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 EDT$3$ -Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number fieldhttps://projecteuclid.org/euclid.dmj/1569290597<strong>Manjul Bhargava</strong>, <strong>Zev Klagsbrun</strong>, <strong>Robert J. Lemke Oliver</strong>, <strong>Ari Shnidman</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 15, 2951--2989.</p><p><strong>Abstract:</strong><br/>
For an abelian variety $A$ over a number field $F$ , we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by- $3$ -isogeny on $A$ factors as a composition of $3$ -isogenies over $F$ . This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than $1$ . In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension $1$ , we deduce that if $E/F$ is an elliptic curve admitting a $3$ -isogeny, then the average rank of its quadratic twists is bounded. If $F$ is totally real, we moreover show that a positive proportion of twists have rank $0$ and a positive proportion have $3$ -Selmer rank $1$ . These results on bounded average ranks in families of quadratic twists represent new progress toward Goldfeld’s conjecture—which states that the average rank in the quadratic twist family of an elliptic curve over $\mathbb{Q}$ should be $1/2$ —and the first progress toward the analogous conjecture over number fields other than $\mathbb{Q}$ . Our results follow from a computation of the average size of the $\phi $ -Selmer group in the family of quadratic twists of an abelian variety admitting a $3$ -isogeny $\phi $ .
</p>projecteuclid.org/euclid.dmj/1569290597_20191011040220Fri, 11 Oct 2019 04:02 EDTOn the growth of eigenfunction averages: Microlocalization and geometryhttps://projecteuclid.org/euclid.dmj/1571040015<strong>Yaiza Canzani</strong>, <strong>Jeffrey Galkowski</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 16, 2991--3055.</p><p><strong>Abstract:</strong><br/>
Let $(M,g)$ be a smooth, compact Riemannian manifold, and let $\{\phi _{h}\}$ be an $L^{2}$ -normalized sequence of Laplace eigenfunctions, $-h^{2}\Delta _{g}\phi _{h}=\phi _{h}$ . Given a smooth submanifold $H\subset M$ of codimension $k\geq 1$ , we find conditions on the pair $(\{\phi _{h}\},H)$ for which \begin{equation*}\vert \int _{H}\phi _{h}\,d\sigma _{H}\vert =o(h^{\frac{1-k}{2}}),\quad h\to 0^{+}.\end{equation*} One such condition is that the set of conormal directions to $H$ that are recurrent has measure $0$ . In particular, we show that the upper bound holds for any $H$ if $(M,g)$ is a surface with Anosov geodesic flow or a manifold of constant negative curvature. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages.
</p>projecteuclid.org/euclid.dmj/1571040015_20191105040408Tue, 05 Nov 2019 04:04 ESTCurvature estimates for constant mean curvature surfaceshttps://projecteuclid.org/euclid.dmj/1571731217<strong>William H. Meeks III</strong>, <strong>Giuseppe Tinaglia</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 16, 3057--3102.</p><p><strong>Abstract:</strong><br/>
We derive extrinsic curvature estimates for compact disks embedded in $\mathbb{R}^{3}$ with nonzero constant mean curvature.
</p>projecteuclid.org/euclid.dmj/1571731217_20191105040408Tue, 05 Nov 2019 04:04 ESTKirillov’s orbit method: The case of discrete series representationshttps://projecteuclid.org/euclid.dmj/1571882473<strong>Paul-Emile Paradan</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 16, 3103--3134.</p><p><strong>Abstract:</strong><br/>
Let $\pi $ be a discrete series representation of a real reductive Lie group $G'$ , and let $G$ be a reductive subgroup of $G'$ . In this paper, we give a geometric expression of the $G$ -multiplicities in $\pi|_{G}$ when the representation $\pi $ is $G$ -admissible.
</p>projecteuclid.org/euclid.dmj/1571882473_20191105040408Tue, 05 Nov 2019 04:04 ESTDuality and nearby cycles over general baseshttps://projecteuclid.org/euclid.dmj/1571904013<strong>Qing Lu</strong>, <strong>Weizhe Zheng</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 16, 3135--3213.</p><p><strong>Abstract:</strong><br/>
This paper studies the sliced nearby cycle functor and its commutation with duality. Over a Henselian discrete valuation ring, we show that this commutation holds, confirming a prediction of Deligne. As an application we give a new proof of Beilinson’s theorem that the vanishing cycle functor commutes with duality up to twist. Over an excellent base scheme, we show that the sliced nearby cycle functor commutes with duality up to modification of the base. We deduce that duality preserves universal local acyclicity over an excellent regular base. We also present Gabber’s theorem that local acyclicity implies universal local acyclicity over a Noetherian base.
</p>projecteuclid.org/euclid.dmj/1571904013_20191105040408Tue, 05 Nov 2019 04:04 ESTNoncommutative boundaries and the ideal structure of reduced crossed productshttps://projecteuclid.org/euclid.dmj/1572401193<strong>Matthew Kennedy</strong>, <strong>Christopher Schafhauser</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 17, 3215--3260.</p><p><strong>Abstract:</strong><br/>
A C $^{*}$ -dynamical system is said to have the ideal separation property if every ideal in the corresponding crossed product arises from an invariant ideal in the C $^{*}$ -algebra. In this paper we characterize this property for unital C $^{*}$ -dynamical systems over discrete groups. To every C $^{*}$ -dynamical system we associate a “twisted” partial C $^{*}$ -dynamical system that encodes much of the structure of the action. This system can often be “untwisted,” for example, when the algebra is commutative or when the algebra is prime and a certain specific subgroup has vanishing Mackey obstruction. In this case, we obtain relatively simple necessary and sufficient conditions for the ideal separation property. A key idea is a notion of noncommutative boundary for a C $^{*}$ -dynamical system that generalizes Furstenberg’s notion of topological boundary for a group.
</p>projecteuclid.org/euclid.dmj/1572401193_20191112220637Tue, 12 Nov 2019 22:06 ESTCovering systems with restricted divisibilityhttps://projecteuclid.org/euclid.dmj/1572422465<strong>Robert D. Hough</strong>, <strong>Pace P. Nielsen</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 17, 3261--3295.</p><p><strong>Abstract:</strong><br/>
We prove that every distinct covering system has a modulus divisible by either $2$ or $3$ .
