Showing papers in "Journal of the European Mathematical Society in 2020"

Bernstein - von Mises theorems for statistical inverse problems I: Schrödinger equation

Abstract: The inverse problem of determining the unknown potential $f>0$ in the partial differential equation $$\\frac{\\Delta}{2} u - fu =0 \\text{ on } \\mathcal O ~~\\text{s.t. } u = g \\text { on } \\partial \\mathcal O,$$ where $\\mathcal O$ is a bounded $C^\\infty$-domain in $\\mathbb R^d$ and $g>0$ is a given function prescribing boundary values, is considered. The data consist of the solution $u$ corrupted by additive Gaussian noise. A nonparametric Bayesian prior for the function $f$ is devised and a Bernstein - von Mises theorem is proved which entails that the posterior distribution given the observations is approximated in a suitable function space by an infinite-dimensional Gaussian measure that has a `minimal' covariance structure in an information-theoretic sense. As a consequence the posterior distribution performs valid and optimal frequentist statistical inference on $f$ in the small noise limit.

On Kim-independence

Abstract: We study NSOP$_{1}$ theories. We define Kim-independence, which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim's lemma, local character, symmetry, and an independence theorem and that, moreover, these properties individually characterize NSOP$_{1}$ theories. We describe Kim-independence in several concrete theories and observe that it corresponds to previously studied notions of independence in Frobenius fields and vector spaces with a generic bilinear form.

The excitation spectrum of Bose gases interacting through singular potentials

Abstract: We consider systems of N bosons in a box of volume one, interacting through a repulsive two-body potential of the form κN3β−1V(Nβx). For all 0 0, we establish the validity of Bogolyubov theory, identifying the ground state energy and the low-lying excitation spectrum up to errors that vanish in the limit of large N.

Compactness and lower semicontinuity in $GSBD$

Abstract: In this paper, we prove a compactness and semicontinuity result in GSBD for sequences with bounded Griffith energy. This generalises classical results in (G)SBV by Ambrosio [1, 2, 3] and SBD by Bellettini-Coscia-Dal Maso [9]. As a result, the static problem in Francfort-Marigo's variational approach to crack growth [27] admits (weak) solutions. Moreover, we obtain a compactness property for minimisers of suitable Ambrosio-Tortorelli's type energies [6], which have been shown to Γ-converge to Griffith energy in [16].

Dimension, comparison, and almost finiteness

Abstract: We develop a dynamical version of some of the theory surrounding the Toms-Winter conjecture for simple separable nuclear C*-algebras and study its connections to the C*-algebra side via the crossed product. We introduce an analogue of hyperfiniteness for free actions of amenable groups on compact spaces and show that it plays the role of Z-stability in the Toms-Winter conjecture in its relation to dynamical comparison, and also that it implies Z-stability of the crossed product. This property, which we call almost finiteness, generalizes Matui's notion of the same name from the zero-dimensional setting. We also introduce a notion of tower dimension as partial analogue of nuclear dimension and study its relation to dynamical comparison and almost finiteness, as well as to the dynamical asymptotic dimension and amenability dimension of Guentner, Willett, and Yu.

Descent in algebraic $K$-theory and a conjecture of Ausoni–Rognes

Abstract: Let $A \to B$ be a $G$-Galois extension of rings, or more generally of $\mathbb{E}_\infty$-ring spectra in the sense of Rognes. A basic question in algebraic $K$-theory asks how close the map $K(A) \to K(B)^{hG}$ is to being an equivalence, i.e., how close algebraic $K$-theory is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of Thomason, one also expects such a result after periodic localization. We formulate and prove a general result which enables one to promote rational descent statements as above into descent statements after periodic localization. This reduces the localized descent problem to establishing an elementary condition on $K_0(-)\otimes \mathbb{Q}$. As applications, we prove various descent results in the periodic localized $K$-theory, $TC$, $THH$, etc. of structured ring spectra, and verify several cases of a conjecture of Ausoni and Rognes.

Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients

Abstract: We introduce a condition on accretive matrix functions, called $p$-ellipticity, and discuss its applications to the $L^p$ theory of elliptic PDE with complex coefficients. Our examples are: (i) generalized convexity of power functions (Bellman functions), (ii) dimension-free bilinear embeddings, (iii) $L^p$-contractivity of semigroups and (iv) holomorphic functional calculus. Recent work by Dindos and Pipher (arXiv:1612.01568v3) established close ties between $p$-ellipticity and (v) regularity theory of elliptic PDE with complex coefficients. The $p$-ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on one hand, and the Hessian of a power function on the other. Our results regarding contractivity extend earlier theorems by Cialdea and Maz'ya.

