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Showing papers in "Duke Mathematical Journal in 2017"


Journal ArticleDOI
TL;DR: In this article, it was shown that ordV is the unique minimizer among all C∗-invariant quasimonomial valuations on the cone associated to any positive Cartier multiple of −KV.
Abstract: This is a continuation of an earlier work in which we proposed a problem of minimizing normalized volumes over Q-Gorenstein Kawamata log terminal singularities. Here we consider its relation with K-semistability, which is an important concept in the study of Kahler–Einstein metrics on Fano varieties. In particular, we prove that for a Q-Fano variety V, the K-semistability of (V,−KV) is equivalent to the condition that the normalized volume is minimized at the canonical valuation ordV among all C∗-invariant valuations on the cone associated to any positive Cartier multiple of −KV. In this case, we show that ordV is the unique minimizer among all C∗-invariant quasimonomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over V.

178 citations


Journal ArticleDOI
TL;DR: In this article, a 3-manifold invariant called involutive Heegaard Floer homology is defined, which is meant to correspond to Z4-equivariant Seiberg-Witten homology.
Abstract: 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 Z4-equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, d and d¯, and two invariants of smooth knot concordance, V0 and 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 V0 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.

127 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the conditional distribution of the inside points, given the outside points, is mutually absolutely continuous with respect to the Lebesgue measure on its supporting submanifold.
Abstract: Let Π be a translation-invariant point process on the complex plane C, and let D⊂C be a bounded open set. We ask the following: What does the point configuration Πout obtained by taking the points of Π outside D tell us about the point configuration Πin of Π inside D? We show that, for the Ginibre ensemble, Πout determines the number of points in Πin. For the translation-invariant zero process of a planar Gaussian analytic function, we show that Πout determines the number as well as the center of mass of the points in Πin. 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.

88 citations


Journal ArticleDOI
TL;DR: In this paper, the authors give a complete description of the group of exact autoequivalences of coherent sheaves on a K3 surface of Picard rank 1 and prove that a distinguished connected component of the space of stability conditions is preserved by all autoeqivalences, and is contractible.
Abstract: 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.

79 citations


Journal ArticleDOI
TL;DR: In this paper, the phase transition from singular continuous spectrum to pure point spectrum was shown to take place in the almost Mathieu operator (AMO) for the a.i.d. phase.
Abstract: It is known that the spectral type of the almost Mathieu operator (AMO) depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya’s conjecture. Together with a previous work by Avila, this gives the sharp description of phase transitions for the AMO for the a.e. phase.

79 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite non-degenerate energy near the Cauche horizon in the interior of the black hole.
Abstract: It has long been suggested that solutions to the linear scalar wave equation □gϕ=0 on a fixed subextremal Reissner–Nordstrom 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 Wloc1,2. 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–Nordstrom 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.

68 citations


Journal ArticleDOI
TL;DR: In this paper, the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings, and basic structural properties such as local Noetherianity are proved.
Abstract: We study analogues of 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.

68 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the absolute constant A in the inequality of T 1/2≤t ≥ T cannot be taken smaller than 1. The method of proof uses Soundararajan's resonance method and a certain large greatest common divisor sum.
Abstract: It is shown that the maximum of |ζ(1/2+it)| on the interval T1/2≤t≤T is at least exp((1/2+o(1))logTlogloglogT/loglogT). 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≤n1<⋯

59 citations


Journal ArticleDOI
TL;DR: In this paper, the authors apply automorphy lifting techniques to establish new cases of symmetric power functoriality for Hilbert modular forms of regular algebraic weight, based on a novel application of an automorephy lifting theorem for residually reducible Galois representations.
Abstract: We apply automorphy lifting techniques to establish new cases of symmetric power functoriality for Hilbert modular forms of regular algebraic weight. The proof is based on a novel application of an automorphy lifting theorem for residually reducible Galois representations.