</p>projecteuclid.org/euclid.dmj/1572422465_20191112220637Tue, 12 Nov 2019 22:06 ESTOn higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard programhttps://projecteuclid.org/euclid.dmj/1572422464<strong>Weiwei Ao</strong>, <strong>Hardy Chan</strong>, <strong>Azahara DelaTorre</strong>, <strong>Marco A. Fontelos</strong>, <strong>María del Mar González</strong>, <strong>Juncheng Wei</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 17, 3297--3411.</p><p><strong>Abstract:</strong><br/>
We consider the problem of constructing solutions to the fractional Yamabe problem which are singular at a given smooth submanifold, for which we establish the classical gluing method of Mazzeo and Pacard (J. Differential Geom., 1996) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional-order ordinary differential equation (ODE). Thus, our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, we first provide a rigorous construction of radial fast-decaying solutions by a blowup argument and a bifurcation method. Then, second, we use conformal geometry to rewrite this nonlocal ODE, giving a hint of what a nonlocal phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrödinger equation with a Hardy-type critical potential. We construct its Green’s function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a $2$ -dimensional kernel as in the second-order case.
</p>projecteuclid.org/euclid.dmj/1572422464_20191112220637Tue, 12 Nov 2019 22:06 ESTBeyond expansion, III: Reciprocal geodesicshttps://projecteuclid.org/euclid.dmj/1573787142<strong>Jean Bourgain</strong>, <strong>Alex Kontorovich</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 18, 3413--3435.</p><p><strong>Abstract:</strong><br/>
We prove the existence of infinitely many low-lying and fundamental closed geodesics on the modular surface which are reciprocal, that is, invariant under time reversal. The method combines ideas from parts I and II of this series, namely, the dispersion method in bilinear forms, as applied to thin semigroups coming from restricted continued fractions.
</p>projecteuclid.org/euclid.dmj/1573787142_20191125040042Mon, 25 Nov 2019 04:00 ESTNewton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannianshttps://projecteuclid.org/euclid.dmj/1573462819<strong>K. Rietsch</strong>, <strong>L. Williams</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 18, 3437--3527.</p><p><strong>Abstract:</strong><br/>
In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton–Okounkov bodies for Grassmannians. We consider the Grassmannian $\mathbb{X}=\mathit{Gr}_{n-k}(\mathbb{C}^{n})$ , as well as the mirror dual Landau–Ginzburg model $(\check{\mathbb{X}}^{\circ },W:\check{\mathbb{X}}^{\circ }\to \mathbb{C})$ , where $\check{\mathbb{X}}^{\circ }$ is the complement of a particular anticanonical divisor in a Langlands dual Grassmannian $\check{\mathbb{X}}=\mathit{Gr}_{k}((\mathbb{C}^{n})^{*})$ and the superpotential $W$ has a simple expression in terms of Plücker coordinates. Grassmannians simultaneously have the structure of an $\mathcal{A}$ -cluster variety and an $\mathcal{X}$ -cluster variety; roughly speaking, a cluster variety is obtained by gluing together a collection of tori along birational maps. Given a plabic graph or, more generally, a cluster seed $G$ , we consider two associated coordinate systems: a network or $\mathcal{X}$ - cluster chart $\Phi _{G}:(\mathbb{C}^{*})^{k(n-k)}\to {\mathbb{X}}^{\circ }$ and a Plücker cluster or $\mathcal{A}$ - cluster chart $\Phi _{G}^{\vee }:(\mathbb{C}^{*})^{k(n-k)}\to \check{\mathbb{X}}^{\circ }$ . Here ${\mathbb{X}}^{\circ }$ and $\check{\mathbb{X}}^{\circ }$ are the open positroid varieties in $\mathbb{X}$ and $\check{\mathbb{X}}$ , respectively. To each $\mathcal{X}$ -cluster chart $\Phi _{G}$ and ample boundary divisor $D$ in $\mathbb{X}\setminus{\mathbb{X}}^{\circ }$ , we associate a Newton–Okounkov body $\Delta _{G}(D)$ in $\mathbb{R}^{k(n-k)}$ , which is defined as the convex hull of rational points; these points are obtained from the multidegrees of leading terms of the Laurent polynomials $\Phi _{G}^{*}(f)$ for $f$ on $\mathbb{X}$ with poles bounded by some multiple of $D$ . On the other hand, using the $\mathcal{A}$ -cluster chart $\Phi _{G}^{\vee }$ on the mirror side, we obtain a set of rational polytopes—described in terms of inequalities—by writing the superpotential $W$ as a Laurent polynomial in the $\mathcal{A}$ -cluster coordinates and then tropicalizing. Our first main result is that the Newton–Okounkov bodies $\Delta _{G}(D)$ and the polytopes obtained by tropicalization on the mirror side coincide. As an application, we construct degenerations of the Grassmannian to normal toric varieties corresponding to (dilates of ) these Newton–Okounkov bodies. Our second main result is an explicit combinatorial formula in terms of Young diagrams, for the lattice points of the Newton–Okounkov bodies, in the case in which the cluster seed $G$ corresponds to a plabic graph. This formula has an interpretation in terms of the quantum Schubert calculus of Grassmannians.
</p>projecteuclid.org/euclid.dmj/1573462819_20191125040042Mon, 25 Nov 2019 04:00 ESTSplitting of a gap in the bulk of the spectrum of random matriceshttps://projecteuclid.org/euclid.dmj/1573117217<strong>Benjamin Fahs</strong>, <strong>Igor Krasovsky</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 18, 3529--3590.</p><p><strong>Abstract:</strong><br/>
We consider the probability of having two intervals (gaps) without eigenvalues in the bulk scaling limit of the Gaussian unitary ensemble of random matrices. We describe uniform asymptotics for the transition between a single large gap and two large gaps. For the initial stage of the transition, we explicitly determine all the asymptotic terms (up to the decreasing ones) of the logarithm of the probability. We obtain our results by analyzing double-scaling asymptotics of a Toeplitz determinant whose symbol is supported on two arcs of the unit circle.
