Showing papers in "Duke Mathematical Journal in 2015"

FI-modules and stability for representations of symmetric groups

Abstract: In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: • the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold; • the diagonal coinvariant algebra on r sets of n variables; • the cohomology and tautological ring of the moduli space of n -pointed curves; • the space of polynomials on rank varieties of n × n matrices; • the subalgebra of the cohomology of the genus n Torelli group generated by H 1 ; and more. The symmetric group S n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n . In particular, the dimension is eventually a polynomial in n . In this framework, representation stability (in the sense of Church–Farb) for a sequence of S n -representations is converted to a finite generation property for a single FI-module.

G2-manifolds and associative submanifolds via semi-Fano 3-folds

Abstract: We construct many new topological types of compact G 2 -manifolds, that is, Riemannian 7 -manifolds with holonomy group G 2 . To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi–Yau 3 -folds built from semi-Fano 3 -folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 7 -manifolds completely; we find that many 2 -connected 7 -manifolds can be realized as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G 2 -metrics. Many of the G 2 -manifolds we construct contain compact rigid associative 3 -folds, which play an important role in the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of G 2 -metrics. By varying the semi-Fanos used to build different G 2 -metrics on the same 7 -manifold we can change the number of rigid associative 3 -folds we produce.

Baxter's relations and spectra of quantum integrable models

Abstract: Generalized Baxter's relations on the transfer-matrices (also known as Bax- ter's TQ relations) are constructed and proved for an arbitrary untwisted quantum affine algebra. Moreover, we interpret them as relations in the Grothendieck ring of the cate- gory O introduced by Jimbo and the second author in (HJ) involving infinite-dimensional representations constructed in (HJ), which we call here "prefundamental". We define the transfer-matrices associated to the prefundamental representations and prove that their eigenvalues on any finite-dimensional representation are polynomials up to a universal factor. These polynomials are the analogues of the celebrated Baxter polynomials. Com- bining these two results, we express the spectra of the transfer-matrices in the general quantum integrable systems associated to an arbitrary untwisted quantum affine algebra in terms of our generalized Baxter polynomials. This proves a conjecture of Reshetikhin and the first author formulated in 1998 (FR1). We also obtain generalized Bethe Ansatz equations for all untwisted quantum affine algebras.

The Choquet boundary of an operator system

Abstract: We show that every operator system (and hence every unital operator algebra) has sufficiently many boundary representations to generate the C*-envelope.

Crystal bases and Newton–Okounkov bodies

Abstract: Let G be a connected reductive algebraic group. We prove that the string parameterization of a crystal basis for a finite-dimensional irreducible representation of G extends to a natural valuation on the field of rational functions on the flag variety G/B, which is a highest-term valuation corresponding to a coordinate system on a Bott–Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton–Okounkov bodies for the flag variety. This is closely related to an earlier result of Okounkov for the Gelfand–Cetlin polytopes of the symplectic group. As a corollary, we recover a multiplicativity property of the canonical basis due to Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations, follow recovering results by Alexeev and Brion, Caldero, and the author.

Delocalization of eigenvectors of random matrices with independent entries

Abstract: Author(s): Rudelson, Mark; Vershynin, Roman | Abstract: We prove that an n by n random matrix G with independent entries is completely delocalized. Suppose the entries of G have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high probability all unit eigenvectors of G have all coordinates of magnitude O(n^{-1/2}), modulo logarithmic corrections. This comes a consequence of a new, geometric, approach to delocalization for random matrices.

Faber–Krahn inequalities in sharp quantitative form

Abstract: The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincare-Sobolev constant for the embeddings $W^{1,2}_0(\Omega)\hookrightarrow L^q(\Omega)$.

The Möbius function and distal flows

Abstract: We prove that the Mobius function is linearly disjoint from an analytic skew product on the 2 -torus. These flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points. The previous cases for which such disjointness has been proved are all regular. We also establish the linear disjointness of the Mobius function from various distal homogeneous flows.

Toeplitz determinants with merging singularities

Abstract: We study asymptotic behavior for determinants of n×n Toeplitz matrices corresponding to symbols with two Fisher-Hartwig singularities at the distance 2t≥0 from each other on the unit circle. We obtain large n asymptotics which are uniform for 0

Quantum ergodicity on large regular graphs

Abstract: We propose a version of the quantum ergodicity theorem on large regular graphs of fixed valency. This is a property of delocalization of “most” eigenfunctions. We consider expander graphs with few short cycles (for instance random large regular graphs). Our method mimics the proof of quantum ergodicity on manifolds: it uses microlocal analysis on regular trees, as introduced by the second author in an earlier paper.

The brundan-kazhdan-lusztig conjecture for general linear lie superalgebras

Abstract: In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan–Lusztig-type conjecture for the characters of the irreducible and tilting modules in the Bernstein–Gelfand–Gelfand category for the general linear Lie superalgebra for the first time. In this paper, we prove Brundan’s conjecture and its variants associated to all Borel subalgebras in full generality.

