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  • Tightness of Liouville first passage percolation for γ ∈ ( 0 , 2 ) $\gamma \in (0,2)$
    Publ. math. IHES (IF 1.222) Pub Date : 2020-11-04
    Jian Ding, Julien Dubédat, Alexander Dunlap, Hugo Falconet

    We study Liouville first passage percolation metrics associated to a Gaussian free field \(h\) mollified by the two-dimensional heat kernel \(p_{t}\) in the bulk, and related star-scale invariant metrics. For \(\gamma \in (0,2)\) and \(\xi = \frac{\gamma }{d_{\gamma }}\), where \(d_{\gamma }\) is the Liouville quantum gravity dimension defined in Ding and Gwynne (Commun. Math. Phys. 374:1877–1934,

    更新日期:2020-11-04
  • Deviations of ergodic sums for toral translations II. Boxes
    Publ. math. IHES (IF 1.222) Pub Date : 2020-10-19
    Dmitry Dolgopyat, Bassam Fayad

    We study the Kronecker sequence \(\{n\alpha \}_{n\leq N}\) on the torus \({\mathbf {T}}^{d}\) when \(\alpha \) is uniformly distributed on \({\mathbf {T}}^{d}\). We show that the discrepancy of the number of visits of this sequence to a random box, normalized by \(\ln ^{d} N\), converges as \(N\to \infty \) to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the

    更新日期:2020-10-19
  • Generic regularity of free boundaries for the obstacle problem
    Publ. math. IHES (IF 1.222) Pub Date : 2020-07-02
    Alessio Figalli, Xavier Ros-Oton, Joaquim Serra

    The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in \(\mathbf {R}^{n}\). By classical results of Caffarelli, the free boundary is \(C^{\infty }\) outside a set of singular points. Explicit examples show that the singular set could be in general \((n-1)\)-dimensional—that is, as large as the regular set. Our main result establishes that, generically

    更新日期:2020-07-02
  • Explicit spectral gaps for random covers of Riemann surfaces
    Publ. math. IHES (IF 1.222) Pub Date : 2020-06-25
    Michael Magee, Frédéric Naud

    We introduce a permutation model for random degree \(n\) covers \(X_{n}\) of a non-elementary convex-cocompact hyperbolic surface \(X=\Gamma \backslash \mathbf {H}\). Let \(\delta \) be the Hausdorff dimension of the limit set of \(\Gamma \). We say that a resonance of \(X_{n}\) is new if it is not a resonance of \(X\), and similarly define new eigenvalues of the Laplacian. We prove that for any \(\epsilon

    更新日期:2020-06-25
  • Complexity of parabolic systems
    Publ. math. IHES (IF 1.222) Pub Date : 2020-05-15
    Tobias Holck Colding, William P. Minicozzi

    We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. We use this in a major application of our new methods to give the first general bounds on generic singularities of surfaces

    更新日期:2020-05-15
  • Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations
    Publ. math. IHES (IF 1.222) Pub Date : 2020-05-14
    Helge Ruddat, Bernd Siebert

    We give a simple expression for the integral of the canonical holomorphic volume form in degenerating families of varieties constructed from wall structures and with central fiber a union of toric varieties. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the central fiber. One application is a proof that the mirror map for the canonical formal families

    更新日期:2020-05-14
  • Discrete series multiplicities for classical groups over Z$\mathbf {Z}$ and level 1 algebraic cusp forms
    Publ. math. IHES (IF 1.222) Pub Date : 2020-03-05
    Gaëtan Chenevier, Olivier Taïbi

    The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation in the space of level 1 automorphic forms of a split classical group \(G\) over \(\mathbf {Z}\), and provide numerical applications in absolute rank \(\leq 8\). Second, we prove a classification result for the level one cuspidal algebraic automorphic representations

    更新日期:2020-04-22
  • Riemannian hyperbolization
    Publ. math. IHES (IF 1.222) Pub Date : 2020-02-28
    Pedro Ontaneda

    The strict hyperbolization process of Charney and Davis produces a large and rich class of negatively curved spaces (in the geodesic sense). This process is based on an earlier version introduced by Gromov and later studied by Davis and Januszkiewicz. If M is a manifold its Charney-Davis strict hyperbolization is also a manifold, but the negatively curved metric obtained is very far from being Riemannian

    更新日期:2020-04-22
  • Quasimap wall-crossings and mirror symmetry
    Publ. math. IHES (IF 1.222) Pub Date : 2020-02-07
    Ionuţ Ciocan-Fontanine, Bumsig Kim