</p>projecteuclid.org/euclid.dmj/1573117217_20191125040042Mon, 25 Nov 2019 04:00 ESTIrrationality of motivic zeta functionshttps://projecteuclid.org/euclid.dmj/1574326819<strong>Michael J. Larsen</strong>, <strong>Valery A. Lunts</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 1, 1--30.</p><p><strong>Abstract:</strong><br/>
Let $K_{0}(\mathrm{Var}_{\mathbb{Q}})[1/\mathbb{L}]$ denote the Grothendieck ring of $\mathbb{Q}$ -varieties with the Lefschetz class inverted. We show that there exists a K3 surface $X$ over $\mathbb{Q}$ such that the motivic zeta function $\zeta _{X}(t):=\sum _{n}[\mathrm{Sym}^{n}X]t^{n}$ regarded as an element in $K_{0}(\mathrm{Var}_{\mathbb{Q}})[1/\mathbb{L}][[t]]$ is not a rational function in $t$ , thus disproving a conjecture of Denef and Loeser.
</p>projecteuclid.org/euclid.dmj/1574326819_20191227221030Fri, 27 Dec 2019 22:10 ESTThe rank of Mazur’s Eisenstein idealhttps://projecteuclid.org/euclid.dmj/1574305215<strong>Preston Wake</strong>, <strong>Carl Wang-Erickson</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 1, 31--115.</p><p><strong>Abstract:</strong><br/>
We use pseudodeformation theory to study Mazur’s Eisenstein ideal. Given prime numbers $N$ and $p\gt 3$ , we study the Eisenstein part of the $p$ -adic Hecke algebra for $\Gamma _{0}(N)$ . We compute the rank of this Hecke algebra (and, more generally, its Newton polygon) in terms of Massey products in Galois cohomology, thereby answering a question of Mazur and generalizing a result of Calegari and Emerton. We also give new proofs of Merel’s result on this rank and of Mazur’s results on the structure of the Hecke algebra.
</p>projecteuclid.org/euclid.dmj/1574305215_20191227221030Fri, 27 Dec 2019 22:10 ESTDiagonal actions in positive characteristichttps://projecteuclid.org/euclid.dmj/1574845213<strong>Manfred Einsiedler</strong>, <strong>Elon Lindenstrauss</strong>, <strong>Amir Mohammadi</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 1, 117--175.</p><p><strong>Abstract:</strong><br/>
We prove positive-characteristic analogues of certain measure rigidity theorems in characteristic $0$ . More specifically, we give a classification result for positive entropy measures on quotients of $\operatorname{SL}_{d}$ and a classification of joinings for higher-rank actions on simply connected, absolutely almost simple groups.
</p>projecteuclid.org/euclid.dmj/1574845213_20191227221030Fri, 27 Dec 2019 22:10 ESTThe Fyodorov–Bouchaud formula and Liouville conformal field theoryhttps://projecteuclid.org/euclid.dmj/1576573213<strong>Guillaume Remy</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 1, 177--211.</p><p><strong>Abstract:</strong><br/>
In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (subcritical) Gaussian multiplicative chaos (GMC) associated to the Gaussian free field (GFF) on the unit circle. In this paper we will give a proof of this formula. In the mathematical literature this is the first occurrence of an explicit probability density for the total mass of a GMC measure. The key observation of our proof is that the negative moments of the total mass of GMC determine its law and are equal to one-point correlation functions of Liouville conformal field theory in the disk recently defined by Huang, Rhodes, and Vargas. The rest of the proof then consists in implementing rigorously the framework of conformal field theory (Belavin–Polyakov–Zamolodchikov equations for degenerate field insertions) in a probabilistic setting to compute the negative moments. Finally, we will discuss applications to random matrix theory, asymptotics of the maximum of the GFF, and tail expansions of GMC.
</p>projecteuclid.org/euclid.dmj/1576573213_20191227221030Fri, 27 Dec 2019 22:10 ESTLattice envelopeshttps://projecteuclid.org/euclid.dmj/1580288423<strong>Uri Bader</strong>, <strong>Alex Furman</strong>, <strong>Roman Sauer</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 2, 213--278.</p><p><strong>Abstract:</strong><br/>
We introduce a class of countable groups by some abstract group-theoretic conditions. This class includes linear groups with finite amenable radical and finitely generated residually finite groups with some nonvanishing $\ell ^{2}$ -Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group $\Gamma $ in this class, we determine the general structure of the possible lattice embeddings of $\Gamma $ , that is, of all compactly generated, locally compact groups that contain $\Gamma $ as a lattice. This leads to a precise description of possible nonuniform lattice embeddings of groups in this class. Further applications include the determination of possible lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature.
</p>projecteuclid.org/euclid.dmj/1580288423_20200129040042Wed, 29 Jan 2020 04:00 ESTQuantum unique ergodicity for half-integral weight automorphic formshttps://projecteuclid.org/euclid.dmj/1580288424<strong>Stephen Lester</strong>, <strong>Maksym Radziwiłł</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 2, 279--351.</p><p><strong>Abstract:</strong><br/>
We investigate the analogue of the quantum unique ergodicity (QUE ) conjecture for half-integral weight automorphic forms. Assuming the generalized Riemann hypothesis (GRH) , we establish both QUE for half-integral weight Hecke Maaß cusp forms for $\Gamma _{0}(4)$ lying in Kohnen’s plus subspace and mass equidistribution for half-integral weight holomorphic Hecke cusp forms for $\Gamma _{0}(4)$ lying in Kohnen’s plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp forms equidistribute with respect to hyperbolic measure on $\Gamma _{0}(4)\backslash \mathbb{H}$ as the weight tends to infinity.
</p>projecteuclid.org/euclid.dmj/1580288424_20200129040042Wed, 29 Jan 2020 04:00 ESTDifferentials on the arc spacehttps://projecteuclid.org/euclid.dmj/1580288425<strong>Tommaso de Fernex</strong>, <strong>Roi Docampo</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 2, 353--396.</p><p><strong>Abstract:</strong><br/>
We provide a description of the sheaves of Kähler differentials of the arc space and jet schemes of an arbitrary scheme where these sheaves are computed directly from the sheaf of differentials of the given scheme. Several applications on the structure of arc spaces are presented.
