In this paper, we prove a version of Conn’s linearization theorem for the Lie algebra s l 2 ( C ) ≃ s o ( 3 , 1 ) \mathfrak {sl}_2(\mathbb {C})\simeq \mathfrak {so}(3,1) . Namely, we show that any Poisson structure whose linear approximation at a zero is isomorphic to the Poisson structure associated to s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) is linearizable. In the first part, we calculate the Poisson cohomology associated to s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) , and we construct bounded homotopy operators for the Poisson complex of multivector fields that are flat at the origin. In the second part, we obtain the linearization result, which works for a more general class of Lie algebras. For the proof, we develop a Nash-Moser method for functions that are flat at a point.
In this paper, we prove a version of Conn's linearization theorem for the Lie algebra sl2(C) c so(3, 1). Namely, we show that any Poisson structure whose linear approximation at a zero is isomorphic to the Poisson structure associated to sl2(C) is linearizable. In the first part, we calculate the Poisson cohomology associated to sl2(C), and we construct bounded homotopy operators for the Poisson complex of multivector fields that are flat at the origin. In the second part, we obtain the linearization result, which works for a more general class of Lie algebras. For the proof, we develop a Nash-Moser method for functions that are flat at a point.
The integral model of a GU(n-1, 1) Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet L-functions. We also determine the arithmetic volumes of Kudla-Rapoport divisors and relate them to coefficients of Eisenstein series.
This is the first of two papers on the global topology of the space Sub(G) of all closed subgroups of G = PSL2(R), equipped with the Chabauty topology. In this paper, we study the spaces of lattices and elementary subgroups of G, and prove a continuity result for conformal grafting of (possibly infinite type) vectored orbifolds that will be useful in both papers. More specifically, we first identify the homotopy type of the space of elementary subgroups of G, following Baik-Clavier. Then for a fixed finite type hyperbolizable 2-orbifold S, we show that the space SubS(G) of all lattices Gamma < G with Gamma\H-2 congruent to S is a fiber orbibundle over the moduli space M(S). We describe the closure SubS(G) in Sub(G) and show that partial derivative SubS(G) has a neighborhood deformation retract within SubS(G). When S is not one of finitely many low complexity orbifolds, we show that SubS(G) is simply connected. In the simplest exceptional case, when S is a sphere with three total cusps and cone points, we show that SubS(G) is a (usually nontrivial) lens space. Finally, we show that when (X-i, v(i)) -> (X-infinity,v(infinity)) is a (possibly infinite type) smoothly converging sequence of vectored hyperbolic 2-orbifolds, and we graft in Euclidean annuli along suitable collections of simple closed curves in the Xi, then after uniformization, the resulting vectored hyperbolic 2-orbifolds converge smoothly to the expected limit. As part of the proof, we give a new lower bound on the hyperbolic distance between points in a grafted orbifold in terms of their original distance.
We give closed formulas for the abelian Galois cohomology groups H1ab(F, G) and H2ab(F, G) of a connected reductive group G over a global field F in terms of the algebraic fundamental group pi 1(G) introduced earlier by one of us (M.B.). We further give closed formulas for the effects of restriction, corestriction, and localization, in terms of these formulas. Building on this, we give formulas, suitable for computer computations, for the first nonabelian Galois cohomology set H1(F, G) of G and for the second Galois cohomology group H2(F, T) of an F-torus T. As a preparation for the derivation of our formulas, we review the interpretation of Tate cohomology of a finite group in terms of the stable derived category of Z[F]- modules due to Buchweitz, and relate it to the explicit definition via cochains due to Kottwitz-Shelstad. We use this to construct the Tate-Nakayama isomorphisms for bounded complexes of tori over local and global fields, whose specialization to complexes of length 2 is then applied to obtain the desired formulas.
Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a "great unification" of Riemannian and sub-Riemannian geometries in a comprehensive framework of synthetic Ricci curvature lower bounds. With the aim of achieving such a unification program, in this paper we initiate the study of gauge metric measure spaces.
We use instanton gauge theory to prove that if $Y$ is a closed, orientable $3$-manifold such that $H_1(Y;\mathbb{Z})$ is nontrivial and either $2$-torsion or $3$-torsion, and if $Y$ is neither $\#^r \mathbb{RP}^3$ for some $r\geq 1$ nor $\pm L(3,1)$, then there is an irreducible representation $\pi_1(Y) \to \mathrm{SL}(2,\mathbb{C})$. We apply this to show that the Kauffman bracket skein module of a non-prime 3-manifold has nontrivial torsion whenever two of the prime summands are different from $\mathbb{RP}^3$, answering a conjecture of Przytycki (Kirby problem 1.92(F)) unless every summand but one is $\mathbb{RP}^3$. As part of the proof in the $2$-torsion case, we also show that if $M$ is a compact, orientable $3$-manifold with torus boundary whose rational longitude has order 2 in $H_1(M)$, then $M$ admits a degree-1 map onto the twisted $I$-bundle over the Klein bottle.