    We state a wall-crossing formula for the virtual classes of \({\varepsilon }\)-stable quasimaps to GIT quotients and prove it for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing formula relating the genus \(g\) descendant Gromov-Witten potential and the genus \(g\)\({\varepsilon }\)-quasimap descendant potential

    更新日期:2020-04-22
  • E2$E_{2}$ -cells and mapping class groups
    Publ. math. IHES (IF 1.222) Pub Date : 2019-06-17
    Søren Galatius, Alexander Kupers, Oscar Randal-Williams

    We prove a new kind of stabilisation result, “secondary homological stability,” for the homology of mapping class groups of orientable surfaces with one boundary component. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of \(E_{2}\)-algebras, which have no \(E_{2}\)-cells below a certain vanishing line.

    更新日期:2020-04-22
  • The period-index problem for real surfaces
    Publ. math. IHES (IF 1.222) Pub Date : 2019-05-28
    Olivier Benoist

    We study when the period and the index of a class in the Brauer group of the function field of a real algebraic surface coincide. We prove that it is always the case if the surface has no real points (more generally, if the class vanishes in restriction to the real points of the locus where it is well-defined), and give a necessary and sufficient condition for unramified classes. As an application

    更新日期:2020-04-22
  • Separation for the stationary Prandtl equation
    Publ. math. IHES (IF 1.222) Pub Date : 2019-09-05
    Anne-Laure Dalibard, Nader Masmoudi

    In this paper, we prove that separation occurs for the stationary Prandtl equation, in the case of adverse pressure gradient, for a large class of boundary data at \(x=0\). We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at \(x=0\), there exists \(x^{*}>0\) such that \(\partial _{y} u_{|y=0}(x) \sim C \sqrt{x^{*} -x}\) as \(x\to x^{*}\)

    更新日期:2020-04-22
  • A local model for the trianguline variety and applications
    Publ. math. IHES (IF 1.222) Pub Date : 2019-08-22
    Christophe Breuil, Eugen Hellmann, Benjamin Schraen

    We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck’s simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial

    更新日期:2020-04-22
  • Covariantly functorial wrapped Floer theory on Liouville sectors
    Publ. math. IHES (IF 1.222) Pub Date : 2019-08-23
    Sheel Ganatra, John Pardon, Vivek Shende

    We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaid’s generation criterion

    更新日期:2020-04-22
  • Foliations with positive slopes and birational stability of orbifold cotangent bundles
    Publ. math. IHES (IF 1.222) Pub Date : 2019-04-18
    Frédéric Campana, Mihai Păun

    Let \(X\) be a smooth connected projective manifold, together with an snc orbifold divisor \(\Delta \), such that the pair \((X, \Delta )\) is log-canonical. If \(K_{X}+\Delta \) is pseudo-effective, we show, among other things, that any quotient of its orbifold cotangent bundle has a pseudo-effective determinant. This improves considerably our previous result (Campana and Păun in Ann. Inst. Fourier

    更新日期:2020-04-22
  • Joinings of higher rank torus actions on homogeneous spaces
    Publ. math. IHES (IF 1.222) Pub Date : 2019-02-14
    Manfred Einsiedler, Elon Lindenstrauss

    We show that joinings of higher rank torus actions on \(S\)-arithmetic quotients of semi-simple or perfect algebraic groups must be algebraic.

    更新日期:2020-04-22
  • Topological Hochschild homology and integral p $p$ -adic Hodge theory
    Publ. math. IHES (IF 1.222) Pub Date : 2019-04-17
    Bhargav Bhatt, Matthew Morrow, Peter Scholze

    In mixed characteristic and in equal characteristic \(p\) we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic \(K\)-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex \(A\Omega\) constructed in our previous work, and in equal characteristic \(p\) to crystalline cohomology

    更新日期:2020-04-22
  • Categorical actions on unipotent representations of finite unitary groups
    Publ. math. IHES (IF 1.222) Pub Date : 2019-03-08
    O. Dudas, M. Varagnolo, E. Vasserot

    Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincides with

    更新日期:2020-04-22
  • Fourier interpolation on the real line
    Publ. math. IHES (IF 1.222) Pub Date : 2018-09-17
    Danylo Radchenko, Maryna Viazovska

    In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set \(\{0, \pm\sqrt{1}, \pm\sqrt{2}, \pm\sqrt{3},\dots\}\). The functions in the interpolating basis are constructed in a closed form as an integral transform of

    更新日期:2020-04-22
  • Integral p $p$ -adic Hodge theory
    Publ. math. IHES (IF 1.222) Pub Date : 2019-01-16
    Bhargav Bhatt, Matthew Morrow, Peter Scholze

