Geometric and Functional Analysis最新文献

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.

We obtain a homological characterisation of virtually free-by-cyclic groups among groups that are hyperbolic and virtually compact special. As a consequence, we show that many groups known to be coherent actually possess the stronger property of being virtually free-by-cyclic. In particular, we show that all one-relator groups with torsion are virtually free-by-cyclic, solving a conjecture of Baumslag.

We prove that the cylindrical capacity of a dynamically convex domain in ({mathbb{R}}^{4}) agrees with the least symplectic area of a disk-like global surface of section of the Reeb flow on the boundary of the domain. Moreover, we prove the strong Viterbo conjecture for all convex domains in ({mathbb{R}}^{4}) which are sufficiently C³ close to the round ball. This generalizes a result of Abbondandolo-Bramham-Hryniewicz-Salomão establishing a systolic inequality for such domains.

We prove that the singular support of an element in the derived category of sheaves is γ-coisotropic, a notion defined in [Vit22]. We prove that this implies that it is involutive in the sense of Kashiwara-Schapira, but being γ-coisotropic has the advantage to be invariant by symplectic homeomorphisms (while involutivity is only invariant by C¹ diffeomorphisms) and we give an example of an involutive set that is not γ-coisotropic. Along the way we prove a number of results relating the singular support and the spectral norm γ and raise a number of new questions.

In this article we show that the conformal nets corresponding to WZW models are rational, resolving a long-standing open problem. Specifically, we show that the Jones-Wassermann subfactors associated with these models have finite index. This result was first conjectured in the early 90s but had previously only been proven in special cases, beginning with Wassermann’s landmark results in type A. The proof relies on a new framework for the systematic comparison of tensor products (a.k.a. ‘fusion’) of conformal net representations with the corresponding tensor product of vertex operator algebra modules. This framework is based on the geometric technique of ‘bounded localized vertex operators,’ which realizes algebras of observables via insertion operators localized in partially thin Riemann surfaces. We obtain a general method for showing that Jones-Wassermann subfactors have finite index, and apply it to additional families of important examples beyond WZW models. We also consider applications to a class of positivity phenomena for VOAs, and use this to outline a program for identifying unitary tensor product theories of VOAs and conformal nets even for badly-behaved models.

We study growth of absolute and homological k-dimensional systoles of arithmetic n-manifolds along congruence coverings. Our main interest is in the growth of systoles of manifolds whose real rank r≥2. We observe, in particular, that in some cases for k=r the growth function tends to oscillate between a power of a logarithm and a power function of the degree of the covering. This is a new phenomenon. We also prove the expected polylogarithmic and constant power bounds for small and large k, respectively.

Let Γ be a weakly irreducible lattice in a higher rank semisimple algebraic Lie group G without property (T) such as (mathrm {SL}_{2}({mathbb{Z}}[sqrt{2}])) in (mathrm {SL}_{2}({mathbb{R}})times mathrm {SL}_{2}({mathbb{R}})).

In this paper, for such Γ, we prove a cocycle superrigidity, Dynamical cocycle superrigidity, for Γ-actions under L² integrability and irreducibility assumptions. It gives a similar result to Zimmer’s cocycle superrigidity, so we can apply it instead of Zimmer’s cocycle superrigidity in many situations. For instance, in this paper, we obtain global rigidity of Anosov Γ-actions on nilmanifolds under the irreducibility assumption on a fully supported invariant measure.

We develop the asymptotic analysis as ε→0 for the natural gradient flow of the self-dual U(1)-Higgs energies

on Hermitian line bundles over closed manifolds (Mⁿ,g) of dimension n≥3, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows—i.e., integral (n−2)-Brakke flows—generalizing results of (Pigati and Stern in Invent. Math. 223:1027–1095, 2021) from the stationary case. Given any integral (n−2)-cycle Γ₀ in M, these results can be used together with the convergence theory developed in (Parise et al. in Convergence of the self-dual U(1)-Yang–Mills–Higgs energies to the (n−2)-area functional, 2021, arXiv:2103.14615) to produce nontrivial integral Brakke flows starting at Γ₀ with additional structure, similar to those produced via Ilmanen’s elliptic regularization.

We establish the existence of a spectral gap for the transfer operator induced on (mathbb{P}^{k} = mathbb{P}^{k} (mathbb{C})) by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: CLT, Berry-Esseen Theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).

The mixed equation, defined as a combination of the anti-self-duality equation in gauge theory and Cauchy–Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah–Floer conjecture.