsto1/t^{n+s}$ with $s\gt 0$ , and these bounds are sharp to within a constant factor as $n\to \infty$ with $s$ fixed.
</p>projecteuclid.org/euclid.dmj/1534233620_20180914220702Fri, 14 Sep 2018 22:07 EDTStochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion processhttps://projecteuclid.org/euclid.dmj/1535356817<strong>Guillaume Barraquand</strong>, <strong>Alexei Borodin</strong>, <strong>Ivan Corwin</strong>, <strong>Michael Wheeler</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 13, 2457--2529.</p><p><strong>Abstract:</strong><br/>
We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time $\tau$ ) according to the Tracy–Widom Gaussian orthogonal ensemble distribution on the $\tau^{1/3}$ -scale. This is the first example of Kardar–Parisi–Zhang asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models.
Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes , as well as two surprising relations. The first relates a special (Hall–Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to the ASEP) using a Yang–Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogues via a refined Littlewood identity.
</p>projecteuclid.org/euclid.dmj/1535356817_20180914220702Fri, 14 Sep 2018 22:07 EDTGevrey stability of Prandtl expansions for $2$ -dimensional Navier–Stokes flowshttps://projecteuclid.org/euclid.dmj/1534838557<strong>David Gérard-Varet</strong>, <strong>Yasunori Maekawa</strong>, <strong>Nader Masmoudi</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 13, 2531--2631.</p><p><strong>Abstract:</strong><br/>
We investigate the stability of boundary layer solutions of the $2$ -dimensional incompressible Navier–Stokes equations. We consider shear flow solutions of Prandtl type: $u^{\nu}(t,x,y)=(U^{E}(t,y)+U^{\mathrm{BL}}(t,\frac{y}{\sqrt{\nu}}),0)$ , $0\lt \nu\ll1.$ We show that if $U^{\mathrm{BL}}$ is monotonic and concave in $Y=y/\sqrt{\nu}$ , then $u^{\nu}$ is stable over some time interval $(0,T)$ , $T$ independent of $\nu$ , under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$ . We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in $x$ and $y$ ). Moreover, in the case where $U^{\mathrm{BL}}$ is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr–Sommerfeld operator.
</p>projecteuclid.org/euclid.dmj/1534838557_20180914220702Fri, 14 Sep 2018 22:07 EDTAlmost periodicity in time of solutions of the KdV equationhttps://projecteuclid.org/euclid.dmj/1538121917<strong>Ilia Binder</strong>, <strong>David Damanik</strong>, <strong>Michael Goldstein</strong>, <strong>Milivoje Lukic</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 14, 2633--2678.</p><p><strong>Abstract:</strong><br/>
We study the Cauchy problem for the KdV equation $\partial_{t}u-6u\partial_{x}u+\partial_{x}^{3}u=0$ with almost periodic initial data $u(x,0)=V(x)$ . We consider initial data $V$ , for which the associated Schrödinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify, and we show the existence, uniqueness, and almost periodicity in time of solutions. This establishes a conjecture of Deift for this class of initial data. The result is shown to apply to all small analytic quasiperiodic initial data with Diophantine frequency vector.
</p>projecteuclid.org/euclid.dmj/1538121917_20180928040600Fri, 28 Sep 2018 04:06 EDTThe Markoff group of transformations in prime and composite modulihttps://projecteuclid.org/euclid.dmj/1538121918<strong>Chen Meiri</strong>, <strong>Doron Puder</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 14, 2679--2720.</p><p><strong>Abstract:</strong><br/>
The Markoff group of transformations is a group $\Gamma$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^{2}+y^{2}+z^{2}=xyz$ . The fundamental strong approximation conjecture for the Markoff equation states that for every prime $p$ , the group $\Gamma$ acts transitively on the set $X^{*}(p)$ of nonzero solutions to the same equation over ${\mathbb{Z}}/{p\mathbb{Z}}$ . Recently, Bourgain, Gamburd, and Sarnak proved this conjecture for all primes outside a small exceptional set. Here, we study a group of permutations obtained by the action of $\Gamma$ on $X^{*}(p)$ , and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that $\Gamma$ acts transitively also on the set of nonzero solutions in a big class of composite moduli. Finally, our result also translates to a parallel in the case $r=2$ of a well-known theorem of Gilman and Evans regarding “ $T_{r}$ -systems” of $\operatorname{PSL}(2,p)$ .
</p>projecteuclid.org/euclid.dmj/1538121918_20180928040600Fri, 28 Sep 2018 04:06 EDTCounting points of schemes over finite rings and counting representations of arithmetic latticeshttps://projecteuclid.org/euclid.dmj/1538121919<strong>Avraham Aizenbud</strong>, <strong>Nir Avni</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 14, 2721--2743.</p><p><strong>Abstract:</strong><br/>
We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if $\Gamma$ is an arithmetic lattice whose $\mathbb{Q}$ -rank is greater than $1$ , then let $r_{n}(\Gamma)$ be the number of irreducible $n$ -dimensional representations of $\Gamma$ up to isomorphism. We prove that there is a constant $C$ (in fact, any $C\gt 40$ suffices) such that $r_{n}(\Gamma)=O(n^{C})$ for every such $\Gamma$ . This answers a question of Larsen and Lubotzky.
