Volume 29, Issue 3

We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the d -dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on L2${L^{2}}$-space. The Gaussian is a very important example. For rotationally invariant α-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the p -adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula.

We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain Dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square L -functions associated to cuspidal Maass newforms of squarefree level N>1${N>1}$.

In this paper, we do the following two things: (i) We present a formula to compute the rational cohomology ring of a real topological toric manifold, and thus that of a small cover or a real toric manifold, which implies the formula of Suciu and Trevisan. Furthermore, the formula also works for an arbitrary coefficient ring G in which 2 is a unit. (ii) We construct infinitely many real toric manifolds and small covers whose integral cohomology rings have a q -torsion for any positive odd integer q .

Let 𝔤${{\mathfrak{g}}}$ be a simple Lie algebra. Let Aut⁢(𝔤)${{\rm Aut}(\mathfrak{g})}$ be the group of all automorphisms on 𝔤${\mathfrak{g}}$, and let Int⁢(𝔤)${{\rm Int}(\mathfrak{g})}$ be its identity component. The outer automorphism group of 𝔤${\mathfrak{g}}$ is defined as Aut⁢(𝔤)/Int⁢(𝔤)${{\rm Aut}(\mathfrak{g})/{\rm Int}(\mathfrak{g})}$. If 𝔤${\mathfrak{g}}$ is complex and has Dynkin diagram D${{\rm D}}$, then Aut⁢(𝔤)/Int⁢(𝔤)${{\rm Aut}(\mathfrak{g})/{\rm Int}(\mathfrak{g})}$ is isomorphic to Aut⁢(D)${{\rm Aut}({\rm D})}$. We provide an analogous result for the real case. For 𝔤${\mathfrak{g}}$ real, we let 𝔤${\mathfrak{g}}$ be represented by a painted diagram P${{\rm P}}$. Depending on whether the Cartan involution of 𝔤${\mathfrak{g}}$ belongs to Int⁢(𝔤)${{\rm Int}(\mathfrak{g})}$, we show that Aut⁢(𝔤)/Int⁢(𝔤)${{\rm Aut}(\mathfrak{g})/{\rm Int}(\mathfrak{g})}$ is isomorphic to Aut⁢(P)${{\rm Aut}({\rm P})}$ or Aut⁢(P)×ℤ2${{\rm Aut}({\rm P})\times\mathbb{Z}_{2}}$. This result extends to the outer automorphism groups of all real semisimple Lie algebras.

We define a non-abelian tensor product of multiplicative Lie rings. This is a new concept providing a common approach to the non-abelian tensor product of groups defined by Brown and Loday and to the non-abelian tensor product of Lie rings defined by Ellis. We also prove an analogue of Miller’s theorem for multiplicative Lie rings.

In this paper, we look for an explicit lower bound of the smallest value of the spectrum for a relativistic Schrödinger operator in a domain of the Euclidean space.

The Alperin–McKay conjecture relates irreducible characters of a block of an arbitrary finite group to those of its p -local subgroups. A refinement of this conjecture was stated by the author in a previous paper. We prove that this refinement holds for all blocks of symmetric groups. Along the way we identify a “canonical” isometry between the principal block of Sp⁢w${S_{pw}}$ and that of Sp≀Sw${S_{p}\wr S_{w}}$. We also prove a general theorem on expressing virtual characters of wreath products in terms of certain induced characters. Much of the paper generalises character-theoretic results on blocks of symmetric groups with abelian defect and related wreath products to the case of arbitrary defect.

