[en] A one-parameter family of coupled flows depending on a parameter $\mathrm{\kappa <!--\; ??\; -->}>0$ is introduced which reduces when $\mathrm{\kappa <!--\; ??\; -->}=1$ to the coupled flow of a metric $\mathrm{\omega <!--\; ??\; -->}$ with a (1, 1)-form $\mathrm{\alpha <!--\; ??\; -->}$ due recently to Y. Li, Y. Yuan, and Y. Zhang. It is shown in particular that, for $\mathrm{\kappa <!--\; ??\; -->}\ne 1$, estimates for derivatives of all orders would follow from $C^0$ estimates for $\mathrm{\omega <!--\; ??\; -->}$ and $\mathrm{\alpha <!--\; ??\; -->}$. Together with the monotonicity of suitably adapted energy functionals, this can be applied to establish the convergence of the flow in some situations, including on Riemann surfaces. Very little is known as yet about the monotonicity and convergence of flows in presence of couplings, and conditions such as $\mathrm{\kappa <!--\; ??\; -->}\ne 1$ seem new and may be useful in the future.

[en] For an arbitrary quiver $Q=(I,\mathrm{\Omega <!--\; ??\; -->})$ and dimension vector $\mathbf{d}\in <!--\; ???\; -->{\mathbb{N}}^I$ we define the dimension of absolutely cuspidal functions on the moduli stacks of representations of dimension $\mathbf{d}$ of a quiver Q over a finite field ${\mathbb{F}}_q$, and prove that it is a polynomial in q, which we conjecture to be positive and integral. We obtain a closed formula for these dimensions of spaces of cuspidals for totally negative quivers.

[en] We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.

[en] We present a proof of discrete spectrum and dynamical localization for small perturbations of discrete one-dimensional Schrödinger operators with uniform electric fields. The proof of dynamical localization is based on the KAM technique.

[en] Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let $X^H$ denote the set of fixed points of H in X, and $N_G(H)$ the normalizer of H in G. In this paper we study the natural map of quotient varieties ${\mathrm{\psi <!--\; ??\; -->}}_{X,H}:X^H/N_G(H)\to <!--\; ???\; -->X/G$ induced by the inclusion $X^H\subseteq <!--\; ???\; -->X$. We show that, given G and H, ${\mathrm{\psi <!--\; ??\; -->}}_{X,H}$ is a finite morphism for all affine G-varieties X if and only if H is a G-completely reducible subgroup of G (in the sense defined by Serre); this was proved in characteristic 0 by Luna in the 1970s. We discuss some applications and give a criterion for ${\mathrm{\psi <!--\; ??\; -->}}_{X,H}$ to be an isomorphism. We show how to extend some other results in Luna’s paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; then the double coset HgK is closed for generic $g\in <!--\; ???\; -->G$ if and only if $H\cap <!--\; ???\; -->gK{g}^{-<!--\; ???\; -->1}$ is reductive for generic $g\in <!--\; ???\; -->G$.

[en] We study Zariski cancellation problem for noncommutative algebras that are not necessarily domains.

[en] In this work, we study the Willmore submanifolds in a closed connected Riemannian manifold which are orbits for the isometric action of a compact connected Lie group. We call them homogeneous Willmore submanifolds or Willmore orbits. The criteria for these special Willmore submanifolds is much easier than the general theory which requires a complicated Euler-Lagrange equation. Our main theorem claims, when the orbit type stratification for the group action satisfies certain conditions, then we can find a Willmore orbit in each stratified subset. Some classical examples of special importance, like Willmore torus, Veronese surface, etc., can be interpreted as Willmore orbits and easily verified with our method. Our theorems provide a large number of new examples for Willmore submanifolds, as well as estimates for their numbers which are sharp in some classical cases.

[en] The purpose of this article is twofold. The first is to establish a truncated non-integrated defect relation for meromorphic mappings from a complete Kähler manifold quotien of a ball into a projective variety intersecting hypersurfaces in subgeneral position. We also apply it to the Gauss mapping from a closed regular submanifold of ${\mathbb{C}}^m$. The second aim is to establish an above type theorem with truncation level 1 for differentiably nondegenerate meromorphic mappings.

[en] We prove that, given a symmetrically distinguished correspondence of a suitable complex abelian variety (which includes any abelian variety of dimension at most 5, powers of complex elliptic curves, etc.) that vanishes as a morphism on a certain quotient of its middle singular cohomology, then it vanishes as a morphism on the deepest part of a particular filtration on the Chow group of 0-cycles of the abelian variety. As a consequence, we prove that an automorphism of such an abelian variety that acts as the identity on a certain quotient of its middle singular cohomology acts as the identity on the deepest part of this filtration on the Chow group of 0-cycles of the abelian variety. As an application, we prove that for the generalized Kummer variety associated to a complex abelian surface and the automorphism induced from a symplectic automorphism of the complex abelian surface, the automorphism of the generalized Kummer variety acts as the identity on a certain subgroup of its Chow group of 0-cycles.

[en] We study sum-of-squares representations of symmetric univariate real matrix polynomials that are positive semidefinite along the real line. We give a new proof of the fact that every positive semidefinite univariate matrix polynomial of size $n\times <!--\; ??\; -->n$ can be written as a sum of squares $M=Q^{T}Q$, where Q has size $(n+1)\times <!--\; ??\; -->n$, which was recently proved by Blekherman–Plaumann–Sinn–Vinzant. Our new approach using the theory of quadratic forms allows us to prove the conjecture made by these authors that these minimal representations $M=Q^{T}Q$ are generically in one-to-one correspondence with the representations of the nonnegative univariate polynomial $det(M)$ as sums of two squares. In parallel, we will use our methods to prove the more elementary hermitian analogue that every hermitian univariate matrix polynomial M that is positive semidefinite along the real line, is a square, which is known as the matrix Fejér–Riesz Theorem.