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 EDTThe dynamical André–Oort conjecture: Unicritical polynomialshttp://projecteuclid.org/euclid.dmj/1475266423<strong>D. Ghioca</strong>, <strong>H. Krieger</strong>, <strong>K. D. Nguyen</strong>, <strong>H. Ye</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 1, 1--25.</p><p><strong>Abstract:</strong><br/>
We establish equidistribution with respect to the bifurcation measure of postcritically finite (PCF) maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set $\mathcal{M}_{2}$ (or generalized Mandelbrot set $\mathcal{M}_{d}$ for degree $d\gt 2$ ), we classify all curves $C\subset{\mathbb{A}}^{2}$ defined over ${\mathbb{C}}$ with Zariski-dense subsets of points $(a,b)\in C$ , such that both $z^{d}+a$ and $z^{d}+b$ are simultaneously PCF for a fixed degree $d\geq2$ . Our result is analogous to the famous result of André regarding plane curves which contain infinitely many points with both coordinates being complex multiplication parameters in the moduli space of elliptic curves and is the first complete case of the dynamical André–Oort phenomenon studied by Baker and DeMarco.
</p>projecteuclid.org/euclid.dmj/1475266423_20170105220247Thu, 05 Jan 2017 22:02 ESTDerived equivalences for rational Cherednik algebrashttp://projecteuclid.org/euclid.dmj/1472743767<strong>Ivan Losev</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 1, 27--73.</p><p><strong>Abstract:</strong><br/>
Let $W$ be a complex reflection group, and let $H_{c}(W)$ be the rational Cherednik algebra for $W$ depending on a parameter $c$ . One can consider the category $\mathcal{O}$ for $H_{c}(W)$ . We prove a conjecture of Rouquier that the categories $\mathcal{O}$ for $H_{c}(W)$ and $H_{c'}(W)$ are derived-equivalent, provided that the parameters $c,c'$ have integral difference. Two main ingredients of the proof are a connection between the Ringel duality and Harish-Chandra bimodules and an analogue of a deformation technique developed by the author and Bezrukavnikov. We also show that some of the derived equivalences we construct are perverse.
</p>projecteuclid.org/euclid.dmj/1472743767_20170105220247Thu, 05 Jan 2017 22:02 ESTDerived automorphism groups of K3 surfaces of Picard rank $1$http://projecteuclid.org/euclid.dmj/1473854468<strong>Arend Bayer</strong>, <strong>Tom Bridgeland</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 1, 75--124.</p><p><strong>Abstract:</strong><br/>
We give a complete description of the group of exact autoequivalences of the bounded derived category of coherent sheaves on a K3 surface of Picard rank $1$ . We do this by proving that a distinguished connected component of the space of stability conditions is preserved by all autoequivalences and is contractible.
</p>projecteuclid.org/euclid.dmj/1473854468_20170105220247Thu, 05 Jan 2017 22:02 ESTThe Cauchy–Szegő projection for domains in $\mathbb{C}^{n}$ with minimal smoothnesshttp://projecteuclid.org/euclid.dmj/1478919691<strong>Loredana Lanzani</strong>, <strong>Elias M. Stein</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 1, 125--176.</p><p><strong>Abstract:</strong><br/>
We prove the $L^{p}(bD)$ -regularity of the Cauchy–Szegő projection (also known as the Szegő projection) for bounded domains $D\subset\mathbb{C}^{n}$ which are strongly pseudoconvex and whose boundary satisfies the minimal regularity condition of class $C^{2}$ .
</p>projecteuclid.org/euclid.dmj/1478919691_20170105220247Thu, 05 Jan 2017 22:02 ESTK-stability for Fano manifolds with torus action of complexity $1$http://projecteuclid.org/euclid.dmj/1477494164<strong>Nathan Ilten</strong>, <strong>Hendrik Süß</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 1, 177--204.</p><p><strong>Abstract:</strong><br/>
We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension $1$ . Using a recent result of Datar and Székelyhidi, we effectively determine the existence of Kähler–Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kähler–Einstein Fano threefolds and Fano threefolds admitting a nontrivial Kähler–Ricci soliton.
