We introduce a common domain of definition for the loop product and the loop coproduct, reduced loop homology, on which they combine to a unital infinitesimal antisymmetric bialgebra structure. In particular, a relation conjectured by Sullivan holds with an extra term. The structure depends on choices governed by secondary continuation maps. These results on string topology are proved in the more general context of reduced symplectic homology for a suitable class of Weinstein manifolds.
Let G be a locally compact group, K be a compact subgroup of G and 8 be a class of unitary irreducible representations of K. The triple (G, K, 8) is commutative if the convolution algebra Cc(G, F8, 8, 8) of 8-radial functions with compact support is commutative. We prove a Bochner-type theorem for commutative triples using some algebraic properties of 8-radial functions of positive type. This work extends some results of Marouane Rabaoui.
A global representation is a compatible collection of representations of the outer automorphism groups of the groups belonging to some collection of finite groups $\mathscr{U}$. Global representations assemble into an abelian category $\mathsf{A}(\mathscr{U})$, simultaneously generalising classical representation theory and the category of VI-modules appearing in the representation theory of the general linear groups. In this paper we establish homological foundations of its derived category $\mathsf{D}(\mathscr{U})$. We prove that any complex of projective global representations is DG-projective, and hence conclude that the derived category admits an explicit model as the homotopy category of projective global representations. We show that from a tensor-triangular perspective it exhibits some unusual features: for example, there are very few dualizable objects and in general many more compact objects. Under more restrictive conditions on the family $\mathscr{U}$, we then construct torsion-free classes for global representations which encode certain growth properties in $\mathscr{U}$. This lays the foundations for a detailed study of the tensor-triangular geometry of derived global representations which we pursue in forthcoming work.
In this paper, we investigate the link between kinetic equations (including Boltzmann with or without cutoff assumption and Landau equations) and the incompressible Navier-Stokes equation. We work with strong solutions and we treat all the cases in a unified framework. The main purpose of this work is to be as accurate as possible in terms of functional spaces. More precisely, it is well-known that the Navier-Stokes equation can be solved in a lower regularity setting (in the space variable) than kinetic equations. Our main result allows to get a rigorous link between solutions to the Navier-Stokes equation with such low regularity data and kinetic equations.
Given a coherent sheaf E on a scheme of finite type X over a perfect field, we introduce a category of complexes of & eacute;tale sheaves on X with logarithmic conductors bounded by E and study its compatibilities with finite pushforward.
Given a right-angled Artin group G with finite outer automorphism group, we determine which right-angled Artin groups are measure equivalent (or orbit equivalent) to G.
Capistrano-Filho We focus on the linear Kawahara equation in a bounded domain, employing two boundary controls. The controllability of this system has been previously demonstrated over the past decade using the Hilbert uniqueness method, which involves proving an observability inequality, in general, shown via Carleman estimates. Here, we extend this understanding by achieving exact controllability within a space of analytic functions, employing the flatness approach, which is a new approach for higher-order dispersive systems. We also provide a class of reachable states (taking 0 as initial data) which are holomorphic in some disk around 0.
We give a new construction of tensor product gamma factors for a pair of irreducible representations of GLc(Fq) and GLk(Fq). This construction is a finite field analog of a construction of doubling type due to Kaplan in the local field case and due to Ginzburg in the global case, and it only assumes that one of the representations in question is generic. We use this construction to establish a relation between special values of Bessel functions attached to Speh representations of generic principal series representations and twisted matrix Kloosterman sums. Using this relation, we establish the multiplicativity identity of twisted matrix Kloosterman sums.
We study the homotopy type of spaces of commuting elements in connected nilpotent Lie groups, via almost commuting elements in their Lie algebras. We give a necessary and sufficient condition on the fundamental group of such a Lie group G to ensure Hom(7Lk, G) is path-connected. In particular for the reduced upper unitriangular groups and the reduced generalized Heisenberg groups, Hom(7Lk, G) is not path-connected, and we compute the homotopy type of its path-connected components in terms of Stiefel manifolds and the maximal torus of G.