</p>projecteuclid.org/euclid.dmj/1580288425_20200129040042Wed, 29 Jan 2020 04:00 ESTExceptional splitting of reductions of abelian surfaceshttps://projecteuclid.org/euclid.dmj/1576033298<strong>Ananth N. Shankar</strong>, <strong>Yunqing Tang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 3, 397--434.</p><p><strong>Abstract:</strong><br/>
Heuristics based on the Sato–Tate conjecture and the Lang–Trotter philosophy suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces with real multiplication. As in previous work by Charles and Elkies, this shows that a density $0$ set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.
</p>projecteuclid.org/euclid.dmj/1576033298_20200217040111Mon, 17 Feb 2020 04:01 ESTOn the stability of blowup solutions for the critical corotational wave-map problemhttps://projecteuclid.org/euclid.dmj/1578474143<strong>Joachim Krieger</strong>, <strong>Shuang Miao</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 3, 435--532.</p><p><strong>Abstract:</strong><br/>
We show that the finite time blowup solutions for the corotational wave-map problem constructed by the first author along with Gao, Schlag, and Tataru are stable under suitably small perturbations within the corotational class, provided that the scaling parameter $\lambda(t)=t^{-1-\nu}$ is sufficiently close to $t^{-1}$ ; that is, the constant $\nu$ is sufficiently small and positive. The method of proof is inspired by recent work by the first author and Burzio, but takes advantage of geometric structures of the wave-map problem, already used in previous work by the first author, Bejenaru, Tataru, Raphaël, and Rodnianski, to simplify the analysis. In particular, we heavily exploit the fact that the resonance at zero satisfies a natural first-order differential equation.
</p>projecteuclid.org/euclid.dmj/1578474143_20200217040111Mon, 17 Feb 2020 04:01 ESTMatrix factorization of Morse–Bott functionshttps://projecteuclid.org/euclid.dmj/1580202168<strong>Constantin Teleman</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 3, 533--549.</p><p><strong>Abstract:</strong><br/>
For a function $W\in \mathbb{C}[X]$ on a smooth algebraic variety $X$ with Morse–Bott critical locus $Y\subset X$ , Kapustin, Rozansky, and Saulina suggest that the associated matrix factorization category $\operatorname{MF}(X;W)$ should be equivalent to the differential graded category of $2$ -periodic coherent complexes on $Y$ (with a topological twist from the normal bundle). We confirm their conjecture in the special case when the first neighborhood of $Y$ in $X$ is split and establish the corrected general statement. The answer involves the full Gerstenhaber structure on Hochschild cochains. This note was inspired by the failure of the conjecture, observed by Pomerleano and Preygel, when $X$ is a general $1$ -parameter deformation of a $K3$ surface $Y$ .
</p>projecteuclid.org/euclid.dmj/1580202168_20200217040111Mon, 17 Feb 2020 04:01 ESTOn the width of transitive sets: Bounds on matrix coefficients of finite groupshttps://projecteuclid.org/euclid.dmj/1578539084<strong>Ben Green</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 3, 551--578.</p><p><strong>Abstract:</strong><br/>
We say that a finite subset of the unit sphere in $\mathbf{R}^{d}$ is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times $(\log d)^{-1/2}$ .
This is a consequence of the following result: if $G$ is a finite group and $\rho :G\rightarrow \operatorname{U}_{d}(\mathbf{C})$ a unitary representation, and if $v\in \mathbf{C}^{d}$ is a unit vector, then there is another unit vector $w\in \mathbf{C}^{d}$ such that \begin{equation*}\sup _{g\in G}\vert \langle \rho (g)v,w\rangle \vert \leq (1+c\log d)^{-1/2}.\end{equation*}
These results answer a question of Yufei Zhao. An immediate consequence of our result is that the diameter of any quotient $S(\mathbf{R}^{d})/G$ of the unit sphere by a finite group $G$ of isometries is at least $\pi /2-o_{d\rightarrow \infty }(1)$ .
</p>projecteuclid.org/euclid.dmj/1578539084_20200217040111Mon, 17 Feb 2020 04:01 ESTIrreducible polynomials of bounded heighthttps://projecteuclid.org/euclid.dmj/1578646813<strong>Lior Bary-Soroker</strong>, <strong>Gady Kozma</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 4, 579--598.</p><p><strong>Abstract:</strong><br/>
The goal of this paper is to prove that a random polynomial with independent and identically distributed random coefficients taking values uniformly in $\{1,\ldots ,210\}$ is irreducible with probability tending to $1$ as the degree tends to infinity. Moreover, we prove that the Galois group of the random polynomial contains the alternating group, again with probability tending to $1$ .
</p>projecteuclid.org/euclid.dmj/1578646813_20200305220257Thu, 05 Mar 2020 22:02 ESTA McKay correspondence for reflection groupshttps://projecteuclid.org/euclid.dmj/1582167909<strong>Ragnar-Olaf Buchweitz</strong>, <strong>Eleonore Faber</strong>, <strong>Colin Ingalls</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 4, 599--669.</p><p><strong>Abstract:</strong><br/>
We construct a noncommutative desingularization of the discriminant of a finite reflection group $G$ as a quotient of the skew group ring $A=S*G$ . If $G$ is generated by order $2$ reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement $\mathcal{A}(G)$ viewed as a module over the coordinate ring $S^{G}/(\Delta )$ of the discriminant of $G$ . This yields, in particular, a correspondence between the nontrivial irreducible representations of $G$ to certain maximal Cohen–Macaulay modules over the coordinate ring $S^{G}/(\Delta )$ . These maximal Cohen–Macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement ${\mathcal{A}}(G)$ viewed as a module over $S^{G}/(\Delta )$ . We identify some of the corresponding matrix factorizations, namely, the so-called logarithmic (co-)residues of the discriminant.