The Berenstein–Zelevinsky quantum cluster algebra conjecture

Abstract: We prove the Berenstein-Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite dimensional simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [4]. We furthermore prove that the corresponding upper quantum cluster algebras coincide with the constructed quantum cluster algebras and exhibit a large number of explicit quantum seeds. Along the way a detailed study of the properties of quantum double Bruhat cells from the viewpoint of noncommutative UFDs is carried out and a quantum analog of the Fomin-Zelevinsky twist map is constructed and investigated for all double Bruhat cells. The results are valid over base fields of arbitrary characteristic and the deformation parameter is only assumed to be a non-root of unity.

Counting curves, and the stable length of currents

Abstract: Let $\gamma_0$ be a curve on a surface $\Sigma$ of genus $g$ and with $r$ boundary components and let $\pi_1(\Sigma)\curvearrowright X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves $\gamma$ of type $\gamma_0$ with translation length at most $L$ on $X$. For example, as an application, we derive that for any finite generating set $S$ of $\pi_1(\Sigma)$ the limit $$\lim_{L\to\infty}\frac 1{L^{6g-6+2r}}\{\gamma\text{ of type }\gamma_0\text{ with }S\text{-translation length}\le L\}$$ exists and is positive. The main new technical tool is that the function which associates to each curve its stable length with respect to the action on $X$ extends to a (unique) continuous and homogenous function on the space of currents. We prove that this is indeed the case for any action of a torsion free hyperbolic group.

Quantitative multiple mixing

Abstract: We develop a method for providing quantitative estimates for higher order correlations of group actions. In particular, we establish effective mixing of all orders for actions of semisimple Lie groups as well as semisimple S-algebraic groups and semisimple adele groups. As an application, we deduce existence of approximate configurations in lattices of semisimple groups.

Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions

Abstract: In this work, we investigate the small-time global exact controllability of the Navier-Stokes equation , both towards the null equilibrium state and towards weak trajectories. We consider a viscous incompressible fluid evolving within a smooth bounded domain, either in 2D or in 3D. The controls are only located on a small part of the boundary, intersecting all its connected components. On the remaining parts of the boundary, the fluid obeys a Navier slip-with-friction boundary condition. Even though viscous boundary layers appear near these uncontrolled boundaries, we prove that small-time global exact controllability holds. Our analysis relies on the controllability of the Euler equation combined with asymptotic boundary layer expansions. Choosing the boundary controls with care enables us to guarantee good dissipation properties for the residual boundary layers, which can then be exactly canceled using local techniques.

Universal limits of substitution-closed permutation classes

Abstract: We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons. The limit depends on the generating series of the simple permutations in the class. Under a mild sufficient condition, the limit is an elementary one-parameter deformation of the limit of uniform separable permutations, previously identified as the Brownian separable permuton. This limiting object is therefore in some sense universal. We identify two other regimes with different limiting objects. The first one is degenerate; the second one is nontrivial and related to stable trees. These results are obtained thanks to a characterization of the convergence of random permutons through the convergence of their expected pattern densities. The limit of expected pattern densities is then computed by using the substitution tree encoding of permutations and performing singularity analysis on the tree series.

Renormalising SPDEs in regularity structures

Abstract: The formalism recently introduced in [BHZ19] allows one to assign a regularity structure, as well as a corresponding “renormalisation group”, to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was shown in [CH16] that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT. The present work completes this programme by constructing an action of the renormalisation group on a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular non-linear SPDEs.

Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point

Abstract: The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0\le n\le 2$ the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=\tfrac{1}{\sqrt{2+\sqrt{2-n}}}$. For $0

The two-periodic Aztec diamond and matrix valued orthogonal polynomials

Abstract: We analyze domino tilings of the two-periodic Aztec diamond by means of matrix valued orthogonal polynomials that we obtain from a reformulation of the Aztec diamond as a non-intersecting path model with periodic transition matrices. In a more general framework we express the correlation kernel for the underlying determinantal point process as a double contour integral that contains the reproducing kernel of matrix valued orthogonal polynomials. We use the Riemann-Hilbert problem to simplify this formula for the case of the two-periodic Aztec diamond. In the large size limit we recover the three phases of the model known as solid, liquid and gas. We describe fine asymptotics for the gas phase and at the cusp points of the liquid-gas boundary, thereby complementing and extending results of Chhita and Johansson.