After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed ∞${\infty}$-categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every ∞${\infty}$-topos has a hierarchy of “universal” univalent families, indexed by regular cardinals, and that n -topoi have univalent families classifying (n-2)${(n-2)}$-truncated maps. We show that univalent families are preserved (and detected) by right adjoints to locally cartesian localizations, and use this to exhibit certain canonical univalent families in ∞$\infty$-quasitopoi (certain ∞${\infty}$-categories of “separated presheaves”, introduced here). We also exhibit some more exotic examples of univalent families, illustrating that a univalent family in an n -topos need not be (n-2)${(n-2)}$-truncated, as well as some univalent families in the Morel–Voevodsky ∞${\infty}$-category of motivic spaces, an instance of a locally cartesian closed ∞${\infty}$-category which is not an n -topos for any 0≤n≤∞${0\leq n\leq\infty}$. Lastly, we show that any presentable locally cartesian closed ∞${\infty}$-category is modeled by a combinatorial type-theoretic model category, and conversely that the ∞${\infty}$-category underlying a combinatorial type-theoretic model category is presentable and locally cartesian closed. Under this correspondence, univalent families in presentable locally cartesian closed ∞${\infty}$-categories correspond to univalent fibrations in combinatorial type-theoretic model categories.

In this article subordination of random walks in ℝd${{\mathbb{R}}^{d}}$ is considered. We prove that subordination of random walks in the sense of [4] yields the same process as subordination in the sense of Lévy processes. Furthermore, we prove that appropriately scaled subordinate random walk converges to a multiple of a rotationally 2⁢α${2\alpha}$-stable process if and only if the Laplace exponent of the corresponding subordinator varies regularly at zero with index α∈(0,1]${\alpha\in(0,1]}$.

We consider finitely presented groups admitting 0-combings which are both Lipschitz (in the sense of Thurston) and tame (as defined by Mihalik and Tschantz in [8]). What we prove is that such groups are easy (and hence QSF by [11]), in the sense that they admit an easy representation (that is a map from a 2-complex to a singular 3-manifold associated to the group, satisfying several topological conditions with a strong control over singularities). Besides its own interest, one may also try to adapt the proof in a wider context, namely for groups admitting tame 1-combings (as in [8]), in order to prove the easy-representability for a larger class of finitely presented groups (note that there are still no examples of finitely presented groups which are not tame 1-combable).

The space of configurations of n ordered points in the plane serves as a classifying space for the pure braid group P⁢Bn${PB_{n}}$. Elements of Thompson’s group F admit a model similar to braids, except instead of braiding the strands split and merge. In Belk’s thesis, a space 𝒞⁢F${\mathcal{C}F}$ was considered of configurations of points on the real line allowing for splitting and merging, and a sketch of a proof was given that 𝒞⁢F${\mathcal{C}F}$ is a classifying space for F . The idea there was to build the universal cover and construct an explicit contraction to a point. However, this was never written up rigorously. Here we start with an established CAT⁡(0)${\operatorname{CAT}(0)}$ cube complex X on which F acts freely, and construct an explicit homotopy equivalence between X/F${X/F}$ and 𝒞⁢F${\mathcal{C}F}$, proving that 𝒞⁢F${\mathcal{C}F}$ is indeed a K⁢(F,1)${K(F,1)}$.

In this note, we introduce the asymptotic subspace confinement problem, generalizing the usual concept of convergence in discrete-time linear systems. Instead of precise convergence, subspace confinement only requires the convergence of states to a certain subspace of the state space, offering useful flexibility and applicability. We establish a criterion for deciding the asymptotic subspace confinement, drawing upon a general finiteness result for the infinite product of matrices. Our results indicate that the asymptotic subspace confinement problem is algorithmically decidable when an invariant subspace for the set of matrices and some polytope norms are given.

We characterize exceptional vector bundles on ℙn${\mathbb{P}^{n}}$ of arbitrary homological dimension defined by a linear resolution. Moreover, we determine all Betti numbers of such resolution.

We compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are modelled on certain prehomogeneous vector spaces. Our method is based on evaluating 𝔭${\mathfrak{p}}$-adic integrals associated to certain rank varieties of matrices of linear forms.

We give an explicit approach for Bockstein homomorphisms of the Hochschild cohomology of a group algebra and of a block algebra of a finite group and we show some properties. To give explicit definitions for these maps we use an additive decomposition and a product formula for the Hochschild cohomology of group algebras given by Siegel and Witherspoon in 1999. For an algebraically closed field k of characteristic p and a finite group G we prove an additive decomposition and a product formula for the cohomology algebra of a defect group of a block ideal of kG with coefficients in the source algebra of this block, and we define similar Bockstein homomorphisms.