Duke Mathematical Journal Articles (Project Euclid)
http://projecteuclid.org/euclid.dmj
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
http://projecteuclid.org/
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 EDTModular cocycles and linking numbershttp://projecteuclid.org/euclid.dmj/1483758030<strong>W. Duke</strong>, <strong>Ö. Imamoḡlu</strong>, <strong>Á. Tóth</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 6, 1179--1210.</p><p><strong>Abstract:</strong><br/>
It is known that the $3$ -manifold $\operatorname{SL}(2,\mathbb{Z})\backslash\operatorname{SL}(2,\mathbb{R})$ is diffeomorphic to the complement of the trefoil knot in $S^{3}$ . E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind’s eta function under $\operatorname{SL}(2,\mathbb{Z})$ . In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It can be computed in terms of a special value of a certain Dirichlet series and satisfies a reciprocity law. The homogenization of this symbol, which generalizes the Rademacher symbol, gives the linking number between two distinct symmetric links formed from modular knots.
</p>projecteuclid.org/euclid.dmj/1483758030_20170414040041Fri, 14 Apr 2017 04:00 EDTInvolutive Heegaard Floer homologyhttp://projecteuclid.org/euclid.dmj/1484103841<strong>Kristen Hendricks</strong>, <strong>Ciprian Manolescu</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 7, 1211--1299.</p><p><strong>Abstract:</strong><br/>
Using the conjugation symmetry on Heegaard Floer complexes, we define a $3$ -manifold invariant called involutive Heegaard Floer homology , which is meant to correspond to $\mathbb{Z}_{4}$ -equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, $\underline{d}$ and $\bar{d}$ , and two invariants of smooth knot concordance, $\underline{V}_{0}$ and $\overline{V}_{0}$ . We also develop a formula for the involutive Heegaard Floer homology of large integral surgeries on knots. We give explicit calculations in the case of L-space knots and thin knots. In particular, we show that $\underline{V}_{0}$ detects the nonsliceness of the figure-eight knot. Other applications include constraints on which large surgeries on alternating knots can be homology-cobordant to other large surgeries on alternating knots.
</p>projecteuclid.org/euclid.dmj/1484103841_20170512040029Fri, 12 May 2017 04:00 EDTOn the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groupshttp://projecteuclid.org/euclid.dmj/1485227196<strong>Atsushi Ichino</strong>, <strong>Erez Lapid</strong>, <strong>Zhengyu Mao</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 7, 1301--1348.</p><p><strong>Abstract:</strong><br/>
The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint $\gamma$ -factor of its $L$ -parameter. In this article, we prove the formal degree conjecture for odd special orthogonal and metaplectic groups in the generic case, which, combined with Arthur’s work on the local Langlands correspondence, implies the conjecture in the nongeneric case.
</p>projecteuclid.org/euclid.dmj/1485227196_20170512040029Fri, 12 May 2017 04:00 EDTInfinitesimal Newton–Okounkov bodies and jet separationhttp://projecteuclid.org/euclid.dmj/1487818919<strong>Alex Küronya</strong>, <strong>Victor Lozovanu</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 7, 1349--1376.</p><p><strong>Abstract:</strong><br/>
In this article we explore the connection between asymptotic base loci and Newton–Okounkov bodies associated to infinitesimal flags. Analogously to the surface case, we obtain complete characterizations of augmented and restricted base loci. Interestingly enough, an integral part of the argument is a study of the relationship between certain simplices contained in Newton–Okoukov bodies and jet separation; our results also lead to a convex geometric description of moving Seshadri constants.
</p>projecteuclid.org/euclid.dmj/1487818919_20170512040029Fri, 12 May 2017 04:00 EDTVolume entropy of Hilbert metrics and length spectrum of Hitchin representations into $\operatorname{PSL}(3,\mathbb{R})$http://projecteuclid.org/euclid.dmj/1487322015<strong>Nicolas Tholozan</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 7, 1377--1403.</p><p><strong>Abstract:</strong><br/>
This article studies the geometry of proper open convex domains in the projective space $\mathbb{R}\mathbf{P}^{n}$ . These domains carry several projective invariant distances, among which are the Hilbert distance $d^{H}$ and the Blaschke distance $d^{B}$ . We prove a thin inequality between those distances: for any two points $x$ and $y$ in such a domain,
\[d^{B}(x,y)\lt d^{H}(x,y)+1.\]
We then give two interesting consequences. The first one answers a conjecture of Colbois and Verovic on the volume entropy of Hilbert geometries: for any proper open convex domain in $\mathbb{R}\mathbf{P}^{n}$ , the volume of a ball of radius $R$ grows at most like $e^{(n-1)R}$ . The second consequence is the following fact: for any Hitchin representation $\rho$ of a surface group $\Gamma$ into $\operatorname{PSL}(3,\mathbb{R})$ , there exists a Fuchsian representation $j:\Gamma\to\operatorname{PSL}(2,\mathbb{R})$ such that the length spectrum of $j$ is uniformly smaller than that of $\rho$ . This answers positively a conjecture of Lee and Zhang in the $3$ -dimensional case.
</p>projecteuclid.org/euclid.dmj/1487322015_20170512040029Fri, 12 May 2017 04:00 EDTOpen Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifoldshttp://projecteuclid.org/euclid.dmj/1487991810<strong>Kwokwai Chan</strong>, <strong>Siu-Cheong Lau</strong>, <strong>Naichung Conan Leung</strong>, <strong>Hsian-Hua Tseng</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 8, 1405--1462.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a compact toric Kähler manifold with $-K_{X}$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$ . Fukaya, Oh, Ohta, and Ono defined open Gromov–Witten (GW) invariants of $X$ as virtual counts of holomorphic disks with Lagrangian boundary condition $L$ . We prove a formula that equates such open GW invariants with closed GW invariants of certain $X$ -bundles over $\mathbb{P}^{1}$ used by Seidel and McDuff earlier to construct Seidel representations for $X$ . We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disk potential of $X$ , an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya, Oh, Ohta, and Ono.