</p>projecteuclid.org/euclid.dmj/1477494164_20170105220247Thu, 05 Jan 2017 22:02 ESTTransition asymptotics for the Painlevé II transcendenthttp://projecteuclid.org/euclid.dmj/1481879045<strong>Thomas Bothner</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 2, 205--324.</p><p><strong>Abstract:</strong><br/>
We consider real-valued solutions $u=u(x|s)$ , $x\in\mathbb{R}$ , of the second Painlevé equation $u_{xx}=xu+2u^{3}$ which are parameterized in terms of the monodromy data $s\equiv(s_{1},s_{2},s_{3})\subset\mathbb{C}^{3}$ of the associated Flaschka–Newell system of rational differential equations. Our analysis describes the transition, as $x\rightarrow-\infty$ , between the oscillatory power-like decay asymptotics for $|s_{1}|\lt 1$ (Ablowitz–Segur) to the power-like growth behavior for $|s_{1}|=1$ (Hastings–McLeod) and from the latter to the singular oscillatory power-like growth for $|s_{1}|\gt 1$ (Kapaev). It is shown that the transition asymptotics are of Boutroux type; that is, they are expressed in terms of Jacobi elliptic functions. As applications of our results we obtain asymptotics for the Airy kernel determinant $\operatorname{det}(I-\gamma K_{\mathrm{Ai}})|_{L^{2}(x,\infty)}$ in a double scaling limit $x\rightarrow-\infty$ , $\gamma\uparrow1$ , as well as asymptotics for the spectrum of $K_{\mathrm{Ai}}$ .
</p>projecteuclid.org/euclid.dmj/1481879045_20170124040056Tue, 24 Jan 2017 04:00 ESTLevel-raising and symmetric power functoriality, IIIhttp://projecteuclid.org/euclid.dmj/1481252669<strong>Laurent Clozel</strong>, <strong>Jack A. Thorne</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 2, 325--402.</p><p><strong>Abstract:</strong><br/>
The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers $\operatorname{Sym}^{n}$ of a cuspidal representation of $\operatorname{GL}(2)$ over the adèles of $F$ , where $F$ is a number field. In 1978, Gelbart and Jacquet proved the existence of $\operatorname{Sym}^{2}$ . After this, progress was slow, eventually leading, through the work of Kim and Shahidi, to the existence of $\operatorname{Sym}^{3}$ and $\operatorname{Sym}^{4}$ . In this series of articles we revisit this problem using recent progress in the deformation theory of modular Galois representations. As a consequence, our methods apply only to classical modular forms on a totally real number field; the present article proves the existence, in this “classical” case, of $\operatorname{Sym}^{6}$ and $\operatorname{Sym}^{8}$ .
</p>projecteuclid.org/euclid.dmj/1481252669_20170124040056Tue, 24 Jan 2017 04:00 ESTA topological property of asymptotically conical self-shrinkers of small entropyhttp://projecteuclid.org/euclid.dmj/1476450481<strong>Jacob Bernstein</strong>, <strong>Lu Wang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 3, 403--435.</p><p><strong>Abstract:</strong><br/>
For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the round sphere uniquely minimizes the entropy among all nonflat two-dimensional self-shrinkers. This confirms a conjecture of Colding, Ilmanen, Minicozzi, and White in dimension two.
</p>projecteuclid.org/euclid.dmj/1476450481_20170208040454Wed, 08 Feb 2017 04:04 ESTProof of linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbationshttp://projecteuclid.org/euclid.dmj/1477321009<strong>Jonathan Luk</strong>, <strong>Sung-Jin Oh</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 3, 437--493.</p><p><strong>Abstract:</strong><br/>
It has long been suggested that solutions to the linear scalar wave equation
\[\Box_{g}\phi=0\] on a fixed subextremal Reissner–Nordström spacetime with nonvanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W^{1,2}_{\mathrm{loc}}$ . This instability is related to the celebrated blue-shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein–Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner–Nordström spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price’s law decay is generically sharp along the event horizon.
