Showing papers in "Transactions of the American Mathematical Society in 2019"

An $L^p$ theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions

Abstract: We introduce and develop a theory of limits for sequences of sparse graphs based on Lp graphons, which generalizes both the existing L1 theory of dense graph limits and its extension by Bollobas and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the rst broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the Lp theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper.

Surgery principles for the spectral analysis of quantum graphs

Abstract: We present a systematic collection of spectral surgery principles for the Laplacian on a compact metric graph with any of the usual vertex conditions (natural, Dirichlet, or delta-type) which show ...

Descendant log Gromov-Witten invariants for toric varieties and tropical curves

Abstract: Using degeneration techniques, we prove the correspondence of tropical curve counts and log Gromov-Witten invariants with general incidence and psi-class conditions in toric varieties for genus zero curves and all non-superabundant higher-genus situations. We also relate the log invariants to the ordinary ones, in particular explaining the appearance of negative multiplicities in the descendant correspondence result of Mark Gross.

Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

Abstract: We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Furstenberg's theorem. That is, a Schrodinger operator in $\ell^2(\mathbb{Z})$ whose potential is given by independent identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions and its unitary group exhibits exponential off-diagonal decay, uniformly in time. This is achieved by way of a new result: for the Anderson model, one typically has Lyapunov behavior for all generalized eigenfunctions. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogs of these models.

Locally constrained inverse curvature flows

Abstract: We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order--namely, the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to deduce a new Minkowski-type inequality in the anti-de Sitter Schwarzschild manifolds and a weighted isoperimetric-type inequality in hyperbolic space.

Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity

Abstract: We construct a pathwise integration theory, associated with a change of variable formula , for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of p-th variation along a sequence of time partitions. For paths with finite p-th variation along a sequence of time partitions, we derive a change of variable formula for p times continuously differentiable functions and show pointwise convergence of appropriately defined compensated Riemann sums. Results for functions are extended to regular path-dependent functionals using the concept of vertical derivative of a functional. We show that the pathwise integral satisfies an 'isometry' formula in terms of p-th order variation and obtain a 'signal plus noise' decomposition for regular functionals of paths with strictly increasing p-th variation. For less regular functions we obtain a Tanaka-type change of variable formula using an appropriately defined notion of local time. These results extend to multidimensional paths and yield a natural higher-order extension of the concept of 'reduced rough path'. We show that, while our integral coincides with a rough-path integral for a certain rough path, its construction is canonical and does not involve the specification of any rough-path superstructure.

Simplicity Of Algebras Associated To Non-Hausdorff Groupoids

Abstract: The first named author was partially supported by Marsden grant 15-UOO-071 from the Royal Society of New Zealand. The second named author was partially supported by CNPq. The third named author was partially supported by PAI III grant FQM-298 of the Junta de Andaluc ia, and by the DGI-MINECO and European Regional Development Fund, jointly, through grants MTM2014-53644-P and MTM2017- 83487-P. The fourth named author was partially supported by the Australian Research Council grant DP150101595. The fifth named author was partially supported by a Carleton University internal research grant.

A spectral approach for quenched limit theorems for random hyperbolic dynamical systems

Abstract: We extend the recent spectral approach for quenched limit theorems developed for piecewise expanding dynamics under general random driving to quenched random piecewise hyperbolic dynamics. For general ergodic sequences of maps in a neighborhood of a hyperbolic map we prove a quenched large deviations principle (LDP), central limit theorem (CLT), and local central limit theorem (LCLT).

Quenched decay of correlations for slowly mixing systems

Abstract: We study random towers that are suitable to analyse the statistics of slowly mixing random systems. We obtain upper bounds on the rate of quenched correlation decay in a general setting. We apply our results to the random family of LiveraniSaussol-Vaienti maps with parameters in [α0, α1] ⊂ (0, 1) chosen independently with respect to a distribution ν on [α0, α1] and show that the quenched decay of correlation is governed by the fastest mixing map in the family. In particular, we prove that for every δ > 0, for almost every ω ∈ [α0, α1] Z, the upper bound n 1− 1 α0 +δ holds on the rate of decay of correlation for Holder observables on the fibre over ¨ ω. For three different distributions ν on [α0, α1] (discrete, uniform, quadratic), we also derive sharp asymptotics on the measure of return-time intervals for the quenched dynamics, ranging from n − 1 α0 to (log n) 1 α0 · n − 1 α0 to (log n) 2 α0 · n − 1 α0 respectively.