</p>projecteuclid.org/euclid.dmj/1582167909_20200305220257Thu, 05 Mar 2020 22:02 ESTExtreme superposition: Rogue waves of infinite order and the Painlevé-III hierarchyhttps://projecteuclid.org/euclid.dmj/1580202167<strong>Deniz Bilman</strong>, <strong>Liming Ling</strong>, <strong>Peter D. Miller</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 4, 671--760.</p><p><strong>Abstract:</strong><br/>
We study the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently proposed Riemann–Hilbert representation of the rogue wave solution of arbitrary order $k$ , we establish the existence of a limiting profile of the rogue wave in the large- $k$ limit when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schrödinger equation in the rescaled variables—the rogue wave of infinite order—which also satisfies ordinary differential equations with respect to space and time. The spatial differential equations are identified with certain members of the Painlevé-III hierarchy. We compute the far-field asymptotic behavior of the near-field limit solution and compare the asymptotic formulas with the exact solution using numerical methods for solving Riemann–Hilbert problems. In a certain transitional region for the asymptotics, the near-field limit function is described by a specific globally defined tritronquée solution of the Painlevé-II equation. These properties lead us to regard the rogue wave of infinite order as a new special function.
</p>projecteuclid.org/euclid.dmj/1580202167_20200305220257Thu, 05 Mar 2020 22:02 ESTCanonical parameterizations of metric diskshttps://projecteuclid.org/euclid.dmj/1580958161<strong>Alexander Lytchak</strong>, <strong>Stefan Wenger</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 4, 761--797.</p><p><strong>Abstract:</strong><br/>
We use the recently established existence and regularity of area and energy minimizing disks in metric spaces to obtain canonical parameterizations of metric surfaces. Our approach yields a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parameterizations of linearly locally connected, Ahlfors $2$ -regular metric $2$ -spheres. Generalizations and applications to the geometry of such surfaces are described.
</p>projecteuclid.org/euclid.dmj/1580958161_20200305220257Thu, 05 Mar 2020 22:02 ESTErrata to “Discriminants in the Grothendieck ring”https://projecteuclid.org/euclid.dmj/1580958160<strong>Ravi Vakil</strong>, <strong>Melanie Matchett Wood</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 4, 799--800.</p><p><strong>Abstract:</strong><br/>
We correct a definition in “Discriminants in the Grothendieck ring,” Duke Math. J. 164 (2015), no. 6, 1139–1185.
</p>projecteuclid.org/euclid.dmj/1580958160_20200305220257Thu, 05 Mar 2020 22:02 ESTMinimal modularity lifting for nonregular symplectic representationshttps://projecteuclid.org/euclid.dmj/1581995237<strong>Frank Calegari</strong>, <strong>David Geraghty</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 5, 801--896.</p><p><strong>Abstract:</strong><br/>
We prove a minimal modularity lifting theorem for Galois representations (conjecturally) associated to Siegel modular forms of genus $2$ which are holomorphic limits of discrete series at infinity.
</p>projecteuclid.org/euclid.dmj/1581995237_20200324220610Tue, 24 Mar 2020 22:06 EDTOn the analogy between real reductive groups and Cartan motion groups: Contraction of irreducible tempered representationshttps://projecteuclid.org/euclid.dmj/1580958162<strong>Alexandre Afgoustidis</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 5, 897--960.</p><p><strong>Abstract:</strong><br/>
Attached to any reductive Lie group $G$ is a “Cartan motion group” $G_{0}$ —a Lie group with the same dimension as $G$ , but a simpler group structure. A natural one-to-one correspondence between the irreducible tempered representations of $G$ and the unitary irreducible representations of $G_{0}$ , whose existence was suggested by Mackey in the 1970s, has recently been described by the author. In the present article, we use the existence of a family of groups interpolating between $G$ and $G_{0}$ to realize the bijection as a deformation: for every irreducible tempered representation $\pi $ of G, we build, in an appropriate Fréchet space, a family of subspaces, and evolution operators that contract $\pi $ onto the corresponding representation of $G_{0}$ .
</p>projecteuclid.org/euclid.dmj/1580958162_20200324220610Tue, 24 Mar 2020 22:06 EDTDeviation inequalities for random walkshttps://projecteuclid.org/euclid.dmj/1584151212<strong>P. Mathieu</strong>, <strong>A. Sisto</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 5, 961--1036.</p><p><strong>Abstract:</strong><br/>
We study random walks on groups, with the feature that, roughly speaking, successive positions of the walk tend to be “aligned.” We formalize and quantify this property by means of the notion of deviation inequalities . We show that deviation inequalities have several consequences, including central limit theorems , the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point. In the second part of this article, we show that the (exponential) deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups. These include nonelementary (relatively) hyperbolic groups, mapping class groups , many groups acting on CAT(0) spaces, and small cancellation groups.
</p>projecteuclid.org/euclid.dmj/1584151212_20200324220610Tue, 24 Mar 2020 22:06 EDTErrata for “Mapping class group and a global Torelli theorem for hyperkähler manifolds” by Misha Verbitskyhttps://projecteuclid.org/euclid.dmj/1583398898<p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 5, 1037--1038.</p>projecteuclid.org/euclid.dmj/1583398898_20200324220610Tue, 24 Mar 2020 22:06 EDTAn automorphic generalization of the Hermite–Minkowski theoremhttps://projecteuclid.org/euclid.dmj/1584151213<strong>Gaëtan Chenevier</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 6, 1039--1075.</p><p><strong>Abstract:</strong><br/>
We show that, for any integer $N\geq1$ , there are only finitely many cuspidal algebraic automorphic representations of $\mathrm{GL}_{n}$ over $\mathbb{Q}$ , with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,\dots,23\}$ . More generally, we define an explicit sequence $(\mathrm{r}(w))_{w\geq0}$ such that, for any number field $E$ whose root discriminant is less than $\mathrm{r}(w)$ and any ideal $\mathcal{N}$ in the ring of integers of $E$ , there are only finitely many cuspidal algebraic automorphic representations of $\mathrm{GL}_{n}$ over $E$ , with $n$ varying, whose conductor is $\mathcal{N}$ and whose weights are in the interval $\{0,\dots,w\}$ . We also show that, assuming a version of the generalized Riemann hypothesis, we may replace $\mathrm{r}(w)$ with $8\pi e^{-\psi(1+w)}$ in this statement. The proofs here are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin–Selberg $\mathrm{L}$ -functions. Both the effectiveness and the optimality of the methods are discussed.