</p>projecteuclid.org/euclid.dmj/1477321009_20170208040454Wed, 08 Feb 2017 04:04 ESTGeometry of webs of algebraic curveshttp://projecteuclid.org/euclid.dmj/1475602128<strong>Jun-Muk Hwang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 3, 495--536.</p><p><strong>Abstract:</strong><br/>
A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$ . A web of curves on $X$ induces a web-structure (in the sense of local differential geometry) in a neighborhood of a general point of $X$ . We study how the local differential geometry of the web-structure affects the global algebraic geometry of $X$ . Under two geometric assumptions on the web-structure—the pairwise nonintegrability condition and the bracket-generating condition—we prove that the local differential geometry determines the global algebraic geometry of $X$ , up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when $X\subset\mathbb{P}^{N}$ is a Fano submanifold of Picard number $1$ and the family of lines covering $X$ becomes a web. In this special case, we have the stronger result that the local differential geometry of the web-structure determines $X$ up to biregular equivalences. As an application, we show that if $X,X'\subset\mathbb{P}^{N}$ , $\operatorname{dim}X'\geq3$ , are two such Fano manifolds of Picard number $1$ , then any surjective morphism $f:X\to X'$ is an isomorphism.
</p>projecteuclid.org/euclid.dmj/1475602128_20170208040454Wed, 08 Feb 2017 04:04 ESTAlgebraic Birkhoff factorization and the Euler–Maclaurin formula on coneshttp://projecteuclid.org/euclid.dmj/1478660420<strong>Li Guo</strong>, <strong>Sylvie Paycha</strong>, <strong>Bin Zhang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 3, 537--571.</p><p><strong>Abstract:</strong><br/>
We equip the space of lattice cones with a coproduct which makes it a cograded, coaugmented, connnected coalgebra. The exponential generating sum and exponential generating integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties—reminiscent of the inclusion-exclusion principle for the cardinal on finite sets—of such linear maps and show that these properties are compatible with the convolution quotient of maps on the coalgebra. Implementing the algebraic Birkhoff factorization procedure on the linear maps under consideration, we factorize the exponential generating sum as a convolution quotient of two maps, with each of the maps in the factorization satisfying a subdivision property. A direct computation shows that the polar decomposition of the exponential generating sum on a smooth lattice cone yields an Euler–Maclaurin formula. The compatibility with subdivisions of the convolution quotient arising in the algebraic Birkhoff factorization then yields the Euler–Maclaurin formula for any lattice cone. This provides a simple formula for the interpolating factor by means of a projection formula.
</p>projecteuclid.org/euclid.dmj/1478660420_20170208040454Wed, 08 Feb 2017 04:04 ESTBorelian subgroups of simple Lie groupshttp://projecteuclid.org/euclid.dmj/1476893153<strong>Nicolas de Saxcé</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 3, 573--604.</p><p><strong>Abstract:</strong><br/>
We prove that in a simple real Lie group, there is no Borel measurable dense subgroup of intermediate Hausdorff dimension.
</p>projecteuclid.org/euclid.dmj/1476893153_20170208040454Wed, 08 Feb 2017 04:04 ESTNonsqueezing property of contact ballshttp://projecteuclid.org/euclid.dmj/1479179169<strong>Sheng-Fu Chiu</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 4, 605--655.</p><p><strong>Abstract:</strong><br/>
In this paper we solve a contact nonsqueezing conjecture proposed by Eliashberg, Kim, and Polterovich. Let $B_{R}$ be the open ball of radius $R$ in $\mathbb{R}^{2n}$ , and let $\mathbb{R}^{2n}\times\mathbb{S}^{1}$ be the prequantization space equipped with the standard contact structure. Following Tamarkin’s idea, we apply microlocal category methods to prove that if $R$ and $r$ satisfy $1\leq\pi r^{2}\lt \pi R^{2}$ , then it is impossible to squeeze the contact ball $B_{R}\times\mathbb{S}^{1}$ into $B_{r}\times\mathbb{S}^{1}$ via compactly supported contact isotopies.
</p>projecteuclid.org/euclid.dmj/1479179169_20170301040215Wed, 01 Mar 2017 04:02 ESTZero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocyclehttp://projecteuclid.org/euclid.dmj/1477918664<strong>Simion Filip</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 4, 657--706.</p><p><strong>Abstract:</strong><br/>
We describe all the situations in which the Kontsevich–Zorich (KZ) cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, we find this only occurs when the cocycle satisfies additional geometric constraints. We also show that the connected components of the Zariski closure of the monodromy must be from a specific list, and the representations in which they can occur are described. The number of zero exponents of the KZ cocycle is then as small as possible, given its monodromy.
