Geometric and Functional Analysis最新文献

We determine to leading order the maximum of the characteristic polynomial for Wigner matrices and β-ensembles. In the special case of Gaussian-divisible Wigner matrices, our method provides universality of the maximum up to tightness. These are the first universal results on the Fyodorov–Hiary–Keating conjectures for these models, and in particular answer the question of optimal rigidity for the spectrum of Wigner matrices.

Our proofs combine dynamical techniques for universality of eigenvalue statistics with ideas surrounding the maxima of log-correlated fields and Gaussian multiplicative chaos.

We classify the (mathrm{GL}(2,mathbb{R}))-invariant subvarieties (mathcal{M}) in strata of Abelian differentials for which any two (mathcal{M})-parallel cylinders have homologous core curves. As a corollary we show that outside of an explicit list of exceptions, if (mathcal{M}) is a (mathrm{GL}(2,mathbb{R}))-invariant subvariety, then the Kontsevich-Zorich cocycle has nonzero Lyapunov exponents in the symplectic orthogonal of the projection of the tangent bundle of (mathcal{M}) to absolute cohomology.

Erdős’ unit distance problem and Erdős’ distinct distances problem are among the most classical and well-known open problems in discrete mathematics. They ask for the maximum number of unit distances, or the minimum number of distinct distances, respectively, determined by n points in the Euclidean plane. The question of what happens in these problems if one considers normed spaces other than the Euclidean plane has been raised in the 1980s by Ulam and Erdős and attracted a lot of attention over the years. We give an essentially tight answer to both questions for almost all norms on (mathbb{R}^{d}), in a certain Baire categoric sense.

For the unit distance problem we prove that for almost all norms ∥.∥ on (mathbb{R}^{d}), any set of n points defines at most (frac{1}{2} d cdot n log _{2} n) unit distances according to ∥.∥. We also show that this is essentially tight, by proving that for every norm ∥.∥ on (mathbb{R}^{d}), for any large n, we can find n points defining at least (frac{1}{2}(d-1-o(1))cdot n log _{2} n) unit distances according to ∥.∥.

For the distinct distances problem, we prove that for almost all norms ∥.∥ on (mathbb{R}^{d}) any set of n points defines at least (1−o(1))n distinct distances according to ∥.∥. This is clearly tight up to the o(1) term.

We also answer the famous Hadwiger–Nelson problem for almost all norms on (mathbb{R}^{2}), showing that their unit distance graph has chromatic number 4.

Our results settle, in a strong and somewhat surprising form, problems and conjectures of Brass, Matoušek, Brass–Moser–Pach, Chilakamarri, and Robertson. The proofs combine combinatorial and geometric ideas with tools from Linear Algebra, Topology and Algebraic Geometry.

Given a hyperspherical G-variety 𝒳 we consider the zero moment level Λ_𝒳⊂𝒳 of the action of a Borel subgroup B⊂G. We conjecture that Λ_𝒳 is Lagrangian. For the dual G^∨-variety 𝒳^∨, we conjecture that that there is a bijection between the sets of irreducible components (operatorname {Irr}Lambda _{{mathscr{X}}}) and (operatorname {Irr}Lambda _{{mathscr{X}}^{vee }}). We check this conjecture for all the hyperspherical equivariant slices, and for all the basic classical Lie superalgebras.

What is the optimal way to deform a projective hypersurface into another one? In this paper we will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic projective hypersurfaces.

First, a natural topological embedding of the space of hypersurfaces of a given degree into the space of measures on the projective space is constructed. Then, the optimal transport problem between hypersurfaces is defined through a constrained dynamical formulation, minimizing the energy of absolutely continuous curves which lie on the image of this embedding. In this way an inner Wasserstein distance on the projective space of homogeneous polynomials is introduced. This distance is finer than the Fubini–Study one.

The innner Wasserstein distance is complete and geodesic: geodesics corresponds to optimal deformations of one algebraic hypersurface into another one. Outside the discriminant this distance is induced by a smooth Riemannian metric, which is the real part of an explicit Hermitian structure. Moreover, this Hermitian structure is Kähler and the corresponding metric is of Weil–Petersson type.

To prove these results we develop new techniques, which combine complex and symplectic geometry with optimal transport, and which we expect to be relevant on their own.

We discuss applications on the regularity of the zeroes of a family of multivariate polynomials and on the condition number of polynomial systems solving.

We establish the dimension version of Falconer’s distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In dimensions d=2 or 3, we obtain the first explicit improvements over the classical 1/2 bound for the dimensions of distance sets of general Borel sets of dimension d/2. For example, we show that the set of distances spanned by a planar Borel set of Hausdorff dimension 1 has Hausdorff dimension at least ((sqrt{5}-1)/2approx 0.618). In higher dimensions we obtain explicit estimates for the lower Minkowski dimension of the distance sets of sets of dimension d/2. These results rely on new estimates for the dimensions of radial projections that may have independent interest.

A complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on ({mathbb{R}}^{n}) is established. The valuations obtained are functional versions of the classical intrinsic volumes. For their definition, singular Hessian valuations are introduced.

We establish geometric regularity for Type I blow-up limits of the Kähler-Ricci flow based at any sequence of Ricci vertices. As a consequence, the limiting flow is continuous in time in both Gromov-Hausdorff and Gromov-W₁ distances. In particular, the singular sets of each time slice and its tangent cones are closed and of codimension no less than 4.

We show that, under mild assumptions, the spectrum of a sum of independent random matrices is close to that of the Gaussian random matrix whose entries have the same mean and covariance. This nonasymptotic universality principle yields sharp matrix concentration inequalities for general sums of independent random matrices when combined with the Gaussian theory of Bandeira, Boedihardjo, and Van Handel. A key feature of the resulting theory is that it is applicable to a broad class of random matrix models that may have highly nonhomogeneous and dependent entries, which can be far outside the mean-field situation considered in classical random matrix theory. We illustrate the theory in applications to random graphs, matrix concentration inequalities for smallest singular values, sample covariance matrices, strong asymptotic freeness, and phase transitions in spiked models.

In this paper we prove a perturbative version of a remarkable Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) result. They prove non perturbative Birkhoff conjecture for centrally-symmetric convex domains, namely, a centrally-symmetric convex domain with integrable billiard is ellipse. We combine techniques from Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) with a local result by Kaloshin–Sorrentino (Ann. Math. 188(1):315–380, 2018) and show that a domain close enough to a centrally symmetric one with integrable billiard is ellipse. To combine these results we derive a slight extension of Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) by proving that a notion of rational integrability is equivalent to the C⁰-integrability condition used in their paper.