</p>projecteuclid.org/euclid.dmj/1584151213_20200415220213Wed, 15 Apr 2020 22:02 EDTSymmetric Mahler’s conjecture for the volume product in the $3$ -dimensional casehttps://projecteuclid.org/euclid.dmj/1582081313<strong>Hiroshi Iriyeh</strong>, <strong>Masataka Shibata</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 6, 1077--1134.</p><p><strong>Abstract:</strong><br/>
We prove Mahler’s conjecture concerning the volume product of centrally symmetric, convex bodies in $\mathbb{R}^{n}$ in the case where $n=3$ . More precisely, we show that, for every $3$ -dimensional, centrally symmetric, convex body $K\subset\mathbb{R}^{3}$ , the volume product $|{K}||{K^{\circ}}|$ is greater than or equal to $32/3$ with equality if and only if $K$ or $K^{\circ}$ is a parallelepiped.
</p>projecteuclid.org/euclid.dmj/1582081313_20200415220213Wed, 15 Apr 2020 22:02 EDTExplicit equations of a fake projective planehttps://projecteuclid.org/euclid.dmj/1583377257<strong>Lev A. Borisov</strong>, <strong>JongHae Keum</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 6, 1135--1162.</p><p><strong>Abstract:</strong><br/>
Fake projective planes are smooth, complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the $2$ -dimensional ball by explicitly written arithmetic subgroups. In the following, we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order $7$ automorphism.
</p>projecteuclid.org/euclid.dmj/1583377257_20200415220213Wed, 15 Apr 2020 22:02 EDTMaximality of Galois actions for abelian and hyper-Kähler varietieshttps://projecteuclid.org/euclid.dmj/1585706497<strong>Chun Yin Hui</strong>, <strong>Michael Larsen</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 6, 1163--1207.</p><p><strong>Abstract:</strong><br/>
Let $\{\rho _{\ell }\}_{\ell }$ be the system of $\ell $ -adic representations arising from the $i$ th $\ell $ -adic cohomology of a proper smooth variety $X$ defined over a number field $K$ . Let $\Gamma _{\ell }$ and $\mathbf{G}_{\ell }$ be, respectively, the image and the algebraic monodromy group of $\rho _{\ell }$ . We prove that the reductive quotient of $\mathbf{G}_{\ell }^{\circ }$ is unramified over every degree $12$ totally ramified extension of $\mathbb{Q}_{\ell }$ for all sufficiently large $\ell $ . We give a necessary and sufficient condition $(\ast )$ on $\{\rho _{\ell }\}_{\ell }$ such that, for all sufficiently large $\ell $ , the subgroup $\Gamma _{\ell }$ is in some sense maximal compact in $\mathbf{G}_{\ell }(\mathbb{Q}_{\ell })$ . This is used to deduce Galois maximality results for $\ell $ -adic representations arising from abelian varieties (for all $i$ ) and hyper-Kähler varieties ( $i=2$ ) defined over finitely generated fields over $\mathbb{Q}$ .
</p>projecteuclid.org/euclid.dmj/1585706497_20200415220213Wed, 15 Apr 2020 22:02 EDTThe Fourier expansion of modular forms on quaternionic exceptional groupshttps://projecteuclid.org/euclid.dmj/1587456010<strong>Aaron Pollack</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 7, 1209--1280.</p><p><strong>Abstract:</strong><br/>
Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$ , with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross and Wallach). Then there is a notion of modular forms for $G$ , anchored on the so-called quaternionic discrete series representations of $G(\mathbf{R})$ . The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on $G$ , along the unipotent radical $N$ of the Heisenberg parabolic $P$ of $G$ .
</p>projecteuclid.org/euclid.dmj/1587456010_20200506220103Wed, 06 May 2020 22:01 EDTGalois action on universal covers of Kodaira fibrationshttps://projecteuclid.org/euclid.dmj/1585188223<strong>Gabino González-Diez</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 7, 1281--1303.</p><p><strong>Abstract:</strong><br/>
Catanese has recently asked if there exists an element of the absolute Galois group $\sigma \in \operatorname{Gal}(\overline{\mathbb{Q}})$ for which there is a Kodaira fibration $f:S\to B$ defined over a number field such that the universal covers of $S$ and its Galois conjugate surface $S^{\sigma }$ are not isomorphic. The main result of this article is that every element $\sigma \neq \mathrm{Id}$ has this property.
</p>projecteuclid.org/euclid.dmj/1585188223_20200506220103Wed, 06 May 2020 22:01 EDTRegularity of optimal transport between planar convex domainshttps://projecteuclid.org/euclid.dmj/1585188224<strong>Ovidiu Savin</strong>, <strong>Hui Yu</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 7, 1305--1327.</p><p><strong>Abstract:</strong><br/>
For $0\lt p\lt +\infty $ , we prove a global $W^{2,p}$ -estimate for potentials of optimal transport maps between convex domains in the plane. Among the tools developed for that purpose are obliqueness in general convex domains and estimates for the growth of eccentricity of sections of the potentials.
</p>projecteuclid.org/euclid.dmj/1585188224_20200506220103Wed, 06 May 2020 22:01 EDTLower tail of the KPZ equationhttps://projecteuclid.org/euclid.dmj/1586570419<strong>Ivan Corwin</strong>, <strong>Promit Ghosal</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 7, 1329--1395.</p><p><strong>Abstract:</strong><br/>
We provide the first tight bounds on the lower tail probability of the one-point distribution of the Kardar–Parisi–Zhang (KPZ) equation with narrow wedge initial data. Our bounds hold for all sufficiently large times $T$ and demonstrates a crossover between superexponential decay with exponent $\frac{5}{2}$ (and leading prefactor $\frac{4}{15\pi }T^{1/3}$ ) for tail depth greater than $T^{2/3}$ , and exponent $3$ (with leading prefactor at least $\frac{1}{12}$ ) for tail depth less than $T^{2/3}$ .