    We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of \(\mathbf {C}_{p}\). It takes values in a mixed-characteristic analogue of Dieudonné modules, which was previously defined by Fargues as a version of Breuil–Kisin modules. Notably, this cohomology theory specializes to all other known \(p\)-adic cohomology theories, such as crystalline, de Rham and

    更新日期:2020-04-22
  • Measure concentration and the weak Pinsker property
    Publ. math. IHES (IF 1.222) Pub Date : 2018-02-15
    Tim Austin

    Let \((X,\mu)\) be a standard probability space. An automorphism \(T\) of \((X,\mu)\) has the weak Pinsker property if for every \(\varepsilon > 0\) it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less than \(\varepsilon \). This property was introduced by Thouvenot, who asked whether it holds for all ergodic automorphisms. This paper proves that it does

    更新日期:2020-04-22
  • Integral models of Shimura varieties with parahoric level structure
    Publ. math. IHES (IF 1.222) Pub Date : 2018-04-30
    M. Kisin, G. Pappas

    For a prime \(p > 2\), we construct integral models over \(p\) for Shimura varieties with parahoric level structure, attached to Shimura data \((G,X)\) of abelian type, such that \(G\) splits over a tamely ramified extension of \({\mathbf {Q}}_{\,p}\). The local structure of these integral models is related to certain “local models”, which are defined group theoretically. Under some additional assumptions

    更新日期:2020-04-22
  • La conjecture du facteur direct
    Publ. math. IHES (IF 1.222) Pub Date : 2017-12-07
    Yves André

    M. Hochster a conjecturé que pour toute extension finie \(S\) d’un anneau commutatif régulier \(R\), la suite exacte de \(R\)-modules \(0\to R \to S \to S/R\to0\) est scindée. En nous appuyant sur sa réduction au cas d’un anneau local régulier \(R\) complet non ramifié d’inégale caractéristique, nous proposons une démonstration de cette conjecture dans le contexte de la théorie perfectoïde de P. Scholze

    更新日期:2020-04-22
  • Le lemme d’Abhyankar perfectoide
    Publ. math. IHES (IF 1.222) Pub Date : 2017-12-07
    Yves André

    Nous étendons le théorème de presque-pureté de Faltings-Scholze-Kedlaya-Liu sur les extensions étales finies d’algèbres perfectoïdes au cas des extensions ramifiées, sans restriction sur le lieu de ramification. Nous déduisons cette version perfectoïde du lemme d’Abhyankar du théorème de presque-pureté, par un passage à la limite mettant en jeu des versions perfectoïdes du théorème d’extension de Riemann

    更新日期:2020-04-22
  • Invariant and stationary measures for the action on Moduli space
    Publ. math. IHES (IF 1.222) Pub Date : 2018-04-17
    Alex Eskin, Maryam Mirzakhani

    We prove some ergodic-theoretic rigidity properties of the action of on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.

    更新日期:2020-04-22
  • A viscosity method in the min-max theory of minimal surfaces
    Publ. math. IHES (IF 1.222) Pub Date : 2017-10-26
    Tristan Rivière

    We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface \(\Sigma \) into a given closed manifold, we add to the area Lagrangian a term equal to the \(L^{q}\) norm of the second fundamental form of the immersion times a “viscosity” parameter. This relaxation of the area functional satisfies the Palais–Smale condition

    更新日期:2017-10-26
  • Percolation of random nodal lines
    Publ. math. IHES (IF 1.222) Pub Date : 2017-09-18
    Vincent Beffara,Damien Gayet

    We prove a Russo-Seymour-Welsh percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let \(U\) be a smooth connected bounded open set in \(\mathbf{R}^{2}\) and \(\gamma, \gamma '\) two disjoint arcs of positive length in the boundary of \(U\). We prove that there exists a positive constant

    更新日期:2017-09-18
  • Calabi-Yau manifolds with isolated conical singularities
    Publ. math. IHES (IF 1.222) Pub Date : 2017-08-25
    Hans-Joachim Hein,Song Sun

    Let \(X\) be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let \(L\) be an ample line bundle on \(X\). Assume that the pair \((X,L)\) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point \(x \in X\) there exist a Kähler-Einstein Fano manifold \(Z\) and a positive integer \(q\) dividing \(K_{Z}\)

    更新日期:2017-08-25
  • C*-simplicity and the unique trace property for discrete groups
    Publ. math. IHES (IF 1.222) Pub Date : 2017-06-28
    Emmanuel Breuillard,Mehrdad Kalantar,Matthew Kennedy,Narutaka Ozawa

