We prove that for generic parameters, the quantum radial parts map of Varagnolo and Vasserot gives an isomorphism between the spherical double affine Hecke algebra of GL n and a quantized multiplicative quiver variety, as defined by Jordan.
We continue our study of fixed loci of antisymplectic involutions on projective hyper-K & auml;hler manifolds of K3[n]-type induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice. We prove that if the divisibility of the ample class is 2, then one connected component of the fixed locus is a Fano manifold of index 3, thus generalizing to higher dimensions the case of the LLSvS 8-fold associated to a cubic fourfold. We also show that, in the case of the LLSvS 8-fold associated to a cubic fourfold, the second component of the fixed locus is of general type, thus answering a question by Manfred Lehn. R & Eacute;SUM & Eacute; (La g & eacute;om & eacute;trie des involutions anti-symplectiques, II). Nous poursuivons notre & eacute;tude des lieux fixes des involutions anti-symplectiques sur les vari & eacute;t & eacute;s hyper-k & auml;hl & eacute;riennes projectives de type K3[n] induites par une classe ample de carr & eacute; 2 dans le r & eacute;seau de Beauville-Bogomolov-Fujiki. Nous prouvons que si la divisibilit & eacute; de la classe ample est 2, alors une composante connexe du lieu fixe est une vari & eacute;t & eacute; de Fano d'indice 3, g & eacute;n & eacute;ralisant ainsi aux dimensions sup & eacute;rieures le cas de la vari & eacute;t & eacute; de LLSvS de dimension 8 associ & eacute;e & agrave; une vari & eacute;t & eacute; cubique de dimension 4. Nous montrons & eacute;galement que, dans le cas de la vari & eacute;t & eacute; de dimension 8 de LLSvS associ & eacute;e & agrave; une vari & eacute;t & eacute; cubique de dimension 4, la deuxi & egrave;me composante du lieu fixe est de type g & eacute;n & eacute;ral, r & eacute;pondant ainsi & agrave; une question pos & eacute;e par Manfred Lehn.
We extend to higher dimensions the notion of Birkhoff attractor of a dissipative map. We prove that this notion coincides with the classical Birkhoff attractor. We prove that for the dissipative system associated to the discounted Hamilton-Jacobi equation, the graph of the solution is contained in the Birkhoff attractor. The appendix provides instructive counterexamples in the non-Tonelli case. We also study what happens when we perturb a Hamiltonian system to make it dissipative and let the perturbation go to zero. The paper contains two main results on gamma-supports and elements of the gamma-completion of the space of exact Lagrangians. Firstly the gamma-support of a Lagrangian in a cotangent bundle carries the cohomology of the base and secondly given an exact Lagrangian L, any Floer theoretic equivalent Lagrangian is the gamma-limit of Hamiltonian images of L.
We consider the initial value problem for the 2D nonlinear shallow water model in the presence of a fixed partially immersed solid body. In this problem, we have a contact line where the solid body, the water, and the air meet, and whose projection on the horizontal plane moves freely even if the solid body is fixed. This wave-structure interaction problem reduces to an initial boundary value problem for the nonlinear shallow water equations in an exterior domain with a free boundary, for which we prove a priori energy estimates locally in time for solutions at the quasilinear regularity threshold under irrotationality and subcriticality assumptions. We use the weak dissipativity of the system, second order Alinhac good unknowns associated with a regularizing diffeomorphism, and a new type of hidden boundary regularity.
Ennola duality relates the character table of the finite unitary group GU n ( 𝔽 q ) to that of GL n ( 𝔽 q ) where we replace q by - q (see [5] for the original observation and [21] for its proof). The aim of this paper is to investigate Ennola duality for the decomposition of tensor products of irreducible characters. It does not hold just by replacing q by - q . The main result of this paper is the construction of a family of two-variable polynomials 𝒯 μ ( u , q ) indexed by triples of partitions of n which interpolates between multiplicities in decompositions of tensor products of unipotent characters for GL n ( 𝔽 q ) and GU n ( 𝔽 q ) . We give a module theoretic interpretation of these polynomials and deduce that they have non-negative integer coefficients. We also deduce that the coefficient of the term of highest degree in u equals the corresponding Kronecker coefficient for the symmetric group and that the constant term in u give multiplicities in tensor products of generic irreducible characters of unipotent type (i.e., unipotent characters twisted by linear characters of GL 1 ( 𝔽 q ) ).
