We study uniformization problems for compact manifolds that arise as quotients of domains in complex flag varieties by images of Anosov homomorphisms. We focus on Anosov homomorphisms with “small” limit sets, as measured by the Riemannian Hausdorff codimension in the flag variety. Under such a codimension hypothesis, we show that all first-order deformations of complex structure on the associated compact

A typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs \(2n+1\) times into the complete graph \(K_{2n+1}\). In this

We prove that generalised Monge-Ampère equations (a family of equations which includes the inverse Hessian equations like the J-equation, as well as the Monge-Ampère equation) on projective manifolds have smooth solutions if certain intersection numbers are positive. As corollaries of our work, we improve a result of Chen (albeit in the projective case) on the existence of solutions to the J-equation

We show that for every closed Riemannian manifold there exists a continuous family of 1-cycles (defined as finite collections of disjoint closed curves) parametrized by a sphere and sweeping out the whole manifold so that the lengths of all connected closed curves are bounded in terms of the volume (or the diameter) and the dimension n of the manifold, when \(n \ge 3\). An alternative form of this

We study the possible structures of monodromy groups of Kloosterman and hypergeometric sheaves on \({{\mathbb {G}}}_m\) in characteristic p. We show that most such sheaves satisfy a certain condition \(\mathrm {(\mathbf{S+})}\), which has very strong consequences on their monodromy groups. We also classify the finite, almost quasisimple, groups that can occur as monodromy groups of Kloosterman and

In this paper, we prove upper bounds for the volume spectrum of a Riemannian manifold that depend only on the volume, dimension and a conformal invariant.

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual

Let \(M^{n+1}\) be a closed manifold of dimension \(3\le n+1\le 7\). We show that for a \(C^\infty \)-generic metric g on M, to any connected, closed, embedded, 2-sided, stable, minimal hypersurface \(S\subset (M,g)\) corresponds a sequence of closed, embedded, minimal hypersurfaces \(\{\Sigma _k\}\) scarring along S, in the sense that the area and Morse index of \(\Sigma _k\) both diverge to infinity

By associating frequencies to larger scales, we provide a simpler way to derive local uniformity of multiplicative functions on average from the results of Matomäki-Radziwiłł.

We prove two inequalities for the complex orientations of a separating non-singular real algebraic curve in \({\mathbb {RP}}^2\) of any odd degree. We also construct a separating non-singular real (i.e., invariant under the complex conjugation) pseudoholomorphic curve in \({\mathbb {CP}}^2\) of any degree congruent to 9 mod 12 which does not satisfy one of these inequalities. Therefore the oriented

We give a procedure for “reverse engineering" a closed, simply connected, Riemannian manifold with bounded local geometry from a sparse chain complex over \({\mathbb {Z}}\). Applying this procedure to chain complexes obtained by “lifting" recently developed quantum codes, which correspond to chain complexes over \({\mathbb {Z}}_2\), we construct the first examples of power law \({\mathbb {Z}}_2\) systolic

Let \(g_t\) be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold M starting at the hyperbolic metric. We construct foliations of the Grassmann bundle \(Gr_2(M)\) of tangent 2-planes whose leaves are (lifts of) minimal surfaces in \((M,g_t)\). These foliations are deformations of the foliation of \(Gr_2(M)\) by (lifts of) totally geodesic planes projected down

We solve a long-standing conjecture by Barker, proving that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the two cones is generated by a linearly independent set. Here, given two proper cones \({\mathcal {C}}_1\), \({\mathcal {C}}_2\), their minimal tensor product is the cone generated by products of the form \(x_1\otimes x_2\), where

This article provides a non-asymptotic analysis of the singular values (and Lyapunov exponents) of Gaussian matrix products in the regime where N, the number of terms in the product, is large and n, the size of the matrices, may be large or small and may depend on N. We obtain concentration estimates for sums of Lyapunov exponents, a quantitative rate for convergence of the empirical measure of the

The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying

The theory of graph limits is only understood to a somewhat satisfactory degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that one of the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with a stationary distribution). This motivates our

The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form \((a_n \alpha )_{n \ge 1}\) has been pioneered by Rudnick, Sarnak and Zaharescu. Here \(\alpha \) is a real parameter, and \((a_n)_{n \ge 1}\) is an integer sequence, often

We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard

Let \(\omega \) denote an area form on \(S^2\). Consider the closed symplectic 4-manifold \(M=(S^2\times S^2, A\omega \oplus a \omega )\) with \(0

We give upper bounds for \(L^p\) norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus \(g \rightarrow +\infty \), we show that random hyperbolic surfaces X with respect to the Weil-Petersson volume have with high probability at most one such loop of length