</p>projecteuclid.org/euclid.dmj/1487991810_20170601220057Thu, 01 Jun 2017 22:00 EDTOn the cohomological dimension of the moduli space of Riemann surfaceshttp://projecteuclid.org/euclid.dmj/1489802635<strong>Gabriele Mondello</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 8, 1463--1515.</p><p><strong>Abstract:</strong><br/>
The moduli space of Riemann surfaces of genus $g\geq2$ is (up to a finite étale cover) a complex manifold, so it makes sense to speak of its Dolbeault cohomological dimension. The conjecturally optimal bound is $g-2$ . This expectation is verified in low genus and is supported by Harer’s computation of its de Rham cohomological dimension and by vanishing results in the tautological intersection ring. In this article, we prove that such a dimension is at most $2g-2$ . We also prove an analogous bound for the moduli space of Riemann surfaces with marked points. The key step is to show that the Dolbeault cohomological dimension of each stratum of translation surfaces is at most $g$ . In order to do that, we produce an exhaustion function whose complex Hessian has controlled index: the construction of such a function relies on some basic geometric properties of translation surfaces.
</p>projecteuclid.org/euclid.dmj/1489802635_20170601220057Thu, 01 Jun 2017 22:00 EDTLarge-scale rank of Teichmüller spacehttp://projecteuclid.org/euclid.dmj/1490666574<strong>Alex Eskin</strong>, <strong>Howard Masur</strong>, <strong>Kasra Rafi</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 8, 1517--1572.</p><p><strong>Abstract:</strong><br/>
Suppose that ${\mathcal{X}}$ is either the mapping class group equipped with the word metric or Teichmüller space equipped with either the Teichmüller metric or the Weil–Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that for any large box in ${\mathbb{R}}^{n}$ there is a standard model of a flat in ${\mathcal{X}}$ such that the quasi-Lipschitz image of a large sub-box is near the standard flat. As a consequence, we show that, for all these spaces, the geometric rank and the topological rank are equal. The methods are axiomatic and apply to a larger class of metric spaces.
</p>projecteuclid.org/euclid.dmj/1490666574_20170601220057Thu, 01 Jun 2017 22:00 EDTInvariable generation of the symmetric grouphttp://projecteuclid.org/euclid.dmj/1486695668<strong>Sean Eberhard</strong>, <strong>Kevin Ford</strong>, <strong>Ben Green</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 8, 1573--1590.</p><p><strong>Abstract:</strong><br/>
We say that permutations $\pi_{1},\ldots,\pi_{r}\in\mathcal{S}_{n}$ invariably generate $\mathcal{S}_{n}$ if, no matter how one chooses conjugates $\pi'_{1},\ldots,\pi'_{r}$ of these permutations, the $\pi'_{1},\ldots,\pi'_{r}$ permutations generate $\mathcal{S}_{n}$ . We show that if $\pi_{1},\pi_{2}$ , and $\pi_{3}$ are chosen randomly from $\mathcal{S}_{n}$ , then, with probability tending to $1$ as $n\rightarrow\infty$ , they do not invariably generate $\mathcal{S}_{n}$ . By contrast, it was shown recently by Pemantle, Peres, and Rivin that four random elements do invariably generate $\mathcal{S}_{n}$ with probability bounded away from zero. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.
</p>projecteuclid.org/euclid.dmj/1486695668_20170601220057Thu, 01 Jun 2017 22:00 EDTMean curvature flow with surgeryhttp://projecteuclid.org/euclid.dmj/1487818918<strong>Robert Haslhofer</strong>, <strong>Bruce Kleiner</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 9, 1591--1626.</p><p><strong>Abstract:</strong><br/>
We give a new proof for the existence of mean curvature flow with surgery of $2$ -convex hypersurfaces in $\mathbb{R}^{N}$ . Our proof works for all $N\geq3$ , including mean convex surfaces in $\mathbb{R}^{3}$ . We also derive a priori estimates for a more general class of flows in a local and flexible setting.
</p>projecteuclid.org/euclid.dmj/1487818918_20170612220241Mon, 12 Jun 2017 22:02 EDTStrichartz estimates in similarity coordinates and stable blowup for the critical wave equationhttp://projecteuclid.org/euclid.dmj/1491357654<strong>Roland Donninger</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 9, 1627--1683.</p><p><strong>Abstract:</strong><br/>
We establish Strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a (time-dependent) self-similar potential. As an application, we consider the critical wave equation and prove the asymptotic stability of the ordinary differential equations blowup profile in the energy space.
</p>projecteuclid.org/euclid.dmj/1491357654_20170612220241Mon, 12 Jun 2017 22:02 EDTLarge greatest common divisor sums and extreme values of the Riemann zeta functionhttp://projecteuclid.org/euclid.dmj/1485400054<strong>Andriy Bondarenko</strong>, <strong>Kristian Seip</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 9, 1685--1701.</p><p><strong>Abstract:</strong><br/>
It is shown that the maximum of $\vert \zeta(1/2+it)\vert $ on the interval $T^{1/2}\le t\le T$ is at least $\exp ((1/\sqrt{2}+o(1))\sqrt{\log T\log\log\log T/\log\log T})$ . Our proof uses Soundararajan’s resonance method and a certain large greatest common divisor sum. The method of proof shows that the absolute constant $A$ in the inequality
\[\sup_{1\le n_{1}\lt \cdots\lt n_{N}}\sum_{k,{\ell}=1}^{N}\frac{\operatorname{gcd}(n_{k},n_{\ell})}{\sqrt{n_{k}n_{\ell}}}\ll N\exp (A\sqrt{\frac{\log N\log\log\logN}{\log\log N}}),\] established in a recent paper of ours, cannot be taken smaller than $1$ .
</p>projecteuclid.org/euclid.dmj/1485400054_20170612220241Mon, 12 Jun 2017 22:02 EDTSymplectic embeddings and the Lagrangian bidiskhttp://projecteuclid.org/euclid.dmj/1488445213<strong>Vinicius Gripp Barros Ramos</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 9, 1703--1738.</p><p><strong>Abstract:</strong><br/>
In this article we obtain sharp obstructions to the symplectic embedding of the Lagrangian bidisk into four-dimensional balls, ellipsoids, and symplectic polydisks. We prove, in fact, that the interior of the Lagrangian bidisk is symplectomorphic to a concave toric domain by using ideas that come from billiards on a round disk. In particular, we answer a question of Ostrover. We also obtain sharp obstructions to some embeddings of ellipsoids into the Lagrangian bidisk.
