We show that there is no simple (e.g. finite or countable) basis for Borel graphs with infinite Borel chromatic number. In fact, it is proved that the closed subgraphs of the shift graph on \([\mathbb {N}]^\mathbb {N}\) with finite (or, equivalently, \(\le 3\)) Borel chromatic number form a \(\varvec{\Sigma }^1_2\)-complete set. This answers a question of Kechris and Marks and strengthens several earlier

We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type \(H^5\) initial data in the fully unstable regime. The proof combines convex integration, contour dynamics and a basic calculus for non smooth semiclassical type pseudodifferential operators which is developed.

Let M be a complete, connected Riemannian surface and suppose that \({\mathcal {S}}\subset M\) is a discrete subset. What can we learn about M from the knowledge of all Riemannian distances between pairs of points of \({\mathcal {S}}\)? We prove that if the distances in \({\mathcal {S}}\) correspond to the distances in a 2-dimensional lattice, or more generally in an arbitrary net in \({\mathbb {R}}^2\)

We describe the lamination limits of sequences of compact disks \(M_n\) embedded in \({\mathbb {R}}^3\) with constant mean curvature \(H_n\), when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi (Ann Math 160:573–615, 2004) for minimal disks. We apply this theorem to prove the existence of a chord

We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around

We give an algebraic proof of the equivalence of equivariant K-semistability (resp. equivariant K-polystability) with geometric K-semistability (resp. geometric K-polystability). Along the way we also prove the existence and uniqueness of minimal optimal destabilizing centers on K-unstable log Fano pairs.

We study the étale sheafification of algebraic K-theory, called étale K-theory. Our main results show that étale K-theory is very close to a noncommutative invariant called Selmer K-theory, which is defined at the level of categories. Consequently, we show that étale K-theory has surprisingly well-behaved properties, integrally and without finiteness assumptions. A key theoretical ingredient is the

We introduce the notion of log R-maps, and develop a proper moduli stack of stable log R-maps in the case of a hybrid gauged linear sigma model. Two virtual cycles (canonical and reduced) are constructed for these moduli stacks. The main results are two comparison theorems relating the reduced virtual cycle to the cosection localized virtual cycle, as well as the reduced virtual cycle to the canonical

Given \({{\mathbb {V}}}\) a polarizable variation of \({{\mathbb {Z}}}\)-Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for \({{\mathbb {V}}}^\otimes \) is the set of closed points s of S where the fiber \({{\mathbb {V}}}_s\) has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for

Consider the cubic nonlinear Schrödinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.

Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function \(G_s(z_1,z_2)\) for the elliptic modular group at positive integral spectral parameter s are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable \(z_1\) over all CM points of a fixed

We characterize the tempered part of the automorphic Langlands category \(\mathfrak {D}({\text {Bun}}_G)\) using the geometry of the big cell in the affine Grassmannian. We deduce that, for G non-abelian, tempered D-modules have no de Rham cohomology with compact support. The latter fact boils down to a concrete statement, which we prove using the Ran space and some explicit t-structure estimates:

We prove existence of flips, special termination, the base point free theorem and, in the case of log general type, the existence of minimal models for F-dlt foliated pairs of co-rank one on a \({\mathbb {Q}}\)-factorial projective threefold. As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated pairs and we show that foliations with canonical or F-dlt singularities

We establish an explicit Plancherel decomposition for \({{\,\mathrm{\mathrm {GL}}\,}}_n(F){\backslash } {{\,\mathrm{\mathrm {GL}}\,}}_n(E)\) where E/F is a quadratic extension of local fields of characteristic zero by making use of a local functional equation for Asai \(\gamma \)-factors. We also give two applications of this Plancherel formula: first to the global Ichino–Ikeda conjecture for unitary

We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.

In this paper, we prove that for any Kähler metrics \(\omega _0\) and \(\chi \) on M, there exists a Kähler metric \(\omega _\varphi =\omega _0+\sqrt{-1}\partial {\bar{\partial }}\varphi >0\) satisfying the J-equation \({\mathrm {tr}}_{\omega _\varphi }\chi =c\) if and only if \((M,[\omega _0],[\chi ])\) is uniformly J-stable. As a corollary, we find a sufficient condition for the existence of constant

A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure

Let F be a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from \({\mathrm {GL}}_{n+1}(F)\) to \({\mathrm {GL}}_n(F)\). A main result shows that each Bernstein component of an irreducible smooth representation of \({\mathrm {GL}}_{n+1}(F)\) restricted to \({\mathrm {GL}}_n(F)\) is indecomposable. We also classify all irreducible

We address the Uniform Boundedness Conjecture of Morton and Silverman in the case of unicritical polynomials, assuming a generalization of the abc-conjecture. For unicritical polynomials of degree at least five, we require only the standard abc-conjecture.

As was pointed out by Bragg and Lieblich

In a 2006 article Schlichting conjectured that the negative K–theory of any abelian category must vanish. This conjecture was generalized in a 2019 article by Antieau, Gepner and Heller, who hypothesized that the negative K–theory of any category with a bounded t–structure must vanish. Both conjectures will be shown to be false.