On a closed hyperbolic surface, we investigate semiclassical defect measures associated with the magnetic Laplacian in the presence of a constant magnetic field. Depending on the energy level where the eigenfunctions concentrate, three distinct dynamical regimes emerge. In the low-energy regime, we show that any invariant measure of the magnetic flow in phase space can be obtained as a semiclassical measure. At the critical energy level, we establish Quantum Unique Ergodicity, together with a quantitative rate of convergence of eigenfunctions to the Liouville measure. In the high-energy regime, we prove a Shnirelman-type result: a density-one subsequence of eigenfunctions becomes equidistributed with respect to the Liouville measure.
We introduce operations with p-adic integer coefficients, associated to idempotents in the quantum cohomology of a monotone symplectic manifold, and apply them to the structure of the quantum connection.
. - The airplane, the basilica and the Douady rabbit (and, more generally, rabbits with more than two ears) are well-known Julia sets of complex quadratic polynomials. In this paper we study the groups of all homeomorphisms of such fractals and of all automorphisms of their laminations. In particular, we identify them with some kaleidoscopic groups or universal groups and thus realize them as Polish permutation groups. From these identifications, we deduce algebraic, topological and geometric properties of these groups.
In this article, we make use of a weight function capturing the concentration phenomenon of unstable future-trapped causal geodesics. A projection V+, on the tangent space of the null-shell, of the associated symplectic gradient turns out to enjoy good commutation properties with the massless Vlasov operator. This implies that V+f remains bounded, for any smooth solution f to the massless Vlasov equation. By identifying a well-chosen modification of V+, we are able to construct a W1,1 x,p weighted norm for which any smooth solution to the massless Vlasov equation verifies an integrated local energy decay estimate without relative degeneration. Together with the rp-weighted energy method of Dafermos-Rodnianski, we establish time decay for the energy norm. This norm allows for the control of the energy-momentum tensor T[f] as well as all its first order derivatives. The method developed in this paper is in particular compatible with the approach of [Mav24, DHRT22] used to study quasi-linear wave equations on black hole spacetimes.
Given a partition of 2, the stratum (1) parametrizes meromorphic differen-tial one-forms on the Riemann sphere CP & sup1; with zeros and p poles of orders prescribed by p. The isoresidual fibration is defined by assigning to cach differential in (4) its configuration of residues at the poles. In the case of differentials with n = 2 zeros, generic isoresidual fibers are complex curves endowed with a canonical translation structure, which we describe exten-sively in this paper. Quantitative characteristics of the translation structure on isoresidual fiber curves provide rich discrete invariants for these fibers. We determine the Euler characteristic of generic isoresidual fiber curves from intersection-theoretic computations, we describe a wall and chamber structure for the Euler characteristic of generic isoresidual fiber curves in terms of the partition a, and we classify the connected components of generic inoresidual fibers for strata in genus zero with an arbitrary number of zeros.
Let F be a discrete and torsion-free subgroup of PU(n, 1), the group of biholomorphisms of the unit ball in C , denoted by IHInC. We show that if F is Abelian, then IHI C/F is a Stein manifold. If the critical exponent S(F) of F is less than 2, a conjecture of Dey and Kapovich predicts that the quotient IHI C/F is Stein. We confirm this conjecture in the case where F is parabolic or geometrically finite. We also study the case of quotients with S(F) = 2 that contain compact complex curves and confirm another conjecture of Dey and Kapovich. We finally show that IHI C/F is Stein when F is a parabolic or geometrically finite group preserving a totally real and totally geodesic submanifold of IHInC, without any hypothesis on the critical exponent.
We introduce and study the class of primitive Enriques varieties, whose smooth members are Enriques manifolds. We provide several examples and we demonstrate that this class is preserved under the operations of the Minimal Model Program (MMP). In particular, given an Enriques manifold Y and an effective R-divisor BY on Y such that the pair (Y, BY ) is of (Y, BY ), where Y ' is a Q-factorial primitive Enriques variety with canonical singularities. Finally, we investigate the asymptotic theory of Enriques manifolds.
We investigate the equivalence of Sobolev inequalities and the conjunction of Gaussian upper heat kernel bounds and volume doubling on large scales on graphs. For the normalizing measure, we obtain the equivalence up to constants. If arbitrary measures are considered, we incorporate a new local regularity condition. Furthermore, new correction functions for the Gaussian, doubling, and Sobolev dimension are introduced. For the Gaussian and doubling, the variable correction functions always tend to one at infinity. Moreover, the variable Sobolev dimension can be related to the doubling dimension and the vertex degree growth.