</p>projecteuclid.org/euclid.dmj/1488445213_20170612220241Mon, 12 Jun 2017 22:02 EDTThe eigencurve over the boundary of weight spacehttp://projecteuclid.org/euclid.dmj/1491271255<strong>Ruochuan Liu</strong>, <strong>Daqing Wan</strong>, <strong>Liang Xiao</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 9, 1739--1787.</p><p><strong>Abstract:</strong><br/>
We prove that the eigencurve associated to a definite quaternion algebra over $\mathbb{Q}$ satisfies the following properties, as conjectured by Coleman and Mazur as well as Buzzard and Kilford: (a) over the boundary annuli of weight space, the eigencurve is a disjoint union of (countably) infinitely many connected components, each finite and flat over the weight annuli; (b) the $U_{p}$ -slopes of points on each fixed connected component are proportional to the $p$ -adic valuations of the parameter on weight space; and (c) the sequence of the slope ratios forms a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves toward the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.
</p>projecteuclid.org/euclid.dmj/1491271255_20170612220241Mon, 12 Jun 2017 22:02 EDTRigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalueshttp://projecteuclid.org/euclid.dmj/1494295317<strong>Subhroshekhar Ghosh</strong>, <strong>Yuval Peres</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 10, 1789--1858.</p><p><strong>Abstract:</strong><br/>
Let $\Pi$ be a translation-invariant point process on the complex plane $\mathbb{C}$ , and let $\mathcal{D}\subset\mathbb{C}$ be a bounded open set. We ask the following: What does the point configuration $\Pi_{\mathrm{out}}$ obtained by taking the points of $\Pi$ outside $\mathcal{D}$ tell us about the point configuration $\Pi_{\mathrm{in}}$ of $\Pi$ inside $\mathcal{D}$ ? We show that, for the Ginibre ensemble, $\Pi_{\mathrm{out}}$ determines the number of points in $\Pi_{\mathrm{in}}$ . For the translation-invariant zero process of a planar Gaussian analytic function, we show that $\Pi_{\mathrm{out}}$ determines the number as well as the center of mass of the points in $\Pi_{\mathrm{in}}$ . Further, in both models we prove that the outside says “nothing more” about the inside, in the sense that the conditional distribution of the inside points, given the outside, is mutually absolutely continuous with respect to the Lebesgue measure on its supporting submanifold.
</p>projecteuclid.org/euclid.dmj/1494295317_20170711040316Tue, 11 Jul 2017 04:03 EDTGeometry of pseudodifferential algebra bundles and Fourier integral operatorshttp://projecteuclid.org/euclid.dmj/1490061610<strong>Varghese Mathai</strong>, <strong>Richard B. Melrose</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 10, 1859--1922.</p><p><strong>Abstract:</strong><br/>
We study the geometry and topology of (filtered) algebra bundles $\mathbf{\Psi}^{\mathbb{Z}}$ over a smooth manifold $X$ with typical fiber $\Psi^{\mathbb{Z}}(Z;V)$ , the algebra of classical pseudodifferential operators acting on smooth sections of a vector bundle $V$ over the compact manifold $Z$ and of integral order. First, a theorem of Duistermaat and Singer is generalized to the assertion that the group of projective invertible Fourier integral operators $\operatorname{PG}(\mathcal{F}^{\mathbb{C}}(Z;V))$ is precisely the automorphism group of the filtered algebra of pseudodifferential operators. We replace some of the arguments in their work by microlocal ones, thereby removing the topological assumption. We define a natural class of connections and $B$ -fields on the principal bundle to which $\mathbf{\Psi}^{\mathbb{Z}}$ is associated and obtain a de Rham representative of the Dixmier–Douady class in terms of the outer derivation on the Lie algebra and the residue trace of Guillemin and Wodzicki. The resulting formula only depends on the formal symbol algebra $\mathbf{\Psi}^{\mathbb{Z}}/\mathbf{\Psi}^{-\infty}$ . Examples of pseudodifferential algebra bundles are given that are not associated to a finite-dimensional fiber bundle over $X$ .
</p>projecteuclid.org/euclid.dmj/1490061610_20170711040316Tue, 11 Jul 2017 04:03 EDTAffine representability results in ${\mathbb{A}}^{1}$ -homotopy theory, I: Vector bundleshttp://projecteuclid.org/euclid.dmj/1489802634<strong>Aravind Asok</strong>, <strong>Marc Hoyois</strong>, <strong>Matthias Wendt</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 10, 1923--1953.</p><p><strong>Abstract:</strong><br/>
We establish a general “affine representability” result in ${\mathbb{A}}^{1}$ -homotopy theory over a general base. We apply this result to obtain representability results for vector bundles in ${\mathbb{A}}^{1}$ -homotopy theory. Our results simplify and significantly generalize Morel’s ${\mathbb{A}}^{1}$ -representability theorem for vector bundles.
</p>projecteuclid.org/euclid.dmj/1489802634_20170711040316Tue, 11 Jul 2017 04:03 EDTApproximation by subgroups of finite index and the Hanna Neumann conjecturehttp://projecteuclid.org/euclid.dmj/1489629612<strong>Andrei Jaikin-Zapirain</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 10, 1955--1987.</p><p><strong>Abstract:</strong><br/>
Let $F$ be a free group (pro- $p$ group), and let $U$ and $W$ be two finitely generated subgroups (closed subgroups) of $F$ . The Strengthened Hanna Neumann conjecture says that
\[\sum_{x\in U\backslash F/W}\overline{\operatorname{rk}}(U\cap xWx^{-1})\le\overline{\operatorname{rk}}(U)\overline{\mathrm{rk}}(W),\quad \mbox{where }\overline{\operatorname{rk}}(U)=\max\{\operatorname{rk}(U)-1,0\}.\] This conjecture was proved independently in the case of abstract groups by J. Friedman and I. Mineyev in 2011.
In this paper we give the proof of the conjecture in the pro- $p$ context, and we present a new proof in the abstract case. We also show that the Lück approximation conjecture holds for free groups.