Embedding rainbow trees with applications to graph labelling and decomposition

Abstract: A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares Since then rainbow structures have been the focus of extensive research and have found applications in the areas of graph labelling and decomposition An edge-colouring is locally k-bounded if each vertex is contained in at most k edges of the same colour In this paper we prove that any such edge-colouring of the complete graph Kn contains a rainbow copy of every tree with at most (1−o(1))n/k vertices As a locally k-bounded edge-colouring of Kn may have only (n−1)/k distinct colours, this is essentially tight As a corollary of this result we obtain asymptotic versions of two long-standing conjectures in graph theory Firstly, we prove an asymptotic version of Ringel's conjecture from 1963, showing that any n-edge tree packs into the complete graph K(2n+o(n)) to cover all but o(n^2) of its edges Secondly, we show that all trees have an almost-harmonious labelling The existence of such a labelling was conjectured by Graham and Sloane in 1980 We also discuss some additional applications

Quasi-invariant Gaussian measures for the two-dimensional defocusing cubic nonlinear wave equation

Abstract: We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the two-dimensional defocusing cubic nonlinear wave equation (NLW). Under some regularity condition, we prove quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces for the NLW dynamics. We achieve this goal by introducing a simultaneous renormalization on the energy functional and its time derivative and establishing a renormalized energy estimate in the probabilistic setting.

Multi-bumps analysis for Trudinger-Moser nonlinearities I-Quantification and location of concentration points

Abstract: In this paper, we investigate carefully the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension two containing a nonlinearity with Trudinger-Moser growth. A quantification result had been obtained by the first author in [15] but many questions were left open. Similar questions were also explicitly asked in subsequent papers, see Del Pino-Musso-Ruf [12], Malchiodi-Martinazzi [30] or Martinazzi [34]. We answer all of them, proving in particular that blow up phenomenon is very restrictive because of the strong interaction between bubbles in this equation. This work will have a sequel, giving existence results of critical points of the associated functional at all energy levels via degree theory arguments, in the spirit of what had been done for the Liouville equation in the beautiful work of Chen-Lin [8].

Symmetric elliptic functions, IRF models, and dynamic exclusion processes

Abstract: We introduce stochastic Interaction-Round-a-Face (IRF) models that are related to representations of the elliptic quantum group $E_{\tau,\eta}(sl_2)$. For stochasic IRF models in a quadrant, we evaluate averages for a broad family of observables that can be viewed as higher analogs of $q$-moments of the height function for the stochastic (higher spin) six vertex models. In a certain limit, the stochastic IRF models degenerate to (1+1)d interacting particle systems that we call dynamic ASEP and SSEP; their jump rates depend on local values of the height function. For the step initial condition, we evaluate averages of observables for them as well, and use those to investigate one-point asymptotics of the dynamic SSEP. The construction and proofs are based on remarkable properties (branching and Pieri rules, Cauchy identities) of a (seemingly new) family of symmetric elliptic functions that arise as matrix elements in an infinite volume limit of the algebraic Bethe ansatz for $E_{\tau,\eta}(sl_2)$.

Refined global Gross–Prasad conjecture on special Bessel periods and Böcherer’s conjecture

Abstract: In this paper we pursue the refined global Gross-Prasad conjecture for Bessel periods formulated by Yifeng Liu in the case of special Bessel periods for $\\mathrm{SO}\\left(2n+1\\right)\\times\\mathrm{SO}\\left(2\\right)$. Recall that a Bessel period for $\\mathrm{SO}\\left(2n+1\\right)\\times\\mathrm{SO}\\left(2\\right)$ is called special when the representation of $\\mathrm{SO}\\left(2\\right)$ is trivial. Let $\\pi$ be an irreducible cuspidal tempered automorphic representation of a special orthogonal group of an odd dimensional quadratic space over a totally real number field $F$ whose local component $\\pi_v$ at any archimedean place $v$ of $F$ is a discrete series representation. Let $E$ be a quadratic extension of $F$ and suppose that the special Bessel period corresponding to $E$ does not vanish identically on $\\pi$. Then we prove the Ichino-Ikeda type explicit formula conjectured by Liu for the central value $L\\left(1/2,\\pi\\right)L\\left(1/2,\\pi\\times\\chi_E\\right)$, where $\\chi_E$ denotes the quadratic character corresponding to $E$. Our result yields a proof of Boecherer's conecture on holomorphic Siegel cusp forms of degree two which are Hecke eigenforms.

Contraction property for large perturbations of shocks of the barotropic Navier–Stokes system

Abstract: This paper is dedicated to the construction of a pseudo-norm, for which small shock profiles of the barotropic Navier-Stokes equation have a contraction property. This contraction property holds in the class of any large 1D weak solutions to the barotropic Navier-Stokes equation. It implies a stability condition which is independent of the strength of the viscosity. The proof is based on the relative entropy method, and is reminiscent to the notion of a-contraction first introduced by the authors in the hyperbolic case.