</p>projecteuclid.org/euclid.dmj/1586570419_20200506220103Wed, 06 May 2020 22:01 EDTOn sharp rates and analytic compactifications of asymptotically conical Kähler metricshttps://projecteuclid.org/euclid.dmj/1585274499<strong>Chi Li</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 8, 1397--1483.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a complex manifold, and let $S\hookrightarrow X$ be an embedding of a complex submanifold. Assuming that the embedding is $(k-1)$ -linearizable or $(k-1)$ -comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism $F$ from a small neighborhood of the zero section in the normal bundle $N_{S}$ to a small neighborhood of $S$ in $X$ such that $F$ is in a precise sense holomorphic up to the $(k-1)$ th order. Using this $F$ , we obtain optimal estimates on asymptotic rates for asymptotically conical (AC) Calabi–Yau (CY) metrics constructed by Tian and Yau. Furthermore, when $S$ is an ample divisor satisfying an appropriate cohomological condition, we relate the order of comfortable embedding to the weight of the deformation of the normal isolated cone singularity arising from the deformation to the normal cone. We also give an example showing that the condition of comfortable embedding depends on the splitting liftings. We then prove an analytic compactification result for the deformation of the complex structure on an affine cone that decays to any positive order at infinity. This can be seen as an analytic counterpart of Pinkham’s result on deformations of cone singularities with negative weights.
</p>projecteuclid.org/euclid.dmj/1585274499_20200521220042Thu, 21 May 2020 22:00 EDTArithmeticity of discrete subgroups containing horospherical latticeshttps://projecteuclid.org/euclid.dmj/1587088815<strong>Yves Benoist</strong>, <strong>Sébastien Miquel</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 8, 1485--1539.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a semisimple real algebraic Lie group of real rank at least $2$ , and let $U$ be the unipotent radical of a nontrivial parabolic subgroup. We prove that a discrete Zariski-dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$ . This solves a conjecture of Margulis and extends previous works of Selberg and Oh.
</p>projecteuclid.org/euclid.dmj/1587088815_20200521220042Thu, 21 May 2020 22:00 EDTUniqueness of the blowup at isolated singularities for the Alt–Caffarelli functionalhttps://projecteuclid.org/euclid.dmj/1587974463<strong>Max Engelstein</strong>, <strong>Luca Spolaor</strong>, <strong>Bozhidar Velichkov</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 8, 1541--1601.</p><p><strong>Abstract:</strong><br/>
We prove the uniqueness of blowups and $C^{1,\log}$ -regularity for the free-boundary of minimizers of the Alt–Caffarelli functional at points where one blowup has an isolated singularity. We do this by establishing a (log-)epiperimetric inequality for the Weiss energy for traces close to that of a cone with isolated singularity, whose free boundary is graphical and smooth over that of the cone in the sphere. With additional assumptions on the cone, we can prove a classical epiperimetric inequality which can be applied to deduce a $C^{1,\alpha}$ -regularity result. We also show that these additional assumptions are satisfied by the De Silva–Jerison-type cones, which are the only known examples of minimizing cones with isolated singularity. Our approach draws a connection between epiperimetric inequalities and the Łojasiewicz inequality, and, to our knowledge, provides the first regularity result at singular points in the one-phase Bernoulli problem.
</p>projecteuclid.org/euclid.dmj/1587974463_20200521220042Thu, 21 May 2020 22:00 EDTThe Noether inequality for algebraic $3$ -foldshttps://projecteuclid.org/euclid.dmj/1588903314<strong>Jungkai A. Chen</strong>, <strong>Meng Chen</strong>, <strong>Chen Jiang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 9, 1603--1645.</p><p><strong>Abstract:</strong><br/>
We establish the Noether inequality for projective $3$ -folds, and, specifically, we prove that the inequality \begin{equation*}\operatorname{vol}(X)\geq\frac{4}{3}p_{g}(X)-{\frac{10}{3}}\end{equation*} holds for all projective $3$ -folds $X$ of general type with either $p_{g}(X)\leq 4$ or $p_{g}(X)\geq 21$ , where $p_{g}(X)$ is the geometric genus and $\operatorname{vol}(X)$ is the canonical volume. This inequality is optimal due to known examples found by Kobayashi in 1992.
</p>projecteuclid.org/euclid.dmj/1588903314_20200602040025Tue, 02 Jun 2020 04:00 EDTHigher coherent cohomology and $p$ -adic modular forms of singular weightshttps://projecteuclid.org/euclid.dmj/1588903316<strong>V. Pilloni</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 9, 1647--1807.</p><p><strong>Abstract:</strong><br/>
We investigate the $p$ -adic properties of higher coherent cohomology of automorphic vector bundles of singular weights on Siegel threefolds.
</p>projecteuclid.org/euclid.dmj/1588903316_20200602040025Tue, 02 Jun 2020 04:00 EDTPrime numbers in two baseshttps://projecteuclid.org/euclid.dmj/1588903315<strong>Michael Drmota</strong>, <strong>Christian Mauduit</strong>, <strong>Joël Rivat</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 10, 1809--1876.</p><p><strong>Abstract:</strong><br/>
We estimate the sums \[\sum _{n\leq x}\Lambda (n)f(n)g(n)\exp (2i\pi \vartheta n)\] and \[\sum _{n\leq x}\mu (n)f(n)g(n)\exp (2i\pi \vartheta n),\] where $\Lambda $ denotes the von Mangoldt function (and $\mu $ the Möbius function) whenever $q_{1}$ and $q_{2}$ are two coprime bases, $f$ (resp., $g$ ) is a strongly $q_{1}$ -multiplicative (resp., strongly $q_{2}$ -multiplicative) function of modulus $1$ , and $\vartheta $ is a real number. The goal of this work is to introduce a new approach to study these sums involving simultaneously two different bases combining Fourier analysis, Diophantine approximation, and combinatorial arguments. We deduce from these estimates a prime number theorem (and Möbius orthogonality) for sequences of integers with digit properties in two coprime bases.