58 citations


Journal ArticleDOI
TL;DR: In this article, a general affine representability result for vector bundles in A1-homotopy theory over a general base is established. But this result is restricted to vector bundles.
Abstract: We establish a general “affine representability” result in A1-homotopy theory over a general base. We apply this result to obtain representability results for vector bundles in A1-homotopy theory. Our results simplify and significantly generalize Morel’s A1-representability theorem for vector bundles.

55 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed and proved that there exists a unique common triangular basis with respect to all seeds.
Abstract: 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.

Journal ArticleDOI
TL;DR: For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder, it was shown in this paper that the link of the asymPTotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the selfshrinker.
Abstract: 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.

Journal ArticleDOI
TL;DR: In this paper, the existence of K-stability of Fano manifolds admitting an algebraic torus action with general orbit of codimension 1 was proved. But this result was based on the notion of equivariant Kstability.
Abstract: We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension 1. Using a recent result of Datar and Szekelyhidi, we effectively determine the existence of Kahler–Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kahler–Einstein Fano threefolds and Fano threefolds admitting a nontrivial Kahler–Ricci soliton.

Journal ArticleDOI
TL;DR: In this paper, the existence of mean curvature flow with surgery of 2-convex hypersurfaces in RN has been shown for all n ≥ 3, including mean convex surfaces in R3.
Abstract: We give a new proof for the existence of mean curvature flow with surgery of 2-convex hypersurfaces in RN. Our proof works for all N≥3, including mean convex surfaces in R3. We also derive a priori estimates for a more general class of flows in a local and flexible setting.

Journal ArticleDOI
TL;DR: In this article, it was shown that the associated C∗-algebra A=C(X)⋊σZ is classifiable when (X,σ) is uniquely ergodic.
Abstract: Let X be an infinite metrizable compact space, and let σ:X→X be a minimal homeomorphism. Suppose that (X,σ) has zero mean topological dimension. The associated C∗-algebra A=C(X)⋊σZ is shown to absorb the Jiang–Su algebra Z tensorially; that is, A≅A⊗Z. This implies that A is classifiable when (X,σ) is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that A⊗A always absorbs the Jiang–Su algebra.

Journal ArticleDOI
TL;DR: In this paper, the exceptional set in Manin's conjecture is shown to be a thin set using the minimal model program and boundedness of log Fano varieties, and it is shown that it is bounded by a constant number of rational points of a bounded height.
Abstract: Manin’s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin’s conjecture is a thin set using the minimal model program and boundedness of log Fano varieties.

Journal ArticleDOI
TL;DR: In this article, Liu, Ruochuan, Wan, Daqing, Xiao, Liang, and Zhang showed that the eigencurve associated to a definite quaternion algebra over QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and Buzzard--Kilford: (a) over the boundary annuli of weight space, the eigcurve is a disjoint union of infinitely many connected components each finite and flat over the weight annuli, and (b) the $U_p$-slopes of
Abstract: Author(s): Liu, Ruochuan; Wan, Daqing; Xiao, Liang | Abstract: We prove that the eigencurve associated to a definite quaternion algebra over $\QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and Buzzard--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 form a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves towards the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.

Journal ArticleDOI
TL;DR: In this paper, a new method for dealing with automatic sequences is presented, which allows us to prove a Mobius randomness principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences.
Abstract: We present in this article a new method for dealing with automatic sequences. This method allows us to prove a Mobius randomness principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences. Furthermore, we can show a prime number theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to 0.

Journal ArticleDOI
TL;DR: In this paper, the authors established a characterization of alternating links in terms of definite spanning surfaces and applied 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.
Abstract: 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 Juhasz and Lackenby, applies the characterization to derive an exponential time algorithm for alternating knot recognition.