Let \(f_1,\dots ,f_k\in \mathbb {R}[X]\) be polynomials of degree at most d with \(f_1(0)=\dots =f_k(0)=0\). We show that there is an \(n

We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length \(H < x^{6/11 - \varepsilon }\) and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with \(q > x^{5/11 + \varepsilon }\). On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges

We prove an almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. The lower bound has the dimension dependency \(d^{-o_d(1)}\). When the dimension is large enough, our lower bound is tighter than the previous best bound which has the dimension dependency \(d^{-1/4}\). Improving the current best lower bound of the isoperimetric coefficient in the KLS conjecture has many

In 1998 Kleinbock conjectured that any set of weighted badly approximable \(d\times n\) real matrices is a winning subset in the sense of Schmidt’s game. In this paper we prove this conjecture in full for vectors in \({\mathbb {R}}^d\) in arbitrary dimensions by showing that the corresponding set of weighted badly approximable vectors is hyperplane absolute winning. The proof uses the Cantor potential

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue

We will study metric measure spaces \((X,\mathsf{d},{\mathfrak {m}})\) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds \(\mathsf{BE}_1(\kappa ,\infty )\) for which we prove the equivalence with sharp gradient estimates, the class of which will be preserved under time changes with arbitrary \(\psi \in \mathrm {Lip}_b(X)\)

The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous

Suppose that a binary operation \(\circ \) on a finite set X is injective in each variable separately and also associative. It is easy to prove that \((X,\circ )\) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples \((x,y,z)\in X^3\) satisfy the equation \(x\circ (y\circ z)=(x\circ y)\circ z\). Other results in additive combinatorics would

We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C\(^*\)-algebras of countable groups with (relative) property (T). We derive that the full C\(^*\)-algebras of the groups \(\mathbb {Z}^2\times \text {SL}_2({\mathbb {Z}})\) and \(\text {SL}_n({\mathbb {Z}})\), for \(n\ge 3\), do not have the local lifting property (LLP)

We establish a first general partial regularity theorem for area minimizing currents \({\mathrm{mod}}(p)\), for every p, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current \({\mathrm{mod}}(p)\) cannot be larger than \(m-1\). Additionally, we show that, when p is odd, the interior singular set

Given an integer \(k \ge 5\), and a \(C^k\) Anosov flow \(\Phi \) on some compact connected 3-manifold preserving a smooth volume, we show that the measure of maximal entropy is the volume measure if and only if \(\Phi \) is \(C^{k-\varepsilon }\)-conjugate to an algebraic flow, for \(\varepsilon >0\) arbitrarily small. Moreover, in the case of dispersing billiards, we show that if the measure of maximal

We prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes

The arithmetic version of Anderson localization (AL), i.e., AL with explicit arithmetic description on both the localization frequency and the localization phase, was first given by Jitomirskaya (Ann Math 150:1159–1175, 1999) for the almost Mathieu operators (AMO). Later, the result was generalized by Bourgain and Jitomirskaya (Invent Math 148:453–463, 2002) to a class of one dimensional quasi-periodic

We develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a “long neck principle” for a compact Riemannian spin n-manifold with boundary X, stating that if \({{\,\mathrm{scal}\,}}(X)\ge n(n-1)\) and there is a nonzero degree map into the

We consider the Calabi–Yau metrics on \(\mathbf {C}^n\) constructed recently by Yang Li, Conlon–Rochon, and the author, that have tangent cone \(\mathbf {C}\times A_1\) at infinity for the \((n-1)\)-dimensional Stenzel cone \(A_1\). We show that up to scaling and isometry this Calabi–Yau metric on \(\mathbf {C}^n\) is unique. We also discuss possible generalizations to other manifolds and tangent cones

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is \(o(\log n)\). However, both results become useless

We use tools of additive combinatorics for the study of subvarieties defined by high rank families of polynomials in high dimensional \({\mathbb {F}}_q\)-vector spaces. In the first, analytic part of the paper we prove a number properties of high rank systems of polynomials. In the second, we use these properties to deduce results in Algebraic Geometry , such as an effective Stillman conjecture over

In this note, we show that sub-Riemannian manifolds can contain branching normal minimizing geodesics. This phenomenon occurs if and only if a normal geodesic has a discontinuity in its rank at a non-zero time, which in particular for a strictly normal geodesic means that it contains a non-trivial abnormal subsegment. The simplest example is obtained by gluing the three-dimensional Martinet flat structure

We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods have failed. We start by verifying the small cap decoupling for the parabola. Then we find sharp estimates for exponential sums with small frequency separation on the moment curve in \(\mathbb {R}^3\). This

Let \(Z=G/H\) be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of G on \(L^2(Z)\). It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of \(L^2(Z)\), have infinitesimal characters which are real and belong to a lattice. Moreover, let K be a maximal compact subgroup