</p>projecteuclid.org/euclid.dmj/1477918664_20170301040215Wed, 01 Mar 2017 04:02 ESTOn the cubical geometry of Higman’s grouphttp://projecteuclid.org/euclid.dmj/1483412430<strong>Alexandre Martin</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 4, 707--738.</p><p><strong>Abstract:</strong><br/>
We investigate the cocompact action of Higman’s group on a $\operatorname{CAT}(0)$ square complex associated to its standard presentation. We show that this action is in a sense intrinsic, which allows for the use of geometric techniques to study the endomorphisms of the group, and we show striking similarities with mapping class groups of hyperbolic surfaces, outer automorphism groups of free groups, and linear groups over the integers. We compute explicitly the automorphism group and outer automorphism group of Higman’s group and show that the group is both Hopfian and co-Hopfian. We actually prove a stronger rigidity result about the endomorphisms of Higman’s group: every nontrivial morphism from the group to itself is an automorphism. We also study the geometry of the action and prove a surprising result: although the $\operatorname{CAT}(0)$ square complex acted upon contains uncountably many flats, the Higman group does not contain subgroups isomorphic to $\mathbb{Z}^{2}$ . Finally, we show that this action possesses features reminiscent of negative curvature, which we use to prove a refined version of the Tits alternative for Higman’s group.
</p>projecteuclid.org/euclid.dmj/1483412430_20170301040215Wed, 01 Mar 2017 04:02 ESTOn the Tate and Mumford–Tate conjectures in codimension $1$ for varieties with $h^{2,0}=1$http://projecteuclid.org/euclid.dmj/1481252670<strong>Ben Moonen</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 4, 739--799.</p><p><strong>Abstract:</strong><br/>
We prove the Tate conjecture for divisor classes and the Mumford–Tate conjecture for the cohomology in degree $2$ for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic $0$ , under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford–Tate conjectures for several classes of algebraic surfaces with $p_{g}=1$ .
</p>projecteuclid.org/euclid.dmj/1481252670_20170301040215Wed, 01 Mar 2017 04:02 ESTThe HOMFLYPT skein algebra of the torus and the elliptic Hall algebrahttp://projecteuclid.org/euclid.dmj/1481252671<strong>Hugh Morton</strong>, <strong>Peter Samuelson</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 5, 801--854.</p><p><strong>Abstract:</strong><br/>
We give a generators and relations presentation of the HOMFLYPT skein algebra $H$ of the torus $T^{2}$ , and we give an explicit description of the module corresponding to the solid torus. Using this presentation, we show that $H$ is isomorphic to the $\sigma=\bar{\sigma}^{-1}$ specialization of the elliptic Hall algebra of Burban and Schiffmann.
As an application, for an iterated cable $K$ of the unknot, we use the elliptic Hall algebra to construct a 3-variable polynomial that specializes to the $\lambda$ -colored HOMFLYPT polynomial of $K$ . We show that this polynomial also specializes to one constructed by Cherednik and Danilenko using the $\mathfrak{gl}_{N}$ double affine Hecke algebra. This proves one of the connection conjectures in their recent work.
</p>projecteuclid.org/euclid.dmj/1481252671_20170320040104Mon, 20 Mar 2017 04:01 EDTLinear differential equations on the Riemann sphere and representations of quivershttp://projecteuclid.org/euclid.dmj/1478919690<strong>Kazuki Hiroe</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 5, 855--935.</p><p><strong>Abstract:</strong><br/>
Our interest in this article is a generalization of the additive Deligne–Simpson problem, which was originally defined for Fuchsian differential equations on the Riemann sphere. We extend this problem to differential equations having an arbitrary number of unramified irregular singular points, and we determine the existence of solutions of the generalized additive Deligne–Simpson problems. Moreover, we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere (namely, open embedding of the moduli spaces into quiver varieties and the nonemptiness condition of the moduli spaces). Furthermore, we detail the connectedness of the moduli spaces.