We study the local structure of the representation variety of a knot group into SL(n,C) at certain diagonal representations. In particular we determine the tangent cone of the representation variety at these diagonal representations, and show that the latter can be deformed into irreducible representations. Furthermore, we use Luna's slice theorem to analyze the local structure of the character variety.
In this note, we investigate a doubly nonlinear diffusion equation in the slow diffusion regime. We prove stability of the pressure of solutions that are close to traveling wave solutions in a homogeneous Lipschitz sense. We derive regularity estimates for arbitrary derivatives of the solution's pressure by extending existing results for the porous medium equation (Ref. 15).
Relying on the formalism developed by Alexander Beilinson and Takeshi Saito, we compute the characteristic cycle of an external symmetric power of a tame étale sheaf on a curve. This generalizes a result of Gérard Laumon in characteristic 0 and leads to a result of local acyclicity of the Abel-Jacobi morphism, due to Pierre Deligne and motivated by his geometric approach to the product formula for the determinant of cohomology (epsilon factor).
We establish both the existence and uniqueness of non-negative global solutions for the nonlinear heat equation $u_t-\Delta u=|x|^{-\gamma}\,u^q$, $00$ in the whole space $\mathbb{R}^N$, and for non-negative initial data $u_0\in C_0(\mathbb{R}^N)$.
This paper studies the homotopy theory of the Grothendieck construction using model categories and semi-model categories, provides a unifying framework for the homotopy theory of operads and their algebras and modules, and uses this framework to produce model structures, rectification results, and properness results in new settings. In contrast to previous authors, we begin with a global (semi-)model structure on the Grothendieck and induce (semi-)model structures on the base and fibers. In a companion paper, we show how to produce such global model structures in general settings. Applications include numerous flavors of operads encoded by polynomial monads and substitudes (symmetric, non-symmetric, cyclic, modular, higher operads, dioperads, properads, and PROPs), (commutative) monoids and their modules, and twisted modular operads. We also prove a general result for upgrading a semi-model structure to a full model structure.
Recently, Kedlaya proves certain formula describing explicitly the Frobenius structure on a hypergeometric equation. In this paper, we give a generalization of it. In our case, the Frobenius matrix is no longer described by p-adic gamma function, and then we describe it by the p-adic polygamma functions. Since the p-adic polygamma values are linear combinations of p-adic L-values of Dirichlet characters, it turns out that the Frobenius matrix is described by p-adic L-values. Our result has an application to the study on Frobenius on p-adic cohomology. We show that, for a projective smooth family such that the Picard-Fuchs equation is a hypergeometric equation, the Frobenius matrix on the log-crystalline cohomology is described by some values of the logarithmic function and p-adic L-functions of Dirichlet characters.
We study the Poisson transform for differential forms on the real hyperbolic space $\mathbb H^n$. For $1<r<\infty$, we prove that the Poisson transform is an isomorphism from the space of $L^r$ $q$-differential forms on the boundary $\partial \mathbb H^n$ onto a Hardy-type subspace of $p$-eigenforms of the de Rham-Hodge Laplacian, for $0\leq p<\frac{n-1}{2}$ and $q=p-1, p$.
We provide a systematic approach to twisting differential KO-theory leading to a construction of the corresponding twisted differential Atiyah-Hirzebruch spectral sequence (AHSS). We relate and contrast the degree two and the degree one twists, whose description involves appropriate local systems. Along the way, we provide a complete and explicit identification of the differentials at the E_2 and E_3 pages in the topological case, which has been missing in the literature and which is needed for the general case. The corresponding differentials in the refined theory reveal an intricate interplay between topological and geometric data, the former involving the flat part and the latter requiring the construction of the twisted differential Pontrjagin character. We illustrate with examples and applications from geometry, topology and physics. For instance, quantization conditions show how to lift differential 4k-forms to twisted differential KO-theory leading to integrality results, while considerations of anomalies in type I string theory allow for characterization of twisted differential Spin structures.