Cartan subalgebras in C*-algebras. Existence and uniqueness

Abstract: We initiate the study of Cartan subalgebras in C*-algebras, with a particular focus on existence and uniqueness questions. For homogeneous C*-algebras, these questions can be analyzed systematically using the theory of fiber bundles. For group C*-algebras, while we are able to find Cartan subalgebras in C*-algebras of many connected Lie groups, there are classes of (discrete) groups, for instance non-abelian free groups, whose reduced group C*-algebras do not have any Cartan subalgebras. Moreover, we show that uniqueness of Cartan subalgebras usually fails for classifiable C*-algebras. However, distinguished Cartan subalgebras exist in some cases, for instance in nuclear uniform Roe algebras.

Nodal area distribution for arithmetic random waves

Abstract: We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3= \mathbb{R}^3/ \mathbb{Z}^3$ ($3$-dimensional 'arithmetic random waves'). We prove that, as the multiplicity of the eigenspace goes to infinity, the nodal area converges to a universal, non-Gaussian, distribution. Universality follows from the equidistribution of lattice points on the sphere. Our arguments rely on the Wiener chaos expansion of the nodal area: we show that, analogous to (Marinucci, Peccati, Rossi and Wigman, 2016), the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results in (Benatar and Maffiucci, 2017) that establish an upper bound for the number of non-degenerate correlations of lattice points on the sphere. We finally discuss higher-dimensional extensions of our result.

A volume preserving flow and the isoperimetric problem in warped product spaces

Abstract: In this article, we continue the work in \cite{GL} and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains.

Del Pezzo surfaces and Mori fiber spaces in positive characteristic

Abstract: We settle a question that originates from results and remarks by Koll\'ar on extremal ray in the minimal model program: In positive characteristics, there are no Mori fibrations on threefolds with only terminal singularities whose generic fibers are geometrically non-normal surfaces. To show this we establish some general structure results for del Pezzo surfaces over imperfect ground fields. This relies on Reid's classification of non-normal del Pezzo surfaces over algebraically closed fields, combined with a detailed analysis of geometrical non-reducedness over imperfect fields of p-degree one.

Asymptotic resurgence via integral closures

Abstract: Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finitely many ratios involving Waldschmidt-like constants (which we call skew Waldschmidt constants) defined in terms of Rees valuations. We use this to prove that the asymptotic resurgence coincides with the resurgence if the ideal is normal (that is, all its powers are integrally closed). For a monomial ideal the skew Waldschmidt constants have an interpretation involving the symbolic polyhedron defined by Cooper, Embree, Ha, and Hoefel. Using this intuition we provide several examples of squarefree monomial ideals whose resurgence and asymptotic resurgence are different.

A metrizable topology on the contracting boundary of a group

Abstract: The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable.

The free splitting complex of a free group, II: Loxodromic outer automorphisms

Abstract: We study the loxodromic elements for the action of $Out(F_n)$ on the free splitting complex of the rank $n$ free group $F_n$. We prove that each outer automorphism is either loxodromic, or has bounded orbits without any periodic point, or has a periodic point; and we prove that all three possibilities can occur. We also prove that two loxodromic elements are either co-axial or independent, meaning that their attracting/repelling fixed point pairs on the Gromov boundary of the free splitting complex are either equal or disjoint as sets. Each of the alternatives in these results is also characterized in terms of the attracting/repelling lamination pairs of an outer automorphism. As an application, each attracting lamination determines its corresponding repelling lamination independent of the outer automorphism. As part of this study we describe the structure of the subgroup of $Out(F_n)$ that stabilizes the fixed point pair of a given loxodromic outer automorphism, and we give examples which show that this subgroup need not be virtually cyclic. As an application, the action of $Out(F_n)$ on the free splitting complex is not acylindrical, and its loxodromic elements do not all satisfy the WPD property of Bestvina and Fujiwara.

