We study intrinsic diophantine approximation on Grassmann varieties. Using a new correspondence between the diophantine properties of a linear subspace x in Rd and certain diagonal orbits in the space of lattices, we are able to solve some problems suggested by Schmidt in 1967. In particular, we obtain a version of Dirichlet's principle in this setting with an optimal exponent.
Let E be a CM elliptic curve defined over Q and p a prime. We show that corankZp Selp infinity (E/Q) = 0 =double right arrow ords=1L(s, E/Q) = 0 for the p infinity- Selmer group Selp infinity(E/Q) and the complex L-function L(s, E/Q). Along with Smith's work on the distribution of 2 infinity-Selmer groups, this leads to the first instance of the even parity Goldfeld conjecture: For 50% of the positive square-free integers n, we have ords=1L(s, E/Q(n)) = 0, where E(n) : ny2 = x3-x is a quadratic twist of the congruent number elliptic curve E:y2=x3-x.
McKay's conjecture (1971) on character degrees was reduced by Isaacs-Malle-Navarro (2007) to a so-called inductive condition on characters of finite quasisimple groups [IMN07], thus opening the way to a proof of McKay's conjecture using the classification of finite simple groups. After [Ma07], [Ma08], [S12], [CS13], [KS16], [MS16], [CS17a], [CS17b], [CS19], [S23a], [S23b], we complete here the last step of a proof with an analysis of the representations of certain normalizers ${\mathrm N}_G({\mathbf S})$ in $G={\mathbf G}^F$ of maximal $d$-tori ${\mathbf S}$ ($d\geq 3$) of the ambient simple simply-connected algebraic group ${\mathbf G}$ of type ${\mathrm D}_l$ ($l\geq 4$) for which $F$ is a Frobenius endomorphism. To establish the so-called local conditions A$(d)$ and B$(d)$, we introduce a certain class of $F$-stable reductive subgroups ${\mathbf M}\leq {\mathbf G}$ of maximal rank where ${\mathbf M}^\circ$ is of type ${\mathrm D}_{l_1}\times {\mathrm D}_{l-l_1}$ with ${\mathbf M}/{\mathbf M}^\circ$ of order 2. They are an efficient substitute for ${\mathrm N}_G({\mathbf S})$ or the local subgroups in non-defining characteristic relevant to McKay's abstract statement. For a general class of those subgroups ${\mathbf M}^F$ we describe their characters and the action of $\operatorname {Aut}({\mathbf G}^F)_{{\mathbf M}^F}$ on them, showing in particular that $\mathrm{Irr}({\mathbf M}^F)$ and $\mathrm {Irr}({\mathbf G}^F)$ share some key features in that regard. With this established, McKay's conjecture is now a theorem stating $\textit{McKay's equality}$: For any prime $\ell$, any finite group has as many irreducible complex characters of degree prime to $\ell$ as the normalizers of its Sylow $\ell$-subgroups.
In this paper we show that iterating Nash blowups or normalized Nash blowups does not resolve the singularities of algebraic varieties of dimension four or higher over an algebraically closed field of arbitrary characteristic.
We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length N taking values of size close to N3/4, which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate N(sigma, T) <= T30(1-sigma )/13+o(1) and asymptotics for primes in short intervals of length x17/30+o(1).
We present a counterexample to Viterbo's volume-capacity conjecture. This implies, in particular, that in contrast with a well-known conjecture, symplectic capacities do not coincide on the class of convex domains in the classical phase space.
A family of random matrices XN = (X1N , ... , XdN ) is said to converge strongly to a family of bounded operators x = (x1, ... , xd) when parallel to P(XN,XN & lowast;)parallel to ->parallel to P(x,x & lowast;)parallel to for every noncommutative polynomial P. This phenomenon plays a key role in several recent breakthroughs on random graphs, geometry, and operator algebras. However, proofs of strong convergence are notoriously delicate and have relied largely on problem-specific methods. In this paper, we develop a new approach to strong convergence that uses only soft arguments. Our method exploits the fact that for many natural models, the expected trace of P(XN, XN & lowast;) is a rational function of N 1 whose lowest order asymptotics are easily understood. We develop a general technique to deduce strong convergence directly from these inputs using the inequality of A. and V. Markov for univariate polynomials and elementary Fourier analysis. To illustrate the method, we develop the following applications: 1. We give a short proof of the result of Friedman that random regular graphs have a near-optimal spectral gap and obtain a sharp understanding of the large deviations probabilities of the second eigenvalue. 2. We prove a strong quantitative form of the strong convergence property of random permutation matrices due to Bordenave and Collins. 3. We extend the above to any stable representation of the symmetric group, providing many new examples of the strong convergence phenomenon.