For a random walk on a finitely generated group G we obtain a generalization of a classical inequality of Ancona. We deduce as a corollary that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This provides new results for Martin compactifications of relatively hyperbolic groups.

We prove a conjecture of the first author relating the Bernstein–Sato ideal of a finite collection of multivariate polynomials with cohomology support loci of rank one complex local systems. This generalizes a classical theorem of Malgrange and Kashiwara relating the b-function of a multivariate polynomial with the monodromy eigenvalues on the Milnor fibers cohomology.

We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use SO(3) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group \({\mathcal {C}}\). Our criterion applies widely;

We study the random field Ising model on \({\mathbb {Z}}^2\) where the external field is given by i.i.d. Gaussian variables with mean zero and positive variance. We show that the effect of boundary conditions on the magnetization in a finite box decays exponentially in the distance to the boundary.

We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schrödinger-type classical field theory, in terms of partition functions and reduced density matrices

In this paper we give a complete algebro-geometric characterization of analytic tangent cones of admissible Hermitian–Yang–Mills connections over any reflexive sheaves.

We compute the nuclear dimension of separable, simple, unital, nuclear, \({\mathcal {Z}}\)-stable \(\mathrm {C}^*\)-algebras. This makes classification accessible from \({\mathcal {Z}}\)-stability and in particular brings large classes of \(\mathrm {C}^*\)-algebras associated to free and minimal actions of amenable groups on finite dimensional spaces within the scope of the Elliott classification programme

The Chow–Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove

We prove the holomorphic rigidity conjecture of Teichmüller space which loosely speaking states that the action of the mapping class group uniquely determines the Teichmüller space as a complex manifold. The method of proof is through harmonic maps. We prove that the singular set of a harmonic map from a smooth n-dimensional Riemannian domain to the Weil–Petersson completion \(\overline{\mathcal {T}}\)

We prove the \(K(\pi ,1)\) conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol’d, Pham, and Thom. Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on several new results and constructions. In particular: we show that all affine noncrossing

Let \(f:C \rightarrow \mathbb {P}^1\) be a degree k genus g cover. The stratification of line bundles \(L \in {{\,\mathrm{Pic}\,}}^d(C)\) by the splitting type of \(f_*L\) is a refinement of the stratification by Brill–Noether loci \(W^r_d(C)\). We prove that for general degree k covers, these strata are smooth of the expected dimension. In particular, this determines the dimensions of all irreducible

This is the second part of our serial work on the classification of poly-\({\mathbb {Z}}\) group actions on Kirchberg algebras. Based on technical results obtained in our previous work, we completely reduce the problem to the classification of continuous fields of Kirchberg algebras over the classifying spaces. As an application, we determine the number of cocycle conjugacy classes of outer \({\mathbb

We show that every sequence of torsion-free arithmetic congruence lattices in \(\mathrm{PGL}(2,{{{\mathbb {R}}}})\) or \(\mathrm{PGL}(2,{{{\mathbb {C}}}})\) satisfies a strong quantitative version of the limit multiplicity property. We deduce that for \(R>0\) in certain range, growing linearly in the degree of the invariant trace field, the volume of the R-thin part of any congruence arithmetic hyperbolic

We prove Kontsevich’s homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev–Borisov’s ‘dual reflexive Gorenstein cones’ construction. In particular we prove HMS for all Greene–Plesser mirror pairs (i.e., Calabi–Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi–Yau complete intersections arising from Borisov’s construction

We determine the Krieger type of nonsingular Bernoulli actions \(G \curvearrowright \prod _{g \in G} (\{0,1\},\mu _g)\). When G is abelian, we do this for arbitrary marginal measures \(\mu _g\). We prove in particular that the action is never of type II\(_\infty \) if G is abelian and not locally finite, answering Krengel’s question for \(G = {\mathbb {Z}}\). When G is locally finite, we prove that

We show the existence of finitely presented torsion-free groups with decidable word problem that cannot be embedded in any finitely generated group with decidable conjugacy problem. This answers a well-known question of Collins from the early 1970’s.

We determine the possible functions that can occur, up to asymptotic equivalence, as growth functions of semigroups, hereditary languages, and algebras.

We show that all groups of a distinguished class of «large» topological groups, that of Roelcke precompact Polish groups, have Kazhdan’s Property (T). This answers a question of Tsankov and generalizes previous results by Bekka (for the infinite-dimensional unitary group) and by Evans and Tsankov (for oligomorphic groups). Further examples include the group \({{\,\mathrm{Aut}\,}}(\mu )\) of measure-preserving

We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist. This gives a complete classification of immersed constant mean curvature spheres in three-dimensional homogeneous manifolds.