Asymptotically efficient estimation of smooth functionals of covariance operators

Abstract: Let $X$ be a centered Gaussian random variable in a separable Hilbert space ${\mathbb H}$ with covariance operator $\Sigma.$ We study a problem of estimation of a smooth functional of $\Sigma$ based on a sample $X_1,\dots ,X_n$ of $n$ independent observations of $X.$ More specifically, we are interested in functionals of the form $\langle f(\Sigma), B\rangle,$ where $f:{\mathbb R}\mapsto {\mathbb R}$ is a smooth function and $B$ is a nuclear operator in ${\mathbb H}.$ We prove concentration and normal approximation bounds for plug-in estimator $\langle f(\hat \Sigma),B\rangle,$ $\hat \Sigma:=n^{-1}\sum_{j=1}^n X_j\otimes X_j$ being the sample covariance based on $X_1,\dots, X_n.$ These bounds show that $\langle f(\hat \Sigma),B\rangle$ is an asymptotically normal estimator of its expectation ${\mathbb E}_{\Sigma} \langle f(\hat \Sigma),B\rangle$ (rather than of parameter of interest $\langle f(\Sigma),B\rangle$) with a parametric convergence rate $O(n^{-1/2})$ provided that the effective rank ${\bf r}(\Sigma):= \frac{{\bf tr}(\Sigma)}{\|\Sigma\|}$ (${\rm tr}(\Sigma)$ being the trace and $\|\Sigma\|$ being the operator norm of $\Sigma$) satisfies the assumption ${\bf r}(\Sigma)=o(n).$ At the same time, we show that the bias of this estimator is typically as large as $\frac{{\bf r}(\Sigma)}{n}$ (which is larger than $n^{-1/2}$ if ${\bf r}(\Sigma)\geq n^{1/2}$). In the case when ${\mathbb H}$ is finite-dimensional space of dimension $d=o(n),$ we develop a method of bias reduction and construct an estimator $\langle h(\hat \Sigma),B\rangle$ of $\langle f(\Sigma),B\rangle$ that is asymptotically normal with convergence rate $O(n^{-1/2}).$ Moreover, we study asymptotic properties of the risk of this estimator and prove minimax lower bounds for arbitrary estimators showing the asymptotic efficiency of $\langle h(\hat \Sigma),B\rangle$ in a semi-parametric sense.

Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms

Abstract: In this paper, we establish stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincare inequalities. In particular, we establish the connection between parabolic Harnack inequalities and two-sided heat kernel estimates, as well as with the Holder regularity of parabolic functions for symmetric non-local Dirichlet forms.

Resolving Stanley's $e$-positivity of claw-contractible-free graphs

Abstract: In Stanley's seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not $e$-positive, namely, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic symmetric functions are not $e$-positive. Moreover, one such family is additionally claw-free, thus establishing that the $e$-positivity of chromatic symmetric functions is in general not dependent on the existence of an induced claw or of a contraction to a claw.

Homotopy finiteness of some DG categories from algebraic geometry

Abstract: In this paper, we prove that the bounded derived category $D^b_{coh}(Y)$ of coherent sheaves on a separated scheme $Y$ of finite type over a field $\mathrm{k}$ of characteristic zero is homotopically finitely presented. This confirms a conjecture of Kontsevich. We actually prove a stronger statement: $D^b_{coh}(Y)$ is equivalent to a DG quotient $D^b_{coh}(\tilde{Y})/T,$ where $\tilde{Y}$ is some smooth and proper variety, and the subcategory $T$ is generated by a single object. The proof uses categorical resolution of singularities of Kuznetsov and Lunts \cite{KL}, and a theorem of Orlov \cite{Or} stating that the class of geometric smooth and proper DG categories is stable under gluing. We also prove the analogous result for $\mathbb{Z}/2$-graded DG categories of coherent matrix factorizations on such schemes. In this case instead of $D^b_{coh}(\tilde{Y})$ we have a semi-orthogonal gluing of a finite number of DG categories of matrix factorizations on smooth varieties, proper over $\mathbb{A}_{\mathrm{k}}^1$.

Pushforwards of pluricanonical bundles under morphisms to abelian varieties

Abstract: Let $f \colon X \to A$ be a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). We show that the sheaves $f_* \omega_X^{\otimes m}$ become globally generated after pullback by an isogeny. We use this to deduce a decomposition theorem for these sheaves when $m \ge 2$, analogous to that obtained by Chen-Jiang when $m = 1$. This is in turn applied to effective results for pluricanonical linear series on irregular varieties with canonical singularities.