In this work, we study the random metric for the critical long-range percolation on $\mathbb{Z}^d$. A recent work by B\"aumler [3] implies the subsequential scaling limit, and our main contribution is to prove that the subsequential limit is uniquely characterized by a natural list of axioms. Our proof method is hugely inspired by recent works of Gwynne and Miller [42], and Ding and Gwynne [25] on the uniqueness of Liouville quantum gravity metrics.
We consider the stochastic Yang-Mills heat equation on the two-dimensional torus. Using regularity structures, Chandra, Chevyrev, Hairer, and Shen previously proved both the local well-posedness and gauge-covariance of this model. In this article, we revisit their results using para-controlled calculus. One of the main ingredients is a new coordinate-invariant perspective on vector-valued stochastic objects.
This paper completes the program started in arXiv:2104.11204 and arXiv:2110.04565 aiming at providing a full rigorous justification of the wave kinetic theory for the nonlinear Schr\"odinger (NLS) equation. Here, we cover the full range of scaling laws for the NLS on an arbitrary periodic rectangular box, and derive the wave kinetic equation up to small multiples of the kinetic time. The proof is based on a diagrammatic expansion and a deep analysis of the resulting Feynman diagrams. The main novelties of this work are three-fold: (1) we present a robust way to identify arbitrarily large "bad" diagrams which obstruct the convergence of the Feynman diagram expansion, (2) we systematically uncover intricate cancellations among these large "bad" diagrams, and (3) we present a new robust algorithm to bound all remaining diagrams and prove convergence of the expansion. These ingredients are highly robust, and constitute a powerful new approach in the geneal mathematical study of Feynman diagrams.
Let $B$ be a smooth projective curve and let $\pi: \mathcal{X} \to B$ be a smooth integral model of a geometrically integral Fano variety over $K(B)$. Geometric Manin's Conjecture predicts the structure of the irreducible components $M \subset \textrm{Sec}(\mathcal{X}/B)$ which parametrize non-relatively free sections of sufficiently large anticanonical degree. Over the complex numbers, we prove that for any such component $M$ the sections come from morphisms $f: \mathcal{Y} \to \mathcal{X}$ such that the generic fiber of $\mathcal{Y}$ has Fujita invariant $\geq 1$. Furthermore, we prove that there is a bounded family of morphisms $f$ which together account for all such components $M$. These results verify the first part of Batyrev's heuristics for Geometric Manin's Conjecture over $\mathbb{C}$. Our result has ramifications for Manin's Conjecture over global function fields: if we start with a Fano fibration over a number field and reduce mod $p$, we obtain upper bounds of the desired form by first letting the prime go to infinity, then the height.
For geometrically finite non-compact developable hyperbolic orbisurfaces (including those of infinite volume), we provide transfer operator families whose Fredholm determinants are identical to the Selberg zeta function. Our proof yields an algorithmic and uniform construction. This construction is initiated with an externally provided cross section for the geodesic flow on the considered orbisurface that yields a highly faithful, but non-uniformly expanding discrete dynamical system modelling the geodesic flow. Through a number of algorithmic steps of reduction, extension, translation, induction and acceleration, we turn this cross section into one that yields a still highly faithful, but now uniformly expanding discrete dynamical system. The arising transfer operator family is nuclear of order zero on suitable Banach spaces. In addition, finite-dimensional twists with non-expanding cusp monodromy can be included.
We study families of linear differential equations parametrized by an algebraic variety $\mathcal{X}$ and show that the set of all points $x\in \mathcal{X}$, such that the differential Galois group at the generic fibre specializes to the differential Galois group at the fibre over $x$, is Zariski dense in $\mathcal{X}$. As an application, we prove Matzat's conjecture in full generality: The absolute differential Galois group of a one-variable function field over an algebraically closed field of characteristic zero is a free proalgebraic group.
Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works and provide an open-source Python implementation. As a prerequisite, we build up a new theoretical foundation for combinatorial decomposition strategies and combinatorial specifications. We then apply Combinatorial Exploration to the domain of permutation patterns, to great effect. We rederive hundreds of results in the literature in a uniform manner and prove many new ones. These results can be found in a new public database, the Permutation Pattern Avoidance Library (PermPAL) at https://permpal.com. Finally, we give three additional proofs-of-concept, showing examples of how Combinatorial Exploration can prove results in the domains of alternating sign matrices, polyominoes, and set partitions.
