Geometric and Functional Analysis最新文献

A rational lemniscate is a level set of |r| where (r: widehat {mathbb{C}}rightarrow widehat {mathbb{C}}) is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This generalizes Hilbert’s lemniscate theorem; he proved that any Jordan curve can be approximated (in the same strong sense) by a polynomial lemniscate that is also a Jordan curve. As consequences, we obtain a sharp quantitative version of the classical Runge’s theorem on rational approximation, and we give a new result on the approximation of planar continua by Julia sets of rational maps.

Chemotactic singularity formation in the context of the Patlak-Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of fluid advection can arrest the singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong - this effect is conjectured to hold for more general classes of nonlinear PDEs. In this paper, we consider the Patlak-Keller-Segel equation coupled with a fluid flow that obeys Darcy’s law for incompressible porous media via buoyancy force. We prove that in contrast with passive advection, this active fluid coupling is capable of suppressing singularity formation at arbitrary small coupling strength: namely, the system always has globally regular solutions.

In this paper, we prove that an ancient smooth curve-shortening flow with finite entropy embedded in (mathbb{R}^{2}) has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplicity m≥3 exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.

Let M be an n×n matrix with iid subgaussian entries with mean 0 and variance 1 and let σ_n(M) denote the least singular value of M. We prove that

for all 0⩽ε≪1. This resolves, up to a 1+o(1) factor, a seminal conjecture of Spielman and Teng.

We study conjectures of Ben-Zvi–Sakellaridis–Venkatesh that categorify the relationship between automorphic periods and L-functions in the context of the Geometric Langlands equivalence. We provide evidence for these conjectures in some low-rank examples, by using derived Fourier analysis and the theory of chiral algebras to categorify the Rankin-Selberg unfolding method.

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.