Better bounds for poset dimension and boxicity

Abstract: We prove that the dimension of every poset whose comparability graph has maximum degree $\Delta$ is at most $\Delta\log^{1+o(1)} \Delta$. This result improves on a 30-year old bound of Furedi and Kahn, and is within a $\log^{o(1)}\Delta$ factor of optimal. We prove this result via the notion of boxicity. The "boxicity" of a graph $G$ is the minimum integer $d$ such that $G$ is the intersection graph of $d$-dimensional axis-aligned boxes. We prove that every graph with maximum degree $\Delta$ has boxicity at most $\Delta\log^{1+o(1)} \Delta$, which is also within a $\log^{o(1)}\Delta$ factor of optimal. We also show that the maximum boxicity of graphs with Euler genus $g$ is $\Theta(\sqrt{g \log g})$, which solves an open problem of Esperet and Joret and is tight up to a $O(1)$ factor.

Convergence of positive operator semigroups

Abstract: We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw and Glicksberg with a purely algebraic result about positive group representations. Thus we obtain convergence theorems not only for one-parameter semigroups but for a much larger class of semigroup representations. Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive $C_0$-semigroup containing or dominating a kernel operator converges strongly as $t \to \infty$. We gain new insights into the structure theoretical background of those theorems and generalise them in several respects; especially we drop any kind of continuity or regularity assumption with respect to the time parameter.

Ground states of two-component attractive Bose-Einstein condensates II: Semi-trivial limit behavior

Abstract: As a continuation of [14], we study new pattern formations of ground states $(u_1,u_2)$ for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in $R^2$, where the intraspecies interaction $(-a,-b)$ and the interspecies interaction $-\beta$ are both attractive, $i.e,$ $a$, $b$ and $\beta$ are all positive. If $0

Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term

Abstract: We consider the nonlinear heat equation with a nonlinear gradient term: $\partial_t u =\Delta u+\mu| abla u|^q+|u|^{p-1}u,\; \mu>0,\; q=2p/(p+1),\; p>3,\; t\in (0,T),\; x\in \R^N.$ We construct a solution which blows up in finite time $T>0.$ We also give a sharp description of its blow-up profile and show that it is stable with respect to perturbations in initial data. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. The blow-up profile does not scale as $(T-t)^{1/2}|\log(T-t)|^{1/2},$ like in the standard nonlinear heat equation, i.e. $\mu=0,$ but as $(T-t)^{1/2}|\log(T-t)|^{\beta}$ with $\beta=(p+1)/[2(p-1)]>1/2.$ We also show that $u$ and $ abla u$ blow up simultaneously and at a single point, and give the final profile. In particular, the final profile is more singular than the case of the standard nonlinear heat equation.

The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation

Abstract: For a finite involutive non-degenerate solution $(X,r)$ of the Yang--Baxter equation it is known that the structure monoid $M(X,r)$ is a monoid of I-type, and the structure algebra $K[M(X,r)]$ over a field $K$ share many properties with commutative polynomial algebras, in particular, it is a Noetherian PI-domain that has finite Gelfand--Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid $M(X,r)$ and the algebra $K[M(X,r)]$ is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of $M(X,r)$ as a regular submonoid in the semidirect product $A(X,r)\rtimes\mathrm{Sym}(X)$, where $A(X,r)$ is the structure monoid of the rack solution associated to $(X,r)$, we prove that $K[M(X,r)]$ is a module finite normal extension of a commutative affine subalgebra. In particular, $K[M(X,r)]$ is a Noetherian PI-algebra of finite Gelfand--Kirillov dimension bounded by $|X|$. We also characterize, in ring-theoretical terms of $K[M(X,r)]$, when $(X,r)$ is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of $M(X,r)$. These results allow us to control the prime spectrum of the algebra $K[M(X,r)]$ and to describe the Jacobson radical and prime radical of $K[M(X,r)]$. Finally, we give a matrix-type representation of the algebra $K[M(X,r)]/P$ for each prime ideal $P$ of $K[M(X,r)]$. As a consequence, we show that if $K[M(X,r)]$ is semiprime then there exist finitely many finitely generated abelian-by-finite groups, $G_1,\dotsc,G_m$, each being the group of quotients of a cancellative subsemigroup of $M(X,r)$ such that the algebra $K[M(X,r)]$ embeds into $\mathrm{M}_{v_1}(K[G_1])\times\dotsb\times \mathrm{M}_{v_m}(K[G_m])$.

