Related papers: Remarks on Bianchi sums and Pontrjagin classes

We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp(-{N}\,\mathrm{Tr}(E\Phi^2+(\lambda/4)\Phi^4)) d\Phi$ on Hermitian ${N}\times{N}$-matrices, where $E$ is any positive matrix and $\lambda$ a…

Let $K$ be an imaginary quadratic field and let $\mathcal{O}_K$ be its ring of integers. For an integral ideal $\mathfrak{n}$ of $\mathcal{O}_K$, let $\Gamma_0({\mathfrak{n}})$ be the congruence subgroup of level ${\mathfrak{n}}$ consisting…

In this paper, we prove the following two results: First, we study a class of conformally invariant operators $P$ and their related conformally invariant curvatures $Q$ on even-dimensional Riemannian manifolds. When the manifold is locally…

For a differential form on a manifold, having constant components in suitable local coordinates trivially implies being parallel relative to a torsion-free connection, and the converse implication is known to be true for $p$-forms in…

In earlier work, we constructed counterexamples to Lagrangian Poincar\'e recurrence for many toric symplectic four manifolds. Here we provide a few more examples extending the family of counterexamples to include all non-monotone toric…

We introduce a diffeomorphism invariant of $4$-manifolds, the $\mathrm{Pin}^-(2)$-monopole invariant, defined by using the $\mathrm{Pin}^-(2)$-monopole equations. We compute the invariants of several $4$-manifolds, and prove gluing…

We prove rigidity and gap theorems for self-dual and even Poincar\'e-Einstein metrics in dimension four. As a corollary, we give an obstruction to the existence of self-dual Poincar\'e-Einstein metrics in terms of conformal invariants of…

We study manifolds satisfying a weighed Poincare inequality, which was first introduced by Li-Wang. We generalized one of their results by relaxing the Ricci curvature bound condition only being satisfied outside a compact set and…

We study Lagrangian and orthogonal splittings\textbf{\ }of quadratic vector spaces establishing an equivalence with complex product structures. Then we show that a Manin triple equipped with generalized metric $\mathcal{G}+% \mathcal{B}$…

The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for $\zeta(2)$ and $\zeta(3),$ and those of the second author for Euler's constant $\gamma$ and its alternating analog $\ln(4/\pi),$…

We consider the existence of invariant curves of real analytic reversible mappings which are quasi-periodic in the angle variables. By the normal form theorem, we prove that under some assumptions, the original mapping is changed into its…

We investigate strong maximum (and minimum) principles for fully nonlinear second order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of…

We establish closed-form expressions for the infinite series sum from n=2 to infinity of arctanh(n^-k) for all integers k >= 2 by connecting these sums to infinite product formulas involving the gamma function. Our approach uses logarithmic…

We investigate the curvature operator of the second kind on product Riemannian manifolds and obtain some optimal rigidity results. For instance, we prove that the universal cover of an $n$-dimensional non-flat complete locally reducible…

Let $M$ be a connected smooth manifold, let $\operatorname{Aut}(p)$ be the group automorphisms of the bundle $p\colon \mathbb{R}\times M\to \mathbb{R}$, and let $q\colon J^1(\mathbb{R},M)\times \mathbb{R\to }J^1(\mathbb{R},M)$ be the…

We show that compact, $n$-dimensional Riemannian manifolds with $\frac{n+2}{2}$-nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the $p$-th Betti number…

In this paper, we study the alternating Euler $T$-sums and $\S$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of…

Making use of its smooth structure only, out of a connected oriented smooth $4$-manifold a von Neumann algebra is constructed. It is geometric in the sense that is generated by local operators and as a special four dimensional phenomenon it…

Two methods can be used to calculate explicitly the Killing form on the Lie algebras. The first one is a direct calculation of the traces of the generators in a matrix representation of the algebra, and the second one is the usage of the…

We discuss several well known results about Schur functions that can be proved using cancellations in alternating summations; notably we shall discuss the Pieri and Murnaghan-Nakayama rules, the Jacobi-Trudi identity and its dual (Von…