</p>projecteuclid.org/euclid.dmj/1489629612_20170711040316Tue, 11 Jul 2017 04:03 EDTThe annihilator of the Lefschetz motivehttp://projecteuclid.org/euclid.dmj/1493344842<strong>Inna Zakharevich</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 11, 1989--2022.</p><p><strong>Abstract:</strong><br/>
In this article, we study a spectrum $K(\mathcal{V}_{k})$ such that $\pi_{0}K(\mathcal{V}_{k})$ is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric information about the geometry of varieties. We use the topology of this spectrum to analyze the structure of $K_{0}[\mathcal{V}_{k}]$ and to show that classes in the kernel of multiplication by $[\mathbb{A}^{1}]$ can always be represented as $[X]-[Y]$ , where $[X]\neq[Y]$ , $X\times\mathbb{A}^{1}$ , and $Y\times\mathbb{A}^{1}$ are not piecewise-isomorphic, but $[X\times\mathbb{A}^{1}]=[Y\times\mathbb{A}^{1}]$ in $K_{0}[\mathcal{V}_{k}]$ . Along the way, we present a new proof of the result of Larsen–Lunts on the structure on $K_{0}[\mathcal{V}_{k}]/([\mathbb{A}^{1}])$ .
</p>projecteuclid.org/euclid.dmj/1493344842_20170810040140Thu, 10 Aug 2017 04:01 EDTII $_{1}$ factors with nonisomorphic ultrapowershttp://projecteuclid.org/euclid.dmj/1490666575<strong>Rémi Boutonnet</strong>, <strong>Ionuţ Chifan</strong>, <strong>Adrian Ioana</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 11, 2023--2051.</p><p><strong>Abstract:</strong><br/>
We prove that there exist uncountably many separable $\mathrm{II}_{1}$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are nonisomorphic. In fact, we prove that the families of nonisomorphic $\mathrm{II}_{1}$ factors originally introduced by McDuff are such examples. This entails the existence of a continuum of nonelementarily equivalent $\mathrm{II}_{1}$ factors, thus settling a well-known open problem in the continuous model theory of operator algebras.
</p>projecteuclid.org/euclid.dmj/1490666575_20170810040140Thu, 10 Aug 2017 04:01 EDTLocal Langlands correspondence for $\mathrm{GL}_{n}$ and the exterior and symmetric square $\varepsilon$ -factorshttp://projecteuclid.org/euclid.dmj/1493863448<strong>J. W. Cogdell</strong>, <strong>F. Shahidi</strong>, <strong>T.-L. Tsai</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 11, 2053--2132.</p><p><strong>Abstract:</strong><br/>
Let $F$ be a $p$ -adic field, that is, a finite extension of $\mathbb{Q}_{p}$ for some prime $p$ . The local Langlands correspondence (LLC) attaches to each continuous $n$ -dimensional $\Phi$ -semisimple representation $\rho$ of $W'_{F}$ , the Weil–Deligne group for $\overline{F}/F$ , an irreducible admissible representation $\pi(\rho)$ of $\mathrm{GL}_{n}(F)$ such that, among other things, the local $L$ - and $\varepsilon$ -factors of pairs are preserved. This correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides, such as taking the exterior square or symmetric square. In this article, we show that this is the case for the local arithmetic and analytic symmetric square and exterior square $\varepsilon$ -factors, that is, that $\varepsilon(s,\Lambda^{2}\rho,\psi)=\varepsilon(s,\pi(\rho),\Lambda^{2},\psi)$ and $\varepsilon(s,\operatorname{Sym}^{2}\rho,\psi)=\varepsilon(s,\pi(\rho),\operatorname{Sym}^{2},\psi)$ . The agreement of the $L$ -functions also follows by our methods, but this was already known by Henniart. The proof is a robust deformation argument, combined with local/global techniques, which reduces the problem to the stability of the analytic $\gamma$ -factor $\gamma(s,\pi,\Lambda^{2},\psi)$ under highly ramified twists when $\pi$ is supercuspidal. This last step is achieved by relating the $\gamma$ -factor to a Mellin transform of a partial Bessel function attached to the representation and then analyzing the asymptotics of the partial Bessel function, inspired in part by the theory of Shalika germs for Bessel integrals. The stability for every irreducible admissible representation $\pi$ then follows from those of the corresponding arithmetic $\gamma$ -factors as a corollary.
</p>projecteuclid.org/euclid.dmj/1493863448_20170810040140Thu, 10 Aug 2017 04:01 EDTAlternating links and definite surfaceshttp://projecteuclid.org/euclid.dmj/1493971214<strong>Joshua Evan Greene</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 11, 2133--2151.</p><p><strong>Abstract:</strong><br/>
We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait’s conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and of Hirasawa and Sakuma about Seifert surfaces for special alternating links. The appendix, written by Juhász and Lackenby, applies the characterization to derive an exponential time algorithm for alternating knot recognition.
</p>projecteuclid.org/euclid.dmj/1493971214_20170810040140Thu, 10 Aug 2017 04:01 EDTFree Hilbert transformshttp://projecteuclid.org/euclid.dmj/1493344841<strong>Tao Mei</strong>, <strong>Éric Ricard</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 11, 2153--2182.</p><p><strong>Abstract:</strong><br/>
We study Fourier multipliers of Hilbert transform type on free groups. We prove that they are completely bounded on noncommutative $L^{p}$ -spaces associated with the free group von Neumann algebras for all $1\lt p\lt \infty$ . This implies that the decomposition of the free group $\mathbf{F}_{\infty}$ into reduced words starting with distinct free generators is completely unconditional in $L^{p}$ . We study the case of Voiculescu’s amalgamated free products of von Neumann algebras as well. As by-products, we obtain a positive answer to a compactness problem posed by Ozawa, a length-independent estimate for Junge–Parcet–Xu’s free Rosenthal’s inequality, a Littlewood–Paley–Stein-type inequality for geodesic paths of free groups, and a length reduction formula for $L^{p}$ -norms of free group von Neumann algebras.