M ¯ 0 , n is not a Mori dream space

Abstract: Building on the work of Goto, Nishida, and Watanabe on symbolic Rees algebras of monomial primes, we prove that the moduli space of stable rational curves with n punctures is not a Mori dream space for n>133. This answers a question posed by Hu and Keel.

The boundary of the complex of free factors

Abstract: We give a description of the boundary of a complex of free factors that is analogous to E. Klarreich’s description of the boundary of a curve complex. The argument uses the geometry of folding paths developed by Bestvina and Feighn and the structure theory of trees on the boundary of Outer space developed recently by Coulbois, Hilion, Lustig, and Reynolds.

Finite energy global well-posedness of the Yang–Mills equations on $\mathbb{R}^{1+3}$: An approach using the Yang–Mills heat flow

Abstract: In this work, we propose a novel approach to the problem of gauge choice for the Yang–Mills equations on the Minkowski space $\mathbb{R}^{1+3}$. A crucial ingredient is the associated Yang–Mills heat flow. As this approach avoids the drawbacks of previous approaches, it is expected to be more robust and easily adaptable to other settings. Building on the author’s previous results, we prove, as the first application of our approach, finite energy global well-posedness of the Yang–Mills equations on $\mathbb{R}^{1+3}$. This is a classical result first proved by Klainerman and Machedon using local Coulomb gauges. As opposed to their method, the present approach avoids the use of Uhlenbeck’s lemma and hence does not involve localization in space-time.

Discriminants in the Grothendieck ring

Abstract: We consider the limiting behavior of discriminants, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X and linear systems on X. These are connected—we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motivated by our results, we ask whether the symmetric powers of geometrically irreducible varieties stabilize in the Grothendieck ring (in an appropriate sense). Our results extend parallel results in both arithmetic and topology. We give a number of reasons for considering these questions, and we propose a number of new conjectures, both arithmetic and topological.

A link-splitting spectral sequence in Khovanov homology

Abstract: We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the disjoint union of its components. The page at which the sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed through one another in order to completely separate them. In addition, we build on work of Kronheimer and Mrowka and Hedden and Ni to show that Khovanov homology detects the unlink.

Serrin’s overdetermined problem and constant mean curvature surfaces

Abstract: For all N≥9, we find smooth entire epigraphs in RN, namely, smooth domains of the form Ω:={x∈RN|xN>F(x1,…,xN−1)}, which are not half-spaces and in which a problem of the form Δu+f(u)=0 in Ω has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on ∂Ω. This answers negatively for large dimensions a question by Berestycki, Caffarelli, and Nirenberg. In 1971, Serrin proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy to a famous result by Alexandrov that states that an embedded compact surface with constant mean curvature (CMC) in Euclidean space must be a sphere. In lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given CMC surface where Serrin’s overdetermined problem is solvable.

Periodic points of birational transformations on projective surfaces

Abstract: We give a classification of birational transformations on smooth projective surfaces which have a Zariski-dense set of noncritical periodic points. In particular, we show that if the first dynamical degree is greater than one, the union of all noncritical periodic orbits is Zariski-dense.

Hypersurfaces in projective schemes and a moving lemma

Abstract: Let X / S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H / S of X / S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X / S containing a given closed subscheme C and intersecting properly a closed set F . Assume now that the base S is the spectrum of a ring R such that for any finite morphism Z → S , Pic ( Z ) is a torsion group. This condition is satisfied if R is the ring of integers of a number field or the ring of functions of a smooth affine curve over a finite field. We prove in this context a moving lemma pertaining to horizontal 1 -cycles on a regular scheme X quasi-projective and flat over S . We also show the existence of a finite surjective S -morphism to P S d for any scheme X projective over S when X / S has all its fibers of a fixed dimension d .

Ordinary representations of G ( Q p ) and fundamental algebraic representations

Abstract: Let G be a split connected reductive algebraic group over Q p such that both G and its dual group G ˆ have connected centers. Motivated by a hypothetical p -adic Langlands correspondence for G ( Q p ) , we associate to an n -dimensional ordinary (i.e., Borel-valued) representation ρ : Gal ( Q ¯ p / Q p ) → G ˆ ( E ) a unitary Banach space representation Π ( ρ ) ord of G ( Q p ) over E that is built out of principal series representations. (Here, E is a finite extension of Q p .) Our construction is inspired by the “ordinary part” of the tensor product of all fundamental algebraic representations of G . There is an analogous construction over a finite extension of F p . When G = G L n , we show under suitable hypotheses that Π ( ρ ) ord occurs in the ρ -part of the cohomology of a compact unitary group.