Journal ArticleDOI
TL;DR: In this article, it was shown that every generator of a symmetric contraction semigroup on a σ-finite measure space admits, for 1
Abstract: We prove that every generator of a symmetric contraction semigroup on a σ-finite measure space admits, for 1

Journal ArticleDOI
TL;DR: In this article, the authors give a generator and relations presentation of the HOMFLYPT skein algebra H of the torus T2, and give an explicit description of the module corresponding to the solid torus.
Abstract: We give a generators and relations presentation of the HOMFLYPT skein algebra H of the torus T2, 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 σ=σ¯−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 λ-colored HOMFLYPT polynomial of K. We show that this polynomial also specializes to one constructed by Cherednik and Danilenko using the glN double affine Hecke algebra. This proves one of the connection conjectures in their recent work.

Journal ArticleDOI
TL;DR: The Tate conjecture for divisor classes and the Mumford-tate conjecture for the cohomology in degree 2 for varieties with h2,0=1 over a finitely generated field of characteristic 0, under a mild assumption on their moduli were proved in this article.
Abstract: We prove the Tate conjecture for divisor classes and the Mumford–Tate conjecture for the cohomology in degree 2 for varieties with h2,0=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 pg=1.

Journal ArticleDOI
TL;DR: In this paper, the authors considered real-valued solutions of the second Painleve equation u=u(x|s), x∈R, which are parameterized in terms of the monodromy data s≡(s1,s2,s3)⊂C3 of the associated Flaschka-Newell system of rational differential equations.
Abstract: We consider real-valued solutions u=u(x|s), x∈R, of the second Painleve equation uxx=xu+2u3 which are parameterized in terms of the monodromy data s≡(s1,s2,s3)⊂C3 of the associated Flaschka–Newell system of rational differential equations. Our analysis describes the transition, as x→−∞, between the oscillatory power-like decay asymptotics for |s1| 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 det(I−γKAi)|L2(x,∞) in a double scaling limit x→−∞, γ↑1, as well as asymptotics for the spectrum of KAi.

Journal ArticleDOI
TL;DR: In this article, it was shown that π1,π2, and π3 do not always generate Sn, with probability tending to 1 as n→∞, even when π 1,π 2, π 3 is chosen randomly from Sn.
Abstract: We say that permutations π1,…,πr∈Sn invariably generate Sn if, no matter how one chooses conjugates π'1,…,π'r of these permutations, the π'1,…,π'r permutations generate Sn. We show that if π1,π2, and π3 are chosen randomly from Sn, then, with probability tending to 1 as n→∞, they do not invariably generate Sn. By contrast, it was shown recently by Pemantle, Peres, and Rivin that four random elements do invariably generate Sn 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.

Journal ArticleDOI
TL;DR: In this article, the Strengthened Hanna Neumann conjecture was shown to hold for free groups in the context of abstract groups, and it was shown that the Luck approximation conjecture also holds for abstract groups.
Abstract: 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 ∑x∈U\F/Wrk¯(U∩xWx−1)≤rk¯(U)rk¯(W),whererk¯(U)=max {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 Luck approximation conjecture holds for free groups.

Journal ArticleDOI
TL;DR: In this article, the authors established Strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a time-dependent self-similar potential, and proved the asymptotic stability of the ordinary differential equations blowup profile in the energy space.
Abstract: 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.

Journal ArticleDOI
TL;DR: In this paper, the formal degree conjecture for odd special orthogonal and metaplectic groups in the generic case was shown to imply the conjecture in the nongeneric case.
Abstract: 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 γ-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.

Journal ArticleDOI
TL;DR: In this article, a local conjecture for 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, was formulated.
Abstract: 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.

Journal ArticleDOI
TL;DR: In this article, the authors studied the geometry of proper open convex domains in the projective space RPn and proved a thin inequality between the Hilbert distance dH and the Blaschke distance dB.
Abstract: This article studies the geometry of proper open convex domains in the projective space RPn. These domains carry several projective invariant distances, among which are the Hilbert distance dH and the Blaschke distance dB. We prove a thin inequality between those distances: for any two points x and y in such a domain, dB(x,y)

Journal ArticleDOI
TL;DR: The equivariant cohomology ring of the moduli space of framed instantons over the affine plane was shown to be a Rees algebra associated with the center of cyclotomic degenerate affine Hecke algebras of type A in this article.
Abstract: 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.