</p>projecteuclid.org/euclid.dmj/1493344841_20170810040140Thu, 10 Aug 2017 04:01 EDTOn the arithmetic transfer conjecture for exotic smooth formal moduli spaceshttps://projecteuclid.org/euclid.dmj/1496995226<strong>M. Rapoport</strong>, <strong>B. Smithling</strong>, <strong>W. Zhang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 12, 2183--2336.</p><p><strong>Abstract:</strong><br/>
In the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of $p$ -divisible groups, associated to a unitary group relative to a ramified quadratic extension of a $p$ -adic field. We prove our conjecture in the case of a unitary group in three variables.
</p>projecteuclid.org/euclid.dmj/1496995226_20170824040044Thu, 24 Aug 2017 04:00 EDTTriangular bases in quantum cluster algebras and monoidal categorification conjectureshttps://projecteuclid.org/euclid.dmj/1495764415<strong>Fan Qin</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 12, 2337--2442.</p><p><strong>Abstract:</strong><br/>
We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a unique common triangular basis with respect to all seeds. This basis is parameterized by tropical points as expected in the Fock–Goncharov conjecture.
As an application, we prove the existence of the common triangular bases for the quantum cluster algebras arising from representations of quantum affine algebras and partially for those arising from quantum unipotent subgroups. This result implies monoidal categorification conjectures of Hernandez and Leclerc and Fomin and Zelevinsky in the corresponding cases: all cluster monomials correspond to simple modules.
</p>projecteuclid.org/euclid.dmj/1495764415_20170824040044Thu, 24 Aug 2017 04:00 EDTCounterexamples to a conjecture of Woodshttps://projecteuclid.org/euclid.dmj/1496995227<strong>Oded Regev</strong>, <strong>Uri Shapira</strong>, <strong>Barak Weiss</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 13, 2443--2446.</p><p><strong>Abstract:</strong><br/>
A conjecture of Woods from 1972 is disproved: for $d\geq30$ , there are well-rounded unimodular lattices in ${\mathbb{R}}^{d}$ with covering radius greater than that of ${\mathbb{Z}}^{d}$ .
</p>projecteuclid.org/euclid.dmj/1496995227_20170918220520Mon, 18 Sep 2017 22:05 EDTCM values of regularized theta lifts and harmonic weak Maaß forms of weight $1$https://projecteuclid.org/euclid.dmj/1496995225<strong>Stephan Ehlen</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 13, 2447--2519.</p><p><strong>Abstract:</strong><br/>
We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak Maaß forms of weight $1$ . As it turns out, these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve. Our results imply a conjecture of Duke and Li and give a new proof of the modularity of a certain arithmetic generating series of weight $1$ studied by Kudla, Rapoport, and Yang.
</p>projecteuclid.org/euclid.dmj/1496995225_20170918220520Mon, 18 Sep 2017 22:05 EDTRepresentation stability and finite linear groupshttps://projecteuclid.org/euclid.dmj/1497924228<strong>Andrew Putman</strong>, <strong>Steven V Sam</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 13, 2521--2598.</p><p><strong>Abstract:</strong><br/>
We study analogues of $\operatorname{{\mathtt {FI}}}$ -modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings, and we prove basic structural properties such as local Noetherianity. Applications include a proof of the Lannes–Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with twisted coefficients for the general linear and symplectic groups over finite rings, and representation-theoretic versions of homological stability for congruence subgroups of the general linear group, the automorphism group of a free group, the symplectic group, and the mapping class group.
</p>projecteuclid.org/euclid.dmj/1497924228_20170918220520Mon, 18 Sep 2017 22:05 EDTBounded height in pencils of finitely generated subgroupshttps://projecteuclid.org/euclid.dmj/1500364839<strong>F. Amoroso</strong>, <strong>D. Masser</strong>, <strong>U. Zannier</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 13, 2599--2642.</p><p><strong>Abstract:</strong><br/>
In this article we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric Mordell–Lang theorem by replacing finiteness with emptiness , for the intersection of varieties and subgroups, all moving in a pencil, except for bounded height values of the parameters (and excluding identical relations). More precisely, an instance of the result is as follows. Consider the torus scheme ${{\mathbb{G}_{m}^{r}}_{/\mathcal{C}}}$ over a curve $\mathcal{C}$ defined over $\overline{\mathbb{Q}}$ , and let $\Gamma$ be a subgroup scheme generated by finitely many sections (satisfying some necessary conditions). Further, let $V$ be any subscheme. Then there is a bound for the height of the points $P\in\mathcal{C}(\overline{\mathbb{Q}})$ such that, for some $\gamma\in\Gamma$ which does not generically lie in $V$ , $\gamma(P)$ lies in the fiber $V_{P}$ . We further offer some direct Diophantine applications, to illustrate once again that the results implicitly contain information absent from the previous bounds in this context.
</p>projecteuclid.org/euclid.dmj/1500364839_20170918220520Mon, 18 Sep 2017 22:05 EDTComplex projective structures: Lyapunov exponent, degree, and harmonic measurehttps://projecteuclid.org/euclid.dmj/1503712833<strong>Bertrand Deroin</strong>, <strong>Romain Dujardin</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 14, 2643--2695.</p><p><strong>Abstract:</strong><br/>
We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developing map; and a family of harmonic measures on the Riemann sphere, previously introduced by Hussenot. We show that the degree and the Lyapunov exponent are related by a simple formula and give estimates for the Hausdorff dimension of the harmonic measures in terms of the Lyapunov exponent. In accordance with the famous Sullivan dictionary, this leads to a description of the space of such projective structures that is reminiscent of that of the space of polynomials in holomorphic dynamics.
</p>projecteuclid.org/euclid.dmj/1503712833_20170929040133Fri, 29 Sep 2017 04:01 EDTSharp phase transitions for the almost Mathieu operatorhttps://projecteuclid.org/euclid.dmj/1495764414<strong>Artur Avila</strong>, <strong>Jiangong You</strong>, <strong>Qi Zhou</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 14, 2697--2718.</p><p><strong>Abstract:</strong><br/>
It is known that the spectral type of the almost Mathieu operator (AMO) depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya’s conjecture. Together with a previous work by Avila, this gives the sharp description of phase transitions for the AMO for the a.e. phase.