A new variational principle, convexity, and supercritical Neumann problems

Abstract: Utilizing a new variational principle that allows dealing with problems beyond the usual locally compactness structure, we study problems with a supercritical nonlinearity of the type $ -\Delta u + u= a(x) f(u)$ in $ \Omega$ with $\partial_ u u=0$ on $ \partial \Omega$. Here $\Omega$ is a bounded domain with certain symmetry assumptions. We find positive nontrivial solutions in the case of suitable supercritical nonlinearities $f$ by finding critical points of $I$ where \[ I(u)=\int_\Omega \left\{ a(x) F^* \left( \frac{-\Delta u + u}{a(x)} \right) - a(x) F(u) \right\} dx, \] over the closed convex cone $K_m$ of nonnegative, symmetric and monotonic functions in $H^1(\Omega)$ where $F'=f$ and where $ F^*$ is the Fenchel dual of $F$. We mention two important comments: firstly that there is a hidden symmetry in the functional $I$ due to the presence of a convex function and its Fenchel dual that makes it ideal to deal with super-critical problems lacking the necessary compactness requirement. Secondly the energy $I$ is not at all related to the classical Euler-Lagrange energy associated with equation. After we have proven the existence of critical points $u$ of $I$ on $K_m$ we then unitize a new abstract variational approach (developed by one of the present authors in \cite{Mo,Mo2}) to show these critical points in fact satisfy $-\Delta u + u = a(x) f(u)$. In the particular case of $ f(u)=|u|^{p-2} u$ we show the existence of positive nontrivial solutions beyond the usual Sobolev critical exponent.

A flag variety for the Delta Conjecture

Abstract: The Delta Conjecture of Haglund, Remmel, and Wilson predicts the monomial expansion of the symmetric function $\Delta'_{e_{k-1}} e_n$, where $k \leq n$ are positive integers and $\Delta'_{e_{k-1}}$ is a Macdonald eigenoperator. When $k = n$, the specialization $\Delta'_{e_{n-1}} e_n|_{t = 0}$ is the Frobenius image of the graded $S_n$-module afforded by the cohomology ring of the {\em flag variety} consisting of complete flags in $\mathbb{C}^n$. We define and study a variety $X_{n,k}$ which carries an action of $S_n$ whose cohomology ring $H^{\bullet}(X_{n,k})$ has Frobenius image given by $\Delta'_{e_{k-1}} e_n|_{t = 0}$, up to a minor twist. The variety $X_{n,k}$ has a cellular decomposition with cells $C_w$ indexed by length $n$ words $w = w_1 \dots w_n$ in the alphabet $\{1, 2, \dots, k\}$ in which each letter appears at least once. When $k = n$, the variety $X_{n,k}$ is homotopy equivalent to the flag variety. We give a presentation for the cohomology ring $H^{\bullet}(X_{n,k})$ as a quotient of the polynomial ring $\mathbb{Z}[x_1, \dots, x_n]$ and describe polynomial representatives for the classes $[ \overline{C}_w]$ of the closures of the cells $C_w$; these representatives generalize the classical Schubert polynomials.