The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 - \varepsilon)^k \] for some constant $\varepsilon > 0$. This is the first exponential improvement over the upper bound of Erd\H{o}s and Szekeres, proved in 1935.
An explicit construction of locally testable codes of constant rate, constant distance and constant number of queries is given. Hence answering affirmatively the c^3-problem.
In this paper, we prove that the admissible canonical bundle of the universal family of curves is a big adelic line bundle, and apply it to prove a uniform Bogomolov-type theorem for curves over global fields of all characteristics. This gives a different approach to the uniform Mordell-Lang type of result of Dimitrov-Gao-Habegger and Kuhne. The treatment is based on the recent theory of adelic line bundles of Yuan-Zhang.
We describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel–Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel–Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at the same point. Our approach automatically applies to other self-adjoint systems with 2 × 2 transfer matrices such as continuum Schrödinger and Dirac operators.
We consider Schro & uml;dinger operators H = -triangle + V (x) in Rd, d >= 2, with quasi-periodic potentials V (x). We prove that the absolutely continuous spectrum of a generic H contains a semi-axis [lambda & lowast;, +infinity). We also construct a family of eigenfunctions of the absolutely continuous spectrum; these eigenfunctions are small perturbations of the exponentials. The proof is based on a version of the multi-scale analysis in the momentum space with several new ideas introduced along the way.
AbstractWe study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of$\mathfrak {gl}_2(\mathbb {Q}_p)$. As applications, we prove a classicality result for overconvergent eigenforms of weight 1 and give a new proof of the Fontaine–Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weightk, we show its corresponding Galois representation has Hodge–Tate–Sen weights$0,k-1$and prove a converse result.
We prove that there exists a residual set of (non-rational) polygons such the billiard flow is weakly mixing with respect to the Liouville measure (on the unit tangent bundle to the billiard). This follows, via a Baire category argument, from showing that for any translation surface the product of the flows in almost every pair of directions is ergodic with respect to Lebesgue measure. This in turn is proven by showing that for every translation surface the flows in almost every pair of directions do not share non-trivial common eigenvalues.
A method for determining quantum variance asymptotics on compact quotients attached to non-split quaternion algebras is developed in general and applied to "microlocal lifts" in the non-archimedean setting. The results obtained are in the spirit of recent work of Sarnak–Zhao. The arguments involve a careful analytic study of the theta correspondence, the interplay between additive and multiplicative harmonic analysis on quaternion algebras, the equidistribution of translates of elementary theta functions, and the Rallis inner product formula.
The characterization of global solutions to the obstacle problems in $\mathbb{R}^N$, or equivalently of null quadrature domains, has been studied over more than 90 years. In this paper we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.
This paper gives geometric characterizations of the Weil-Petersson class of rectifiable quasicircles, i.e., the closure of the smooth planar curves in the Weil-Petersson metric on universal Teichmuller space defined by Takhtajan and Teo. Although motivated by the planar case, many of our characterizations make sense for curves in Rn and remain equivalent in all dimensions. We prove that Gamma is Weil-Petersson if and only if it is well approximated by polygons in a precise sense, has finite Mobius energy or has arclength parametrization in H3/2(T). Other results say that a curve is Weil-Petersson if and only if local curvature is square integrable over all locations and scales, where local curvature is measured using various quantities such as Jones's beta-numbers, nonlinearity of conformal weldings, Menger curvature, the "thickness" of the hyperbolic convex hull of Gamma, and the total curvature of minimal surfaces in hyperbolic space. Finally, we prove that planar Weil-Petersson curves are exactly the asymptotic boundaries of minimal surfaces in H3 with finite renormalized area.
In this paper we consider the inductive Alperin-McKay condition for quasi-isolated 2-blocks of exceptional groups of Lie type. Thereby, we complete the proof of the Alperin-McKay conjecture and Brauer's height zero conjecture for the prime 2.
We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsvath-Szabo mixed invariants of closed four-manifolds, in terms of grid diagrams.