</p>projecteuclid.org/euclid.dmj/1495764414_20170929040133Fri, 29 Sep 2017 04:01 EDTSobolev trace inequalities of order fourhttps://projecteuclid.org/euclid.dmj/1502697757<strong>Antonio G. Ache</strong>, <strong>Sun-Yung Alice Chang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 14, 2719--2748.</p><p><strong>Abstract:</strong><br/>
We establish sharp trace Sobolev inequalities of order four on Euclidean $d$ -balls for $d\ge4$ . When $d=4$ , our inequality generalizes the classical second-order Lebedev–Milin inequality on Euclidean $2$ -balls. Our method relies on the use of scattering theory on hyperbolic $d$ -balls. As an application, we characterize the extremal metric of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean $4$ -balls, which surprisingly is not the flat metric on the ball.
</p>projecteuclid.org/euclid.dmj/1502697757_20170929040133Fri, 29 Sep 2017 04:01 EDTA tropical approach to a generalized Hodge conjecture for positive currentshttps://projecteuclid.org/euclid.dmj/1504684816<strong>Farhad Babaee</strong>, <strong>June Huh</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 14, 2749--2813.</p><p><strong>Abstract:</strong><br/>
In 1982, Demailly showed that the Hodge conjecture follows from the statement that all positive closed currents with rational cohomology class can be approximated by positive linear combinations of integration currents. Moreover, in 2012, he showed that the Hodge conjecture is equivalent to the statement that any $(p,p)$ -dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents. In this article, we find a current which does not verify the former statement on a smooth projective variety for which the Hodge conjecture is known to hold. To construct this current, we extend the framework of “tropical currents”—recently introduced by the first author—from tori to toric varieties. We discuss extremality properties of tropical currents and show that the cohomology class of a tropical current is the recession of its underlying tropical variety. The counterexample is obtained from a tropical surface in $\mathbb{R}^{4}$ whose intersection form does not have the right signature in terms of the Hodge index theorem.
</p>projecteuclid.org/euclid.dmj/1504684816_20170929040133Fri, 29 Sep 2017 04:01 EDTErrata for Stephan Ehlen, “CM values of regularized theta lifts and harmonic weak Maaß forms of weight $1$ ,” Duke Math. J., Volume 166, Number 13 (2017), 2447–2519https://projecteuclid.org/euclid.dmj/1506672071<p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 14</p>projecteuclid.org/euclid.dmj/1506672071_20170929040133Fri, 29 Sep 2017 04:01 EDTOn the geometry of thin exceptional sets in Manin’s conjecturehttps://projecteuclid.org/euclid.dmj/1504252913<strong>Brian Lehmann</strong>, <strong>Sho Tanimoto</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 15, 2815--2869.</p><p><strong>Abstract:</strong><br/>
Manin’s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin’s conjecture is a thin set using the minimal model program and boundedness of log Fano varieties.
</p>projecteuclid.org/euclid.dmj/1504252913_20171012040515Thu, 12 Oct 2017 04:05 EDTModuli of curves as moduli of $A_{\infty}$ -structureshttps://projecteuclid.org/euclid.dmj/1504836224<strong>Alexander Polishchuk</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 15, 2871--2924.</p><p><strong>Abstract:</strong><br/>
We define and study the stack $\mathcal{U}^{ns,a}_{g,g}$ of (possibly singular) projective curves of arithmetic genus $g$ with $g$ smooth marked points forming an ample nonspecial divisor. We define an explicit closed embedding of a natural $\mathbb{G}_{m}^{g}$ -torsor $\widetilde{\mathcal{U}}^{ns,a}_{g,g}$ over $\mathcal{U}^{ns,a}_{g,g}$ into an affine space, and we give explicit equations of the universal curve (away from characteristics $2$ and $3$ ). This construction can be viewed as a generalization of the Weierstrass cubic and the $j$ -invariant of an elliptic curve to the case $g\gt 1$ . Our main result is that in characteristics different from $2$ and $3$ the moduli space $\widetilde{\mathcal{U}}^{ns,a}_{g,g}$ is isomorphic to the moduli space of minimal $A_{\infty}$ -structures on a certain finite-dimensional graded associative algebra $E_{g}$ (introduced by Fisette and Polishchuk).
</p>projecteuclid.org/euclid.dmj/1504836224_20171012040515Thu, 12 Oct 2017 04:05 EDTRank, combinatorial cost, and homology torsion growth in higher rank latticeshttps://projecteuclid.org/euclid.dmj/1504684817<strong>Miklos Abert</strong>, <strong>Tsachik Gelander</strong>, <strong>Nikolay Nikolov</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 15, 2925--2964.</p><p><strong>Abstract:</strong><br/>
We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right-angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute.
Most nonuniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (cocompact) right-angled arithmetic groups in $\mathrm{SL}(n,\mathbb{R})$ , $n\geq3$ , and $\mathrm{SO}(p,q)$ for some values of $p,q$ . This is a class of lattices for which the congruence subgroup property is not known in general. By using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right-angled lattice in a higher rank simple Lie group.
</p>projecteuclid.org/euclid.dmj/1504684817_20171012040515Thu, 12 Oct 2017 04:05 EDTA product for permutation groups and topological groupshttps://projecteuclid.org/euclid.dmj/1502244254<strong>Simon M. Smith</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 15, 2965--2999.</p><p><strong>Abstract:</strong><br/>
We introduce a new product for permutation groups. It takes as input two permutation groups, $M$ and $N$ and produces an infinite group $M\boxtimes N$ which carries many of the permutational properties of $M$ . Under mild conditions on $M$ and $N$ the group $M\boxtimes N$ is simple.
As a permutational product, its most significant property is the following: $M\boxtimes N$ is primitive if and only if $M$ is primitive but not regular, and $N$ is transitive. Despite this remarkable similarity with the wreath product in product action, $M\boxtimes N$ and $M\operatorname{Wr}N$ are thoroughly dissimilar.
The product provides a general way to build exotic examples of nondiscrete, simple, totally disconnected, locally compact, compactly generated topological groups from discrete groups.
We use this to obtain the first construction of uncountably many pairwise nonisomorphic simple topological groups that are totally disconnected, locally compact, compactly generated, and nondiscrete. The groups we construct all contain the same compact open subgroup. The analogous result for discrete groups was proved in 1953 by Ruth Camm.