</p>projecteuclid.org/euclid.dmj/1478919690_20170320040104Mon, 20 Mar 2017 04:01 EDTFunctional calculus for generators of symmetric contraction semigroupshttp://projecteuclid.org/euclid.dmj/1482202830<strong>Andrea Carbonaro</strong>, <strong>Oliver Dragičević</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 5, 937--974.</p><p><strong>Abstract:</strong><br/>
We prove that every generator of a symmetric contraction semigroup on a $\sigma$ -finite measure space admits, for $1\lt p\lt \infty$ , a Hörmander-type holomorphic functional calculus on $L^{p}$ in the sector of angle $\phi^{*}_{p}=\operatorname{arcsin}\vert1-2/p\vert$ . The obtained angle is optimal.
</p>projecteuclid.org/euclid.dmj/1482202830_20170320040104Mon, 20 Mar 2017 04:01 EDTChern slopes of surfaces of general type in positive characteristichttp://projecteuclid.org/euclid.dmj/1481771254<strong>Giancarlo Urzúa</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 5, 975--1004.</p><p><strong>Abstract:</strong><br/>
Let $\mathbf{{k}}$ be an algebraically closed field of characteristic $p\gt 0$ , and let $C$ be a nonsingular projective curve over $\mathbf{{k}}$ . We prove that for any real number $x\geq2$ , there are minimal surfaces of general type $X$ over $\mathbf{{k}}$ such that (a) $c_{1}^{2}(X)\gt 0$ , $c_{2}(X)\gt 0$ , (b) $\pi_{1}^{\acute{e}t}(X)\simeq\pi_{1}^{\acute{e}t}(C)$ , and (c) $c_{1}^{2}(X)/c_{2}(X)$ is arbitrarily close to $x$ . In particular, we show the density of Chern slopes in the pathological Bogomolov–Miyaoka–Yau interval $(3,\infty)$ for any given $p$ . Moreover, we prove that for $C=\mathbb{P}^{1}$ there exist surfaces $X$ as above with $H^{1}(X,\mathcal{O}_{X})=0$ , that is, with Picard scheme equal to a reduced point. In this way, we show that even surfaces with reduced Picard scheme are densely persistent in $[2,\infty)$ for any given $p$ .
</p>projecteuclid.org/euclid.dmj/1481771254_20170320040104Mon, 20 Mar 2017 04:01 EDTOn the center of quiver Hecke algebrashttp://projecteuclid.org/euclid.dmj/1486695669<strong>P. Shan</strong>, <strong>M. Varagnolo</strong>, <strong>E. Vasserot</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 6, 1005--1101.</p><p><strong>Abstract:</strong><br/>
We compute the equivariant cohomology ring of the moduli space of framed instantons over the affine plane. It is a Rees algebra associated with the center of cyclotomic degenerate affine Hecke algebras of type $A$ . We also give some related results on the center of quiver Hecke algebras and the cohomology of quiver varieties.
</p>projecteuclid.org/euclid.dmj/1486695669_20170414040041Fri, 14 Apr 2017 04:00 EDTThe Prym–Green conjecture for torsion line bundles of high orderhttp://projecteuclid.org/euclid.dmj/1481879046<strong>Gavril Farkas</strong>, <strong>Michael Kemeny</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 6, 1103--1124.</p><p><strong>Abstract:</strong><br/>
Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove that the minimal resolution of a general paracanonical curve $C$ of odd genus $g$ and order $\ell\geq\sqrt{\frac{g+2}{2}}$ is natural, thus proving the Prym–Green conjecture. In the process, we confirm the expectation of Barth and Verra concerning the number of curves with $\ell$ -torsion line bundle in a linear system on a special K3 surface.
</p>projecteuclid.org/euclid.dmj/1481879046_20170414040041Fri, 14 Apr 2017 04:00 EDTEquivariant indices of $\operatorname{Spin}^{c}$ -Dirac operators for proper moment mapshttp://projecteuclid.org/euclid.dmj/1484276627<strong>Peter Hochs</strong>, <strong>Yanli Song</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 6, 1125--1178.</p><p><strong>Abstract:</strong><br/>
We define an equivariant index of $\operatorname{Spin}^{c}$ -Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. Our main result is that the index decomposes into irreducible representations according to the quantization commutes with reduction principle.
</p>projecteuclid.org/euclid.dmj/1484276627_20170414040041Fri, 14 Apr 2017 04:00 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 EDT