Degenerate Poincaré–Sobolev inequalities

Abstract: We study weighted Poincaré and Poincaré-Sobolev type inequalities with an explicit analysis on the dependence on the Ap constants of the involved weights. We obtain inequalities of the form ( 1 w(Q) ∫ Q |f − fQ|w ) 1 q ≤ Cw`(Q) ( 1 w(Q) ∫ Q |∇f |w ) 1 p , with different quantitative estimates for both the exponent q and the constant Cw. We will derive those estimates together with a large variety of related results as a consequence of a general selfimproving property shared by functions satisfying the inequality − ∫ Q |f − fQ|dμ ≤ a(Q), for all cubes Q ⊂ R and where a is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolevtype exponent pw > p associated to the weight w and obtain further improvements involving L ∗ w norms on the left hand side of the inequality above. For the endpoint case of A1 weights we reach the classical critical Sobolev exponent p∗ = pn n−p which is the largest possible and provide different type of quantitative estimates for Cw. We also show that this best possible estimate cannot hold with an exponent on the A1 constant smaller than 1/p. As a consequence of our results (and the method of proof) we obtain further extensions to two weights Poincaré inequalities and to the case of higher order derivatives. Some other related results in the spirit of the work of Keith and Zhong on the open ended condition of Poincaré inequality are obtained using extrapolation methods. We also apply our method to obtain similar estimates in the scale of Lorentz spaces. We also provide an argument based on extrapolation ideas showing that there is no (p, p), p ≥ 1, Poincaré inequality valid for the whole class of RH∞ weights by showing their intimate connection with the failure of Poincaré inequalities, (p, p) in the range 0 < p < 1. March 18, 2019

Necessary optimality conditions for local minimizers of stochastic optimal control problems with state constraints

Abstract: The main purpose of this work is to establish some first and second order necessary optimality conditions for local minimizers of stochastic optimal control problems with state constraints. The control may affect both the drift and the diffusion terms of the systems and the control regions are allowed to be nonconvex. A stochastic inward pointing condition is proposed to ensure the normality of the corresponding necessary conditions.

Perturbation of normal quaternionic operators

Abstract: The theory of quaternionic operators has applications in several different fields such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of spectrum. In fact, in quaternionic operator theory the classical notion of resolvent operator and the one of spectrum need to be replaced by the two $S$-resolvent operators and the $S$-spectrum. This is a consequence of the non-commutativity of the quaternionic setting. Indeed, the $S$-spectrum of a quaternionic linear operator $T$ is given by the non invertibility of a second order operator. This presents new challenges which makes our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of $S$-spectrum and of slice hyperholomorphicity of the $S$-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.

Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability

Abstract: We first prove local-in-time well-posedness for the Muskat problem, modeling fluid flow in a two-dimensional inhomogeneous porous media. The permeability of the porous medium is described by a step function, with a jump discontinuity across the fixed-in-time curve $(x_1,-1+f(x_1))$, while the interface separating the fluid from the vacuum region is given by the time-dependent curve $(x_1,h(x_1,t))$. Our estimates are based on a new methodology that relies upon a careful study of the PDE system, coupling Darcy's law and incompressibility of the fluid, rather than the analysis of the singular integral contour equation for the interface function $h$. We are able to develop an existence theory for any initial interface given by $h_0 \in H^2$ and any permeability curve-of-discontinuity that is given by $f \in H^{2.5}$. In particular, our method allows for both curves to have (pointwise) unbounded curvature. In the case that the permeability discontinuity is the set $f=0$, we prove global existence and decay to equilibrium for small initial data. This decay is obtained using a new energy-energy dissipation inequality that couples tangential derivatives of the velocity in the bulk of the fluid with the curvature of the interface. To the best of our knowledge, this is the first global existence result for the Muskat problem with discontinuous permeability.

Comparing descent obstruction and Brauer-Manin obstruction for open varieties

Abstract: We provide a relation between Brauer-Manin obstruction and descent obstruction for torsors over open varieties under a connected linear algebraic group or a group of multiplicative type is given. Such a relation is further refined for torsors under a torus. As an appliaction, we prove that the semi-simple part of a connected linear algebraic group will satisfy strong approximation with Brauer-Manin obstruction if G iteself satisfies strong approximation with Brauer-Manin obstruction.The equivalence between descent obstruction and etale Brauer-Manin obstruction for smooth projective varieties is extended to smooth quasi-projective varieties, which provides the perspective to study integral points.