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing. We provide an exact count in the case when the quiver is of finite type or is of affine type and the framing is the coordinate vector at the extending vertex. The latter case precisely covers Etingof’s conjecture on the number

Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and \(p > p_c(G)\) then there exists a positive constant \(c_p\) such that $$\begin{aligned} \mathbf {P}_p(n \le |K| < \infty ) \le e^{-c_p n} \end{aligned}$$ for every \(n\ge 1\), where K is the cluster of the origin. We deduce the following two corollaries: 1. Every

We prove the uniqueness and stability of entropy shocks to the isentropic Euler systems among all vanishing viscosity limits of solutions to associated Navier–Stokes systems. To take into account the vanishing viscosity limit, we show a contraction property for any large perturbations of viscous shocks to the Navier–Stokes system. The contraction estimate does not depend on the strength of the viscosity

Let N and p be primes such that p divides the numerator of \(\frac{N-1}{12}\). In this paper, we study the rank \(g_p\) of the completion of the Hecke algebra acting on cuspidal modular forms of weight 2 and level \(\Gamma _0(N)\) at the p-maximal Eisenstein ideal. We give in particular an explicit criterion to know if \(g_p \ge 3\), thus answering partially a question of Mazur. In order to study \(g_p\)

It is shown that any finite list of smooth closed simply-connected 4-manifolds homeomorphic to a given one X can be obtained by removing a single compact contractible submanifold (or cork) from X, and then regluing it by powers of a boundary diffeomorphism. We then use this result to ‘separate’ finite families of corks embedded in a fixed 4-manifold.

We prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic cycle of a perverse sheaf in characteristic p by Saito. We apply this to prove a conjecture of Shende and Tsimerman on the Betti numbers of

We consider Thurston maps: branched self-coverings of the sphere with ultimately periodic critical points, and prove that the Thurston equivalence problem between them (continuous deformation of maps along with their critical orbits) is decidable. More precisely, we consider the action of mapping class groups, by pre- and post-composition, on branched coverings, and encode them algebraically as mapping

We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed 7-dimensional space with a simple geometric interpretation

For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field \({\mathbf {C}}\) of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal \({\mathbf {C}}\)-representation of a classical

We show that manifolds with \( \lceil \frac{n}{2} \rceil \)-positive curvature operators are rational homology spheres. This follows from a more general vanishing and estimation theorem for the pth Betti number of closed n-dimensional Riemannian manifolds with a lower bound on the average of the lowest \(n-p\) eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot–Meyer

Given a Hermitian line bundle \(L\rightarrow M\) over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as \(\epsilon \rightarrow 0\), of couples \((u_\epsilon ,\nabla _\epsilon )\) critical for the rescalings $$\begin{aligned} E_\epsilon (u,\nabla )=\int _M\Big (|\nabla u|^2+\epsilon ^2|F_\nabla |^2+\frac{1}{4\epsilon ^2}(1-|u|^2)^2\Big ) \end{aligned}$$ of the self-dual

We introduce a relativization of the secant sheaves from Green and Lazarsfeld (A simple proof of Petri’s theorem on canonical curves, Geometry Today, 1984) and Ein and Lazarsfeld (Inventiones Math 190:603-646, 2012) and apply this construction to the study of syzygies of canonical curves. As a first application, we give a simpler proof of Voisin’s Theorem for general canonical curves. This completely

Let A be an abelian fourfold in characteristic p. We prove the standard conjecture of Hodge type for A, namely that the intersection product $$\begin{aligned} {\mathcal {Z}}^2_{\mathrm {num}}(A)_{{\mathbb {Q}}}\times {\mathcal {Z}}_{\mathrm {num}}^2(A)_{{\mathbb {Q}}} \longrightarrow {\mathbb {Q}}\end{aligned}$$ is of signature \((\rho _2 - \rho _1 +1; \rho _1 - 1)\), with \(\rho _n=\dim {\mathcal

The aim of this paper is to study growth properties of group extensions of hyperbolic dynamical systems, where we do not assume that the extension satisfies the symmetry conditions seen, for example, in the work of Stadlbauer on symmetric group extensions and of the authors on geodesic flows. Our main application is to growth rates of periodic orbits for covers of an Anosov flow: we reduce the problem

We show that for each \(\gamma \in (0,2)\), there is a unique metric (i.e., distance function) associated with \(\gamma \)-Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) h, there is a unique random metric \(D_h\) associated with the Riemannian metric tensor “\(e^{\gamma h} (dx^2 + dy^2)\)” on \({\mathbb {C}}\) which is characterized by a

In the present paper we use twistor theory in order to solve two problems related to harmonic maps from surfaces to Euclidean spheres \({\mathbb {S}}^n\). First, we propose a new approach to isoperimetric eigenvalue inequalities based on energy index. Using this approach we show that for any positive k, the k-th non-zero eigenvalue of the Laplacian on the real projective plane endowed with a metric

New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in \(\mathbb {R}_{\ge 0}\). In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and \(\Theta \)-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a

In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension \(n \ge 3\). For every bounded open subset \(\Omega \subset M\) with smooth boundary, we prove that $$\begin{aligned} \int \limits _{\partial \Omega } \left| \frac{\mathrm{H}}{n-1}\right| ^{n-1} \!\!\!\!\!{\mathrm{d}}\sigma \,\,\ge \,\,{\mathrm{AVR}}(g)\

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-\(A_\infty \) property) of harmonic measure with respect to surface measure, on the boundary of an open set \( \Omega \subset \mathbb {R}^{n+1}\) with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in \(\Omega \), with data in \(L^p(\partial \Omega )\) for some