We investigate quasilinear discrete PDEs $\partial_t u = \Delta^N \varphi(u)+ Kf(u)$ of reaction-diffusion type with nonlinear diffusion term defined on an $n$-dimensional unit torus discretized with mesh size $\tfrac1N$ for $N\in {\mathbb N}$, where $\Delta^N$ is the discrete Laplacian, $\varphi$ is a strictly increasing $C^5$ function and $f$ is a $C^1$ function. We establish $L^\infty$ bounds and space-time H\"older estimates, both uniform in $N$, of the first and second spatial discrete derivatives of the solutions. In the equation, $K>0$ is a large constant and we show how these estimates depend on $K$. The motivation for this work stems originally from the study of hydrodynamic scaling limits of interacting particle systems. Our method is a two steps approach in terms of the H\"older estimate and Schauder estimate, which is known for continuous parabolic PDEs. We first show the discrete H\"older estimate uniform in $N$ for the solutions of the associated linear discrete PDEs with continuous coefficients, based on the Nash estimate. We next establish the discrete Schauder estimate for linear discrete PDEs with uniform H\"older coefficients. The link between discrete and continuous settings is given by the polylinear interpolations. Since this operation has a non-local nature, the method requires proper modifications. We also discuss another method based on the study of the corresponding fundamental solutions.
We consider Klein-Gordon equations with an external potential $V$ and a quadratic nonlinearity in $3+1$ space dimensions. We assume that $V$ is regular and decaying and that the (massive) Schr\"odinger operator $H=-\Delta+V+m^2$ has a positive eigenvalue $\lambda^2<m^2$ with associated eigenfunction $\phi.$ This is a so-called internal mode and gives rise to time-periodic and spatially localized solutions of the linear flow. We address the classical question of whether such solutions persist under the full nonlinear flow, and describe the behavior of all solutions in a suitable neighborhood of zero. Provided a natural Fermi-Golden rule holds, our main result shows that a solution to the nonlinear Klein-Gordon equation can be decomposed into a discrete component $a(t)\phi$ where $a(t)$ decays over time, and a continuous component $v$ which has some weak dispersive properties. We obtain precise asymptotic information on these components such as the sharp rates of decay $\vert a(t) \vert \approx t^{-1/2}$ and ${\| v(t) \|}_{L^\infty_x} \approx t^{-1}$, (where the implicit constants are independent of the small size of the data) as well as the growth of a natural weighted norm of the profile of $v.$ In particular, our result extends the seminal work of Soffer-Weinstein for the cubic Klein-Gordon, and shows that radiation damping also occurs in the quadratic case.
A quaternary quartic form, a quartic form in four variables, is the dual socle generator of an Artinian Gorenstein ring of codimension and regularity 4. We present a classification of quartic forms in terms of rank and powersum decompositions which corresponds to the classification by the Betti tables of the corresponding Artinian Gorenstein rings. This gives a stratification of the space of quaternary quartic forms which we compare with the Noether-Lefschetz stratification. We discuss various phenomena related to this stratification. We study the geometry of powersum varieties for a general form in each stratum. In particular, we show that the powersum variety VSP(F,9) of a general quartic with singular middle catalecticant is again a quartic surface, thus giving a rational map between two divisors in the space of quartics. Finally, we provide various explicit constructions of general Artinian Gorenstein rings corresponding to each stratum and discuss their lifting to higher dimension. These provide constructions of codimension four varieties, which include canonical surfaces, Calabi-Yau threefolds and Fano fourfolds. In the particular case of quaternary quartics, our results yield answers to questions posed by Geramita, Iarrobino-Kanev, and Reid.
This is the first of two papers on the global topology of the space Sub(G) of all closed subgroups of G=PSL_2(ℝ), equipped with the Chabauty topology. In this paper, we study the spaces of lattices and elementary subgroups of G, and prove a continuity result for conformal grafting of (possibly infinite type) vectored orbifolds that will be useful in both papers. More specifically, we first identify the homotopy type of the space of elementary subgroups of G, following Baik-Clavier. Then for a fixed finite type hyperbolizable 2-orbifold S, we show that the space Sub_S(G) of all lattices Γ < G with Γ\ℍ^2 ≅ S is a fiber orbibundle over the moduli space ℳ(S). We describe the closure Sub_S(G) in Sub(G) and show that ∂Sub_S(G) has a neighborhood deformation retract within Sub_S(G). When S is not one of finitely many low complexity orbifolds, we show that Sub_S(G) is simply connected. In the simplest exceptional case, when S is a sphere with three total cusps and cone points, we show that Sub_S(G) is a (usually nontrivial) lens space. Finally, we show that when (X_i,v_i) → (X_∞,v_∞) is a (possibly infinite type) smoothly converging sequence of vectored hyperbolic 2-orbifolds, and we graft in Euclidean annuli along suitable collections of simple closed curves in the X_i, then after uniformization, the resulting vectored hyperbolic 2-orbifolds converge smoothly to the expected limit. As part of the proof, we give a new lower bound on the hyperbolic distance between points in a grafted orbifold in terms of their original distance.
We initiate the study of Gaitsgory's central sheaves using complexes of Soergel bimodules, in extended affine type A. We conjecture that the complex associated to the standard representation can be flattened to a central complex in finite type A, which is the complex sought after in a conjecture of Gorsky-Negut-Rasmussen.