</p>projecteuclid.org/euclid.dmj/1588903315_20200701220203Wed, 01 Jul 2020 22:02 EDTIsing model and the positive orthogonal Grassmannianhttps://projecteuclid.org/euclid.dmj/1589335768<strong>Pavel Galashin</strong>, <strong>Pavlo Pylyavskyy</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 10, 1877--1942.</p><p><strong>Abstract:</strong><br/>
We use inequalities to completely describe the set of boundary correlation matrices of planar Ising networks embedded in a disk. Specifically, we build on a recent result of Lis to give a simple bijection between such correlation matrices and points in the totally nonnegative part of the orthogonal Grassmannian, which was introduced in 2013 in the study of the scattering amplitudes of Aharony–Bergman–Jafferis–Maldacena (ABJM) theory. We also show that the edge parameters of the Ising model for reduced networks can be uniquely recovered from boundary correlations, solving the inverse problem. Under our correspondence, the Kramers–Wannier high/low temperature duality transforms into the cyclic symmetry of the Grassmannian, and using this cyclic symmetry, we prove that the spaces under consideration are homeomorphic to closed balls.
</p>projecteuclid.org/euclid.dmj/1589335768_20200701220203Wed, 01 Jul 2020 22:02 EDTIntegral and rational mapping classeshttps://projecteuclid.org/euclid.dmj/1591689610<strong>Fedor Manin</strong>, <strong>Shmuel Weinberger</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 10, 1943--1969.</p><p><strong>Abstract:</strong><br/>
Let $X$ and $Y$ be finite complexes. When $Y$ is a nilpotent space, it has a rationalization $Y\to Y_{(0)}$ which is well understood. Early on it was found that the induced map $[X,Y]\to[X,Y_{(0)}]$ on sets of mapping classes is finite-to-one. The sizes of the preimages need not be bounded; we show, however, that, as the complexity (in a suitable sense) of a rational mapping class increases, these sizes are at most polynomial. This “torsion” information about $[X,Y]$ is in some sense orthogonal to rational homotopy theory but is nevertheless an invariant of the rational homotopy type of $Y$ in at least some cases. The notion of complexity is geometric, and we also prove a conjecture of Gromov regarding the number of mapping classes that have Lipschitz constant at most $L$ .
</p>projecteuclid.org/euclid.dmj/1591689610_20200701220203Wed, 01 Jul 2020 22:02 EDTOn the Hitchin morphism for higher-dimensional varietieshttps://projecteuclid.org/euclid.dmj/1591149667<strong>T. H. Chen</strong>, <strong>B. C. Ngô</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 10, 1971--2004.</p>projecteuclid.org/euclid.dmj/1591149667_20200701220203Wed, 01 Jul 2020 22:02 EDTA proof of the multiplicity $1$ conjecture for min-max minimal surfaces in arbitrary codimensionhttps://projecteuclid.org/euclid.dmj/1594195214<strong>Alessandro Pigati</strong>, <strong>Tristan Rivière</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 11, 2005--2044.</p><p><strong>Abstract:</strong><br/>
Given any admissible $k$ -dimensional family of immersions of a given closed oriented surface into an arbitrary closed Riemannian manifold, we prove that the corresponding min-max width for the area is achieved by a smooth (possibly branched) immersed minimal surface with multiplicity $1$ and Morse index bounded by $k$ .
</p>projecteuclid.org/euclid.dmj/1594195214_20200731220114Fri, 31 Jul 2020 22:01 EDTThe density of rational points near hypersurfaceshttps://projecteuclid.org/euclid.dmj/1593741705<strong>Jing-Jing Huang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 11, 2045--2077.</p><p><strong>Abstract:</strong><br/>
We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies on the synthesis of Poisson summation, projective duality, and the method of stationary phase. This has surprising applications to counting rational points lying on the manifold; indeed, we are able to prove an analogue of Serre’s dimension growth conjecture (originally stated for projective varieties) in this general setup. As another consequence of our main counting result, we solve the generalized Baker–Schmidt problem in the simultaneous setting for hypersurfaces.
</p>projecteuclid.org/euclid.dmj/1593741705_20200731220114Fri, 31 Jul 2020 22:01 EDTThe limit shape of convex hull peelinghttps://projecteuclid.org/euclid.dmj/1593223320<strong>Jeff Calder</strong>, <strong>Charles K. Smart</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 11, 2079--2124.</p><p><strong>Abstract:</strong><br/>
We prove that the convex peeling of a random point set in dimension $d$ approximates motion by the $1/(d+1)$ power of Gaussian curvature. We use viscosity solution theory to interpret the limiting partial differential equation (PDE). We use the martingale method to solve the cell problem associated to convex peeling. Our proof follows the program of Armstrong and Cardaliaguet for homogenization of geometric motions, but with completely different ingredients.
</p>projecteuclid.org/euclid.dmj/1593223320_20200731220114Fri, 31 Jul 2020 22:01 EDTThe nonequivariant coherent-constructible correspondence for toric stackshttps://projecteuclid.org/euclid.dmj/1594778753<strong>Tatsuki Kuwagaki</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 11, 2125--2197.</p><p><strong>Abstract:</strong><br/>
The nonequivariant coherent-constructible correspondence is a microlocal-geometric interpretation of homological mirror symmetry for toric varieties conjectured by Fang, Liu, Treumann, and Zaslow. We prove a generalization of this conjecture for a class of toric stacks which includes any toric variety and toric orbifold. Our proof is based on gluing descriptions of $\infty $ -categories of both sides.
</p>projecteuclid.org/euclid.dmj/1594778753_20200731220114Fri, 31 Jul 2020 22:01 EDTAddendum to “The Noether inequality for algebraic $3$ -folds”https://projecteuclid.org/euclid.dmj/1593223321<strong>Jungkai A. Chen</strong>, <strong>Meng Chen</strong>, <strong>Chen Jiang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 169, Number 11, 2199--2204.</p><p><strong>Abstract:</strong><br/>
We provide an improvement to “The Noether inequality for algebraic $3$ -folds,” Duke Math. J. 169 (2020), no. 9, 1603–1645.
</p>projecteuclid.org/euclid.dmj/1593223321_20200731220114Fri, 31 Jul 2020 22:01 EDT