For random d-regular graphs on N vertices with \(1 \ll d \ll N^{2/3}\), we develop a \(d^{-1/2}\) expansion of the local eigenvalue distribution about the Kesten–McKay law up to order \(d^{-3}\). This result is valid up to the edge of the spectrum. It implies that the eigenvalues of such random regular graphs are more rigid than those of Erdős–Rényi graphs of the same average degree. As a first application

We investigate conformal actions of cocompact lattices in higher-rank simple Lie groups on compact pseudo-Riemannian manifolds. Our main result gives a general bound on the real-rank of the lattice, which was already known for the action of the full Lie group by a result of Zimmer. When the real-rank is maximal, we prove that the manifold is conformally flat. This indicates that a global conclusion

Given \(n \ge 2\) and \(1

On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel–Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsion in cohomology of arithmetic groups.

The main proposition, Theorem 1.2, is the existence for excellent Deligne–Mumford champ of characteristic zero of a resolution functor independent of the resolution process itself. Received wisdom was that this was impossible, but the counterexamples overlooked the possibility of using weighted blow ups. The fundamental local calculations take place in complete local rings, and are elementary in nature

In this article we define new flows on the Hitchin components for \(\mathrm {PGL}(V)\). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when \(n=2\). We determine a global coordinate system

We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety X, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification \((M,{\mathbf {D}})\) of X. We exhibit a broad class of pairs \((M,{\mathbf {D}})\) (characterized by the absence of relative holomorphic spheres or vanishing of certain

Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a one-to-one correspondence between such manifold submetries and maximal Laplacian algebras, thus solving the Inverse Invariant Theory problem for this class of partitions

Singular vectors are those for which the quality of rational approximations provided by Dirichlet’s Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular vectors lying on self-similar fractals in \({\mathbb {R}}^d\) satisfying the open set condition. The bound is in terms of quantities which are closely tied to Frostman

We establish Anderson localization for general analytic k-frequency quasi-periodic operators on \({\mathbb {Z}}^d\) for arbitraryk, d.

We give a short proof of a theorem of Guth relating volume of balls and Uryson width. The same approach applies to Hausdorff content implying a recent result of Liokumovich–Lishak–Nabutovsky–Rotman. We show also that for any \(C>0\) there is a Riemannian metric g on a 3-sphere such that \({\hbox {vol}}(S^3,g)=1\) and for any map \(f:S^3\rightarrow {\mathbb {R}}^2\) there is some \(x\in {\mathbb {R}}^2\)

I show that quasidiagonality and AF embeddability are equivalent properties for traceless \(\mathrm C^*\)-algebras and are characterised in terms of the primitive ideal space. For nuclear \(\mathrm C^*\)-algebras the same characterisation determines when Connes and Higson’s E-theory can be unsuspended.

We prove two results on arithmetic quantum chaos for dihedral Maaß forms, both of which are manifestations of Berry’s random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level 1 forms, these results were previously known for Eisenstein series and conditionally on the generalised Lindelöf hypothesis for Hecke–Maaß eigenforms. A key aspect of

We prove that all flexible Weinstein fillings of a given contact manifold with vanishing first Chern class have isomorphic integral cohomology. As an application, we show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. We also construct the first known infinite family of almost symplectomorphic Weinstein domains

The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil–Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit

Let \((M^n,g)\) be simply connected, complete, with non-positive sectional curvatures, and \(\Sigma \) a 2-dimensional closed integral current (or flat chain mod 2) with compact support in M. Let S be an area minimising integral 3-current (resp. flat chain mod 2) such that \(\partial S = \Sigma \). We use a weak mean curvature flow, obtained via elliptic regularisation, starting from \(\Sigma \), to

We study a class of discrete-time random dynamical systems with compact phase space. Assuming that the deterministic counterpart of the system in question possesses a dissipation property, its linearisation is approximately controllable, and the driving noise is bounded and has a decomposable structure, we prove that the corresponding family of Markov processes has a unique stationary measure, which

In this article we show that in any dimension there exist infinitely many pairs of formally contact isotopic isocontact embeddings into the standard contact sphere which are not contact isotopic. This is the first example of rigidity for contact submanifolds in higher dimensions. The contact embeddings are constructed via contact push-offs of higher-dimensional Legendrian submanifolds, a construction

In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for \(C^0\) metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from \(C^0\) initial data which is smooth for positive times, and that the weak lower scalar

This paper accomplishes two things. First, we construct a geometric analog of the rational Tits building for general noncompact, complete, finite volume n-manifolds M of bounded nonpositive curvature. Second, we prove that this analog has dimension less than \(\lfloor n/2\rfloor \).