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  • 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
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