Selecta Mathematica最新文献

Inspired by an example of Guéritaud and Kassel (Geom Topol 21(2):693–840, 2017), we construct a family of infinitely generated discontinuous groups (Gamma ) for the 3-dimensional anti-de Sitter space (textrm{AdS}^{3}). These groups are not necessarily sharp (a kind of “strong” proper discontinuity condition introduced by Kassel and Kobayashi (Adv Math 287:123–236, 2016), and we give its criterion. Moreover, we find upper and lower bounds of the counting (N_{Gamma }(R)) of a (Gamma )-orbit contained in a pseudo-ball B(R) as the radius R tends to infinity. We then find a non-sharp discontinuous group (Gamma ) for which there exist infinitely many (L^2)-eigenvalues of the Laplacian on the noncompact anti-de Sitter manifold (Gamma backslash textrm{AdS}^{3}), by applying the method established by Kassel–Kobayashi. We also prove that for any increasing function f, there exists a discontinuous group (Gamma ) for (textrm{AdS}^{3}) such that the counting (N_{Gamma }(R)) of a (Gamma )-orbit is larger than f(R) for a sufficiently large R.

We study a simple geometric model for local transformations of bipartite graphs. The state consists of a choice of a vector at each white vertex made in such a way that the vectors neighboring each black vertex satisfy a linear relation. The evolution for different choices of the graph coincides with many notable dynamical systems including the pentagram map, Q-nets, and discrete Darboux maps. On the other hand, for plabic graphs we prove unique extendability of a configuration from the boundary to the interior, an elegant illustration of the fact that Postnikov’s boundary measurement map is invertible. In all cases there is a cluster algebra operating in the background, resolving the open question for Q-nets of whether such a structure exists.

Let be a compact Kähler Lagrangian in a holomorphic symplectic variety (textrm{X}/textbf{C}). We use deformation quantisation to show that the endomorphism differential graded algebra (textrm{RHom}big (i_*textrm{K}_textrm{L}^{1/2},i_*textrm{K}_textrm{L}^{1/2}big )) is formal. We prove a generalisation to pairs of Lagrangians, along with auxiliary results on the behaviour of formality in families of ({text {A}}_{infty })-modules.

Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler–Thorpe trick for sectional curvature bounds in dimension 4.

For each poset P, we construct a polytope ({mathscr {A}}(P)) called the P-associahedron. Similarly to the case of graph associahedra, the faces of ({mathscr {A}}(P)) correspond to certain nested collections of subsets of P. The Stasheff associahedron is a compactification of the configuration space of n points on a line, and we recover ({mathscr {A}}(P)) as an analogous compactification of the space of order-preserving maps (Prightarrow {{mathbb {R}}}). Motivated by the study of totally nonnegative critical varieties in the Grassmannian, we introduce affine poset cyclohedra and realize these polytopes as compactifications of configuration spaces of n points on a circle. For particular choices of (affine) posets, we obtain associahedra, cyclohedra, permutohedra, and type B permutohedra as special cases.

In this paper, we study the cohomology of semisimple local systems in the spirit of classical Hodge theory. On the one hand, we construct a generalized Weil operator from the complex conjugate of the cohomology of a semisimple local system to the cohomology of its dual local system, which is functorial with respect to smooth restrictions. This operator allows us to study the Poincaré pairing, usually not positive definite, in terms of a positive definite Hermitian pairing. On the other hand, we prove a global invariant cycle theorem for semisimple local systems. As an application, we give a new proof of Sabbah’s Decomposition Theorem for the direct images of semisimple local systems under proper algebraic maps, by adapting the method of de Cataldo-Migliorini, without using the category of polarizable twistor ({mathscr {D}})-modules. This answers a question of Sabbah.

K-theoretic Hall algebras (KHAs) of quivers with potential (Q, W) are a generalization of preprojective KHAs of quivers, which are conjecturally positive parts of the Okounkov–Smironov quantum affine algebras. In particular, preprojective KHAs are expected to satisfy a Poincaré–Birkhoff–Witt theorem. We construct semi-orthogonal decompositions of categorical Hall algebras using techniques developed by Halpern-Leistner, Ballard–Favero–Katzarkov, and Špenko–Van den Bergh. For a quotient of (text {KHA}(Q,W)_{{mathbb {Q}}}), we refine these decompositions and prove a PBW-type theorem for it. The spaces of generators of (text {KHA}(Q,0)_{{mathbb {Q}}}) are given by (a version of) intersection K-theory of coarse moduli spaces of representations of Q.

We provide a framework for relating certain q-series defined by sums over partitions to multiple zeta values. In particular, we introduce a space of polynomial functions on partitions for which the associated q-series are q-analogues of multiple zeta values. By explicitly describing the (regularized) multiple zeta values one obtains as (qrightarrow 1), we extend previous results known in this area. Using this together with the fact that other families of functions on partitions, such as shifted symmetric functions, are elements in our space will then give relations among (q-analogues of) multiple zeta values. Conversely, we will show that relations among multiple zeta values can be ‘lifted’ to the world of functions on partitions, which provides new examples of functions for which the associated q-series are quasimodular.