We construct a four-variable p-adic L-function for cuspidal Hida families c CSp(4) x GL(2) and prove a complete interpolation formula for it. The Archimedean zel integrals are computed by using a partial interpolation formula for the four-variable pad L-Function, combined with Yoshida lifts and some previously constructed p-adic L-functions specifically Kubota Leopoldt p-adic L-functions, Rankin-Selberg p-adic I-functions, an p-adic (standard) L-functions for Sp(4).
We show that the mapping class group of a handlebody is a virtual duality group, in the sense of Bieri and Eckmann. In positive genus we give a description of the dualising module of any torsion-free, finite-index subgroup of the handlebody mapping class group as the homology of the complex of non-simple disc systems.
We relate analytically defined deformations of modular curves and modular forms from the literature to motivic periods via cohomological descriptions of deformation theory. Leveraging cohomological vanishing results, we prove the existence and essential uniqueness of deformations, which we make constructive via established Lie algebraic arguments and a notion of formal Lie deformations. Further, we construct a canonical and a totally holomorphic universal family of deformations of modular forms of all weights, which we obtain from the canonical co cycle associated with periods on the moduli space M1,1. Our uniqueness statement shows that non-critical multiple L-values, which appear in our deformations but are a priori non-geometric, are genuinely linked to deformations. Our work thus suggests a new geometric perspective on them.
We study a model of random binary trees grown "by the leaves" in the style of Luczak and Winkler. If τ_n is a uniform plane binary tree of size n, Luczak and Winkler, and later explicitly Caraceni and Stauffer, constructed a measure ν_τ_n such that the tree obtained by adding a cherry on a leaf sampled according to ν_τ_n is still uniformly distributed on the set of all plane binary trees with size n+1. It turns out that the measure ν_τ_n, which we call the leaf-growth measure, is noticeably different from the uniform measure on the leaves of the tree τ_n. In fact, we prove that, as n →∞, with high probability it is almost entirely supported by a subset of only n^3 ( 2 - √(3))+o(1)≈ n^0.8038... leaves. In the continuous setting, we construct the scaling limit of this measure, which is a probability measure on the Brownian Continuum Random Tree supported by a fractal set of dimension 6 (2 - √(3)). We also compute the full (discrete) multifractal spectrum. This work is a first step towards understanding the diffusion limit of the discrete leaf-growth procedure.
We introduce a natural geometric framework for the study of logarithmically divergent integrals on manifolds with corners and algebraic varieties, using the techniques of logarithmic geometry. Key to the construction is a new notion of morphism in logarithmic geometry itself, introduced by Howell, which allows us to interpret the ubiquitous rule of thumb "lim epsilon -> 0 log e := 0" as the restriction to a submanifold. Via a version of de Rham's theorem with logarithmic divergences, we obtain a functorial characterization of the classical theory of "regularized integration": it is the unique way to extend the ordinary integral to the logarithmically divergent context while respecting the basic laws of calculus (change of variables, Fubini's theorem, and Stokes' formula.)
Inspired by [Pan22], we give a new proof that for an overconvergent modular eigenform $f$ of weight $1+k$ with $k\in\mathbb{Z}_{\ge1}$, assuming that its associated global Galois representation $\rho_{f}$ is irreducible, then $f$ is classical if and only if $\rho_{f}$ is de Rham at $p$. For the proof, we prove that theta operator $\theta^{k}$ coincides with Fontaine operator in a suitable sense.
We prove that for generic parameters, the quantum radial parts map of Varagnolo and Vasserot gives an isomorphism between the spherical double affine Hecke algebra of GLn and a quantized multiplicative quiver variety, as defined by Jordan. R & Eacute;SUM & Eacute; (Isomorphisme de Harish-Chandra quantique pour l'alg & egrave;bre de Hecke affine double de GLn) Nous prouvons que pour des param & egrave;tres g & eacute;n & eacute;riques, l'application des parties radiales quantiques de Varagnolo et Vasserot donne un isomorphisme entre l'alg & egrave;bre de Hecke affine double sph & eacute;rique de GLn et une vari & eacute;t & eacute; carquois multiplicative quantifi & eacute;e, telle que d & eacute;finie par Jordan.