</p>projecteuclid.org/euclid.dmj/1538121919_20180928040600Fri, 28 Sep 2018 04:06 EDTMöbius disjointness for homogeneous dynamicshttps://projecteuclid.org/euclid.dmj/1538121920<strong>Ryan Peckner</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 14, 2745--2792.</p><p><strong>Abstract:</strong><br/>
We prove Sarnak’s Möbius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $\Gamma\subset G$ a lattice, and $u\in G$ an Ad-unipotent element, then for every $x\in\Gamma\backslash G$ and every function $f$ continuous on the $1$ -point compactification of $\Gamma\backslash G$ , the sequence $f(xu^{n})$ cannot correlate with the Möbius function on average.
</p>projecteuclid.org/euclid.dmj/1538121920_20180928040600Fri, 28 Sep 2018 04:06 EDTEffective finiteness of irreducible Heegaard splittings of non-Haken $3$ -manifoldshttps://projecteuclid.org/euclid.dmj/1538532049<strong>Tobias Holck Colding</strong>, <strong>David Gabai</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 15, 2793--2832.</p><p><strong>Abstract:</strong><br/>
The main result is a short effective proof of Tao Li’s theorem that a closed non-Haken hyperbolic $3$ -manifold $N$ has at most finitely many irreducible Heegaard splittings. Along the way we show that $N$ has finitely many branched surfaces of pinched negative sectional curvature carrying all closed index- $\le1$ minimal surfaces. This effective result, together with the sequel with Daniel Ketover, solves the classification problem for Heegaard splittings of non-Haken hyperbolic $3$ -manifolds.
</p>projecteuclid.org/euclid.dmj/1538532049_20181018040118Thu, 18 Oct 2018 04:01 EDTOn the classification of Heegaard splittingshttps://projecteuclid.org/euclid.dmj/1538532048<strong>Tobias Holck Colding</strong>, <strong>David Gabai</strong>, <strong>Daniel Ketover</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 15, 2833--2856.</p><p><strong>Abstract:</strong><br/>
The long-standing classification problem in the theory of Heegaard splittings of $3$ -manifolds is to exhibit for each closed $3$ -manifold a complete list, without duplication, of all its irreducible Heegaard surfaces, up to isotopy. We solve this problem for non-Haken hyperbolic $3$ -manifolds.
</p>projecteuclid.org/euclid.dmj/1538532048_20181018040118Thu, 18 Oct 2018 04:01 EDTLarge deviations and the Lukic conjecturehttps://projecteuclid.org/euclid.dmj/1538532052<strong>Jonathan Breuer</strong>, <strong>Barry Simon</strong>, <strong>Ofer Zeitouni</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 15, 2857--2902.</p><p><strong>Abstract:</strong><br/>
We use the large deviation approach to sum rules pioneered by Gamboa, Nagel, and Rouault to prove higher-order sum rules for orthogonal polynomials on the unit circle. In particular, we prove one half of a conjectured sum rule of Lukic in the case of two singular points, one simple and one double. This is important because it is known that the conjecture of Simon fails in exactly this case, so this article provides support for the idea that Lukic’s replacement for Simon’s conjecture might be true.
</p>projecteuclid.org/euclid.dmj/1538532052_20181018040118Thu, 18 Oct 2018 04:01 EDTApproximate latticeshttps://projecteuclid.org/euclid.dmj/1538532050<strong>Michael Björklund</strong>, <strong>Tobias Hartnick</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 15, 2903--2964.</p><p><strong>Abstract:</strong><br/>
In this article we introduce and study uniform and nonuniform approximate lattices in locally compact second countable (lcsc) groups. These are approximate subgroups (in the sense of Tao) which simultaneously generalize lattices in lcsc group and mathematical quasicrystals (Meyer sets) in lcsc Abelian groups. We show that envelopes of strong approximate lattices are unimodular and that approximate lattices in nilpotent groups are uniform. We also establish several results relating properties of approximate lattices and their envelopes. For example, we prove a version of the Milnor–Schwarz lemma for uniform approximate lattices in compactly generated lcsc groups, which we then use to relate the metric amenability of uniform approximate lattices to the amenability of the envelope. Finally we extend a theorem of Kleiner and Leeb to show that the isometry groups of irreducible higher-rank symmetric spaces of noncompact type are quasi-isometrically rigid with respect to finitely generated approximate groups.
</p>projecteuclid.org/euclid.dmj/1538532050_20181018040118Thu, 18 Oct 2018 04:01 EDT