Geometric and Functional Analysis最新文献

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.

Given a closed symplectic manifold X, we construct Gromov-Witten-type invariants valued both in (complex) K-theory and in any complex-oriented cohomology theory (mathbb{K}) which is K_p(n)-local for some Morava K-theory K_p(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantum K-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantum K-theory and quantum (mathbb{K})-theory as commutative deformations of the corresponding (generalised) cohomology rings of X; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus to X. On the algebraic side, in order to establish a common framework covering both ordinary K-theory and K_p(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts.

We give a complete description of the embeddings of direct products of nonabelian free groups into Aut(F_N) and Out(F_N) when the number of direct factors is maximal. To achieve this, we prove that the image of each such embedding has a canonical fixed point of a particular type in the boundary of Outer space.

In this work, we obtain the first upper bound on the multiplicity of Laplacian eigenvalues for negatively curved surfaces which is sublinear in the genus g. Our proof relies on a trace argument for the heat kernel, and on the idea of leveraging an r-net in the surface to control this trace. This last idea was introduced in 2021 for similar spectral purposes in the context of graphs of bounded degree. Our method is robust enough to also yield an upper bound on the “approximate multiplicity” of eigenvalues, i.e., the number of eigenvalues in windows of size 1/log^β(g), β>0. This work provides new insights on a conjecture by Colin de Verdière and new ways to transfer spectral results from graphs to surfaces.

Let F be a holomorphic cuspidal Hecke eigenform for (mathrm{Sp}_{4}({mathbb{Z}})) of weight k that is a Saito–Kurokawa lift. Assuming the Generalized Riemann Hypothesis (GRH), we prove that the mass of F equidistributes on the Siegel modular variety as k⟶∞. As a corollary, we show under GRH that the zero divisors of Saito–Kurokawa lifts equidistribute as their weights tend to infinity.