To build the product, we describe a group $\mathcal{U}(M,N)$ that acts on a biregular tree $T$ . This group has a natural universal property and is a generalization of the iconic universal group construction of Marc Burger and Shahar Mozes for locally finite regular trees.
</p>projecteuclid.org/euclid.dmj/1502244254_20171012040515Thu, 12 Oct 2017 04:05 EDTA metric interpretation of reflexivity for Banach spaceshttps://projecteuclid.org/euclid.dmj/1505959221<strong>P. Motakis</strong>, <strong>T. Schlumprecht</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 16, 3001--3084.</p><p><strong>Abstract:</strong><br/>
We define two metrics $d_{1,\alpha}$ and $d_{\infty,\alpha}$ on each Schreier family $\mathcal{S}_{\alpha}$ , $\alpha\lt \omega_{1}$ , with which we prove the following metric characterization of the reflexivity of a Banach space $X$ : $X$ is reflexive if and only if there is an $\alpha\lt \omega_{1}$ such that there is no mapping $\Phi:\mathcal{S}_{\alpha}\to X$ for which \begin{equation*}cd_{\infty,\alpha}(A,B)\le\Vert \Phi(A)-\Phi(B)\Vert \le Cd_{1,\alpha}(A,B)\quad \text{for all }A,B\in\mathcal{S}_{\alpha}.\end{equation*} Additionally we prove, for separable and reflexive Banach spaces $X$ and certain countable ordinals $\alpha$ , that $\max(\operatorname{Sz}(X),\operatorname{Sz}(X^{*}))\le\alpha$ if and only if $(\mathcal{S}_{\alpha},d_{1,\alpha})$ does not bi-Lipschitzly embed into $X$ . Here $\operatorname{Sz}(Y)$ denotes the Szlenk index of a Banach space $Y$ .
</p>projecteuclid.org/euclid.dmj/1505959221_20171020040059Fri, 20 Oct 2017 04:00 EDTThe Coolidge–Nagata conjecturehttps://projecteuclid.org/euclid.dmj/1499911237<strong>Mariusz Koras</strong>, <strong>Karol Palka</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 16, 3085--3145.</p><p><strong>Abstract:</strong><br/>
Let $E\subseteq\mathbb{P}^{2}$ be a complex rational cuspidal curve contained in the projective plane. The Coolidge–Nagata conjecture asserts that $E$ is Cremona-equivalent to a line, that is, it is mapped onto a line by some birational transformation of $\mathbb{P}^{2}$ . The second author recently analyzed the log minimal model program run for the pair $(X,\frac{1}{2}D)$ , where $(X,D)\to(\mathbb{P}^{2},E)$ is a minimal resolution of singularities, and as a corollary he proved the conjecture in the case when more than one irreducible curve in $\mathbb{P}^{2}\setminus E$ is contracted by the process of minimalization. We prove the conjecture in the remaining cases.
</p>projecteuclid.org/euclid.dmj/1499911237_20171020040059Fri, 20 Oct 2017 04:00 EDTK-semistability is equivariant volume minimizationhttps://projecteuclid.org/euclid.dmj/1505527493<strong>Chi Li</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 16, 3147--3218.</p><p><strong>Abstract:</strong><br/>
This is a continuation of an earlier work in which we proposed a problem of minimizing normalized volumes over $\mathbb{Q}$ -Gorenstein Kawamata log terminal singularities. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler–Einstein metrics on Fano varieties. In particular, we prove that for a $\mathbb{Q}$ -Fano variety $V$ , the K-semistability of $(V,-K_{V})$ is equivalent to the condition that the normalized volume is minimized at the canonical valuation $\mathrm{ord}_{V}$ among all $\mathbb{C}^{*}$ -invariant valuations on the cone associated to any positive Cartier multiple of $-K_{V}$ . In this case, we show that $\mathrm{ord}_{V}$ is the unique minimizer among all $\mathbb{C}^{*}$ -invariant quasimonomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over $V$ .
</p>projecteuclid.org/euclid.dmj/1505527493_20171020040059Fri, 20 Oct 2017 04:00 EDTAutomatic sequences fulfill the Sarnak conjecturehttps://projecteuclid.org/euclid.dmj/1507169019<strong>Clemens Müllner</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 17, 3219--3290.</p><p><strong>Abstract:</strong><br/>
We present in this article a new method for dealing with automatic sequences. This method allows us to prove a Möbius randomness principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences. Furthermore, we can show a prime number theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to $0$ .
</p>projecteuclid.org/euclid.dmj/1507169019_20171116040048Thu, 16 Nov 2017 04:00 ESTEnumeration of real curves in $\mathbb{C}\mathbb{P}^{2n-1}$ and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariantshttps://projecteuclid.org/euclid.dmj/1507687598<strong>Penka Georgieva</strong>, <strong>Aleksey Zinger</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 17, 3291--3347.</p><p><strong>Abstract:</strong><br/>
We establish a homology relation for the Deligne–Mumford moduli spaces of real curves which lifts to a Witten–Dijkgraaf–Verlinde–Verlinde (WDVV)-type relation for a class of real Gromov–Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus $0$ real invariants with conjugate pairs of constraints to genus $0$ invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all cases when they are nonzero and thus provide nontrivial lower bounds in high-dimensional real algebraic geometry. We also show that the real invariants of the $3$ -dimensional projective space with conjugate point constraints are congruent to their complex analogues modulo $4$ .
</p>projecteuclid.org/euclid.dmj/1507687598_20171116040048Thu, 16 Nov 2017 04:00 ESTHasse principle for three classes of varieties over global function fieldshttps://projecteuclid.org/euclid.dmj/1505808016<strong>Zhiyu Tian</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 17, 3349--3424.</p><p><strong>Abstract:</strong><br/>
We give a geometric proof that the Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces, smooth cubic hypersurfaces of dimension at least $4$ in characteristic at least $7$ , and smooth complete intersections of two quadrics, which are of dimension at least $3$ , in odd characteristics.