Symmetric quiver Hecke algebras and R -matrices of quantum affine algebras, II

Abstract: Let g be an untwisted affine Kac–Moody algebra of type A n ( 1 ) ( n ≥ 1 ) or D n ( 1 ) ( n ≥ 4 ), and let g 0 be the underlying finite-dimensional simple Lie subalgebra of g . For each Dynkin quiver Q of type g 0 , Hernandez and Leclerc introduced a tensor subcategory C Q of the category of finite-dimensional integrable U ' q ( g ) -modules and proved that the Grothendieck ring of C Q is isomorphic to C [ N ] , the coordinate ring of the unipotent group N associated with g 0 . We apply the generalized quantum affine Schur–Weyl duality to construct an exact functor F from the category of finite-dimensional graded R -modules to the category C Q , where R denotes the symmetric quiver Hecke algebra associated to g 0 . We prove that the homomorphism induced by the functor F coincides with the homomorphism of Hernandez and Leclerc and show that the functor F sends the simple modules to the simple modules.

On the mean number of $2$-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields

Abstract: Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2 -torsion elements in the class groups and narrow class groups of these cubic fields when they are ordered by their absolute discriminants. For an order O in a cubic field, we study three groups: Cl 2 ( O ) , the group of ideal classes of O of order 2 ; Cl 2 + ( O ) , the group of narrow ideal classes of O of order 2 ; and I 2 ( O ) , the group of ideals of O of order 2 . We prove that the mean value of the difference | Cl 2 ( O ) | - 1 4 | I 2 ( O ) | is always equal to 1 , regardless of whether one averages over the maximal orders in real cubic fields, over all orders in real cubic fields, or indeed over any family of real cubic orders defined by local conditions. For the narrow class group, we prove that the average value of the difference | Cl 2 + ( O ) | - | I 2 ( O ) | is equal to 1 for any such family. Also, for any family of complex cubic orders defined by local conditions, we prove similarly that the mean value of the difference | Cl 2 ( O ) | - 1 2 | I 2 ( O ) | is always equal to 1 , independent of the family. The determination of these mean numbers allows us to prove a number of further results as by-products. Most notably, we prove—in stark contrast to the case of quadratic fields—that (1) a positive proportion of cubic fields have odd class number, (2) a positive proportion of real cubic fields have isomorphic 2 -torsion in the class group and the narrow class group, and (3) a positive proportion of real cubic fields contain units of mixed real signature. We also show that a positive proportion of real cubic fields have narrow class group strictly larger than the class group, and thus a positive proportion of real cubic fields do not possess units of every possible real signature.

Expanders with respect to Hadamard spaces and random graphs

Abstract: It is shown that there exist a sequence of 3 -regular graphs { G n } n = 1 ∞ and a Hadamard space X such that { G n } n = 1 ∞ forms an expander sequence with respect to X , yet random regular graphs are not expanders with respect to X . This answers a question of the second author and Silberman. The graphs { G n } n = 1 ∞ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear-time constant-factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.

Prime polynomials in short intervals and in arithmetic progressions

Abstract: In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+xϵ] is about xϵ/logx. The second says that the number of primes p

The complex volume of SL(n,C)-representations of 3-manifolds

Abstract: For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parameterization of the set of conjugacy classes of boundary-unipotent representations of π1(M) into SL(n,C). Our parameterization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmuller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann’s extended Bloch group Bˆ(C), and we use this to obtain an efficient formula for the Cheeger–Chern–Simons invariant, and, in particular, for the volume. Computations for the census manifolds show that boundary-unipotent representations are abundant, and numerical comparisons with census volumes suggest that the volume of a representation is an integral linear combination of volumes of hyperbolic 3-manifolds. This is in agreement with a conjecture of Walter Neumann, stating that the Bloch group is generated by hyperbolic manifolds.

Coisotropic rigidity and $C^{0}$-symplectic geometry

Abstract: We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov–Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov–Eliashberg theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach and Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C0-dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C0-Hamiltonian dynamics: uniqueness of generators for continuous analogues of Hamiltonian flows.

Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian

Abstract: We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.

Around the stability of KAM tori

Abstract: We study the accumulation of an invariant quasi-periodic torus of a Hamiltonian flow by other quasi-periodic invariant tori. We show that an analytic invariant torus T 0 with Diophantine frequency ω 0 is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at T 0 satisfies a Russmann transversality condition, the torus T 0 is accumulated by Kolmogorov–Arnold–Moser (KAM) tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least d + 1 that is foliated by analytic invariant tori with frequency ω 0 . For frequency vectors ω 0 having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian H satisfies a Kolmogorov nondegeneracy condition at T 0 , then T 0 is accumulated by KAM tori of positive total measure. In four degrees of freedom or more, we construct for any ω 0 ∈ R d , C ∞ (Gevrey) Hamiltonians H with a smooth invariant torus T 0 with frequency ω 0 that is not accumulated by a positive measure of invariant tori.

Global well-posedness for the maxwell-klein gordon equation in 4 + 1 dimensions. small energy.

Abstract: We prove that the critical Maxwell-Klein-Gordon equation on R4+1 is globally well-posed for smooth initial data which are small in the energy norm. This reduces the problem of global regularity for large, smooth initial data to precluding concentration of energy.

On the Breuil–Mézard conjecture

Abstract: We give a new local proof of the Breuil–Mezard conjecture for 2-dimensional representations of the absolute Galois group of Qp, when p≥5 and the representation has scalar endomorphisms.