    A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take

    更新日期:2017-06-28
  • Meromorphic tensor equivalence for Yangians and quantum loop algebras
    Publ. math. IHES (IF 1.222) Pub Date : 2017-06-28
    Sachin Gautam,Valerio Toledano Laredo

    Let \(\mathfrak{g}\) be a complex semisimple Lie algebra, and \(Y_{\hbar }(\mathfrak{g})\), \(U_{q}(L\mathfrak{g})\) the corresponding Yangian and quantum loop algebra, with deformation parameters related by \(q=e^{\pi \iota \hbar }\). When \(\hbar \) is not a rational number, we constructed in Gautam and Toledano Laredo (J. Am. Math. Soc. 29:775, 2016) a faithful functor \(\Gamma \) from the category

    更新日期:2017-06-28
  • On the hyperbolicity of general hypersurfaces
    Publ. math. IHES (IF 1.222) Pub Date : 2017-06-27
    Damian Brotbek

    In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in \(\mathbf {P}^{n}\) are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove this statement, we construct hypersurfaces satisfying a property which is Zariski open and which implies hyperbolicity. These hypersurfaces are chosen

    更新日期:2017-06-27
  • Double ramification cycles on the moduli spaces of curves
    Publ. math. IHES (IF 1.222) Pub Date : 2017-05-10
    F. Janda,R. Pandharipande,A. Pixton,D. Zvonkine

    Curves of genus \(g\) which admit a map to \(\mathbf {P}^{1}\) with specified ramification profile \(\mu\) over \(0\in \mathbf {P}^{1}\) and \(\nu\) over \(\infty\in \mathbf {P}^{1}\) define a double ramification cycle \(\mathsf{DR}_{g}(\mu,\nu)\) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated

    更新日期:2017-05-10
  • Geometric presentations of Lie groups and their Dehn functions
    Publ. math. IHES (IF 1.222) Pub Date : 2016-12-20
    Yves Cornulier,Romain Tessera

    We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic groups over local fields, uses and extends Abels’ theory of multiamalgams of graded Lie algebras, in order to provide workable presentations of these

    更新日期:2016-12-20
  • Diffeomorphisms with positive metric entropy
    Publ. math. IHES (IF 1.222) Pub Date : 2016-10-18
    A. Avila,S. Crovisier,A. Wilkinson

    We obtain a dichotomy for \(C^{1}\)-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.

    更新日期:2016-10-18
  • Teichmüller curves in genus three and just likely intersections in \(\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}\)
    Publ. math. IHES (IF 1.222) Pub Date : 2016-06-15
    Matt Bainbridge,Philipp Habegger,Martin Möller

    We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum \(\Omega\mathcal{M}_{3}(4)\), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmüller curve to obtain new information on the locations of

    更新日期:2016-06-15
  • Gaussian asymptotics of discrete \(\beta \)-ensembles
    Publ. math. IHES (IF 1.222) Pub Date : 2016-06-14
    Alexei Borodin,Vadim Gorin,Alice Guionnet

    We introduce and study stochastic \(N\)-particle ensembles which are discretizations for general-\(\beta \) log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, \((z,w)\)-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as \(N\to \infty

    更新日期:2016-06-14
  • Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums
    Publ. math. IHES (IF 1.222) Pub Date : 2016-03-23
    Karim A. Adiprasito,Raman Sanyal

    In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical

    更新日期:2016-03-23
  • Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces
    Publ. math. IHES (IF 1.222) Pub Date : 2016-03-02
    Mohammed Abouzaid,Denis Auroux,Ludmil Katzarkov

    We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface \(H\) in a toric variety \(V\) we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of \(V\times \mathbf {C}\) along \(H\times0\), under a positivity assumption. This construction also

    更新日期:2016-03-02
  • On the Fukaya category of a Fano hypersurface in projective space
    Publ. math. IHES (IF 1.222) Pub Date : 2016-02-15
    Nick Sheridan

    This paper is about the Fukaya category of a Fano hypersurface \(X \subset \mathbf {CP}^{n}\). Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed–open string maps, weak

    更新日期:2016-02-15
  • Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs
    Publ. math. IHES (IF 1.222) Pub Date : 2016-01-18
    Caucher Birkar,De-Qi Zhang

    For every smooth complex projective variety \(W\) of dimension \(d\) and nonnegative Kodaira dimension, we show the existence of a universal constant \(m\) depending only on \(d\) and two natural invariants of the very general fibres of an Iitaka fibration of \(W\) such that the pluricanonical system \(|mK_{W}|\) defines an Iitaka fibration. This is a consequence of a more general result on polarized

    更新日期:2016-01-18
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