</p>projecteuclid.org/euclid.dmj/1505808016_20171116040048Thu, 16 Nov 2017 04:00 ESTQuantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaceshttps://projecteuclid.org/euclid.dmj/1507860019<strong>Etienne Le Masson</strong>, <strong>Tuomas Sahlsten</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 18, 3425--3460.</p><p><strong>Abstract:</strong><br/>
We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and Le Masson. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Le Masson, and Lindenstrauss on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalized averaging operators over disks, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.
</p>projecteuclid.org/euclid.dmj/1507860019_20171204040052Mon, 04 Dec 2017 04:00 ESTEquidistribution in $\operatorname{Bun}_{2}(\mathbb{P}^{1})$https://projecteuclid.org/euclid.dmj/1510887861<strong>Vivek Shende</strong>, <strong>Jacob Tsimerman</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 18, 3461--3504.</p><p><strong>Abstract:</strong><br/> Fix a finite field. The set of $\operatorname{PGL}_{2}$ bundles over $\mathbb{P}^{1}$ is in bijection with the natural numbers, and carries a natural measure assigning to each bundle the inverse of the number of automorphisms. A branched double cover $\pi:C\to\mathbb{P}^{1}$ determines another measure, given by counting the number of line bundles over $C$ whose image on $\mathbb{P}^{1}$ has a given sheaf of endomorphisms. We show the measures induced by a sequence of such hyperelliptic curves tends to the canonical measure on the space of $\operatorname{PGL}_{2}$ bundles. This is a function field analogue of Duke’s theorem on the equidistribution of Heegner points, and can be proven similarly. Our real interest is the corresponding analogue of the “Mixing Conjecture” of Michel and Venkatesh. This amounts to considering measures on the space of pairs of $\operatorname{PGL}_{2}$ bundles induced by taking a fixed line bundle $\mathcal{L}$ over $C$ , and looking at the distribution of pairs $(\pi_{*}\mathcal{M},\pi_{*}(\mathcal{L}\otimes\mathcal{M}))$ . As in the number field setting, ergodic theory classifies limiting measures for sufficiently special $\mathcal{L}$ . The heart of this work is a geometric attack on the general case. We count points on intersections of translates of loci of special divisors in the Jacobian of a hyperelliptic curve. To prove equidistribution, we would require two results. The first, we prove: in high degree, the cohomologies of these loci match the cohomology of the Jacobian. The second, we establish in characteristic zero and conjecture in characteristic $p$ : the cohomology of these spaces grows at most exponentially in the genus of the curve $C$ . </p>projecteuclid.org/euclid.dmj/1510887861_20171204040052Mon, 04 Dec 2017 04:00 ESTOn the Lagrangian structure of transport equations: The Vlasov–Poisson systemhttps://projecteuclid.org/euclid.dmj/1504836225<strong>Luigi Ambrosio</strong>, <strong>Maria Colombo</strong>, <strong>Alessio Figalli</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 18, 3505--3568.</p><p><strong>Abstract:</strong><br/>
The Vlasov–Poisson system is an important nonlinear transport equation, used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of classical solutions is limited to dimensions $d\leq3$ under strong assumptions on the initial data, whereas weak solutions are known to exist under milder conditions. However, in the setting of weak solutions it is unclear whether the Eulerian description provided by the equation physically corresponds to a Lagrangian evolution of the particles. In this article we develop several general tools concerning the Lagrangian structure of transport equations with nonsmooth vector fields, and we apply these results to show that weak/renormalized solutions of Vlasov–Poisson are Lagrangian and actually that the concepts of renormalized and Lagrangian solutions are equivalent. As a corollary, we prove that finite-energy solutions in dimension $d\leq4$ are transported by a global flow (in particular, they preserve all the natural Casimir invariants), and we obtain the global existence of weak solutions in any dimension under minimal assumptions on the initial data.
</p>projecteuclid.org/euclid.dmj/1504836225_20171204040052Mon, 04 Dec 2017 04:00 ESTThe C∗-algebra of a minimal homeomorphism of zero mean dimensionhttps://projecteuclid.org/euclid.dmj/1511233446<strong>George A. Elliott</strong>, <strong>Zhuang Niu</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 18, 3569--3594.</p><p><strong>Abstract:</strong><br/>
Let $X$ be an infinite metrizable compact space, and let $\sigma:X\toX$ be a minimal homeomorphism. Suppose that $(X,\sigma)$ has zero mean topological dimension. The associated C∗-algebra $A=\mathrm{C}(X)\rtimes_{\sigma}\mathbb{Z}$ is shown to absorb the Jiang–Su algebra $\mathcal{Z}$ tensorially; that is, $A\cong A\otimes\mathcal{Z}$ . This implies that $A$ is classifiable when $(X,\sigma)$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang–Su algebra.
</p>projecteuclid.org/euclid.dmj/1511233446_20171204040052Mon, 04 Dec 2017 04:00 ESTFull-rank affine invariant submanifoldshttps://projecteuclid.org/euclid.dmj/1512529217<strong>Maryam Mirzakhani</strong>, <strong>Alex Wright</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 1, 1--40.</p><p><strong>Abstract:</strong><br/>
We show that every $\operatorname{GL}(2,R)$ orbit closure of translation surfaces is a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. We use this result to give applications related to polygonal billiards. For example, we exhibit infinitely many rational triangles whose unfoldings have dense $\operatorname{GL}(2,R)$ orbit.
</p>projecteuclid.org/euclid.dmj/1512529217_20180104040138Thu, 04 Jan 2018 04:01 ESTQuantitative nonorientability of embedded cycleshttps://projecteuclid.org/euclid.dmj/1510736537<strong>Robert Young</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 1, 41--108.</p><p><strong>Abstract:</strong><br/>
We introduce an invariant linked to some foundational questions in geometric measure theory and provide bounds on this invariant by decomposing an arbitrary cycle into uniformly rectifiable pieces. Our invariant measures the difficulty of cutting a nonorientable closed manifold or mod- $2$ cycle in $\mathbb{R}^{n}$ into orientable pieces, and we use it to answer some simple but long-open questions on filling volumes and mod- $\nu$ currents.
</p>projecteuclid.org/euclid.dmj/1510736537_20180104040138Thu, 04 Jan 2018 04:01 ESTSpectral 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 EST