Let $f:M\to M$ be a $C^1$ map of a compact manifold $M$, with dimension at least $2$, admitting some point whose future trajectory has only negative Lyapunov exponents. Then this trajectory converges to a periodic sink. We need only assume that $Df$ is never the null map at any point (in particular, we need no extra smoothness assumption on $Df$ nor the existence of a invariant probability measure), encompassing a wide class of possible critical behavior. Similarly, a trajectory having only positive Lyapunov exponents for a $C^1$ diffeomorphism is itself a periodic repeller (source). Analogously for a $C^1$ open and dense subset of vector field on finite dimensional manifolds: for a flow $\phi_t$ generated by such a vector field, if a trajectory admits weak asymptotic sectional contraction (the extreme rates of expansion of the Linear Poincaré Flow are all negative), then this trajectory belongs either to the basin of attraction of a periodic hyperbolic attracting orbit (a periodic sink or an attracting equilibrium); or the trajectory accumulates a codimension one saddle singularity. Similar results hold for weak sectional expanding trajectories. Both results extend part of the non-uniform hyperbolic theory (Pesin's Theory) from the $C^{1+}$ diffeomorphism setting to $C^1$ endomorphisms and $C^1$ flows. Some ergodic theoretical consequences are discussed. The proofs use versions of Pliss' Lemma for maps and flows translated as (reverse) hyperbolic times, and a result ensuring that certain subadditive cocycles over $C^1$ vector fields are in fact additive.

The twisted Alexander polynomial is defined as a rational function, not necessarily a polynomial. It is shown that for a ribbon 2-knot, the twisted Alexander polynomial associated to an irreducible representation of the knot group to $\mathrm{SL}(2, \mathbb{F})$ is always a polynomial. Furthermore, the twisted Alexander polynomial of a fibered ribbon 2-knot of 1-fusion has the lowest and highest degree coefficients $1$ with breadth $2m-2$, where $m$ is the breadth of its Alexander polynomial.

In this note we consider homogeneous Willmore surfaces in $S^{n+2}$. The main result is that a homogeneous Willmore two-sphere is conformally equivalent to a homogeneous minimal two-sphere in $S^{n+2}$, i.e., either a round two-sphere or one of the Borůka-Veronese 2-spheres in $S^{2m}$. This entails a classification of all Willmore $\mathbb{C} P^1$ in $S^{2m}$. As a second main result we show that there exists no homogeneous Willmore upper-half plane in $S^{n+2}$ and we give, in terms of special constant potentials, a simple loop group characterization of all homogeneous surfaces which have an abelian transitive group.

Our aim in this paper is to deal with boundary limits of monotone Sobolev functions for the double phase functional $\Phi_{p,q}(x,t) = t^{p} + (b(x) t)^{q}$ in the unit ball ${\bf B}$ of ${\mathbb R}^n$, where 1 < $p$ < $q$ < $\infty$ and $b(\cdot)$ is a non-negative bounded function on ${\bf B}$ which is Hölder continuous of order $\theta \in (0,1]$.

Let $\mathcal{F}_1(n,m)$ be the ${\rm PSp}(n,1)$-configuration space of ordered $m$-tuple of pairwise distinct points in the boundary of quaternionic hyperbolic n-space $\partial\mathbf{H}_\mathbb{H}^n$ , i.e., the $m$-tuple of pairwise distinct points in $\partial\mathbf{H}_\mathbb{H}^n$ up to the diagonal action of ${\rm PSp}(n,1)$. In terms of Cartan's angular invariant and cross-ratio invariants, the moduli space of $\mathcal{F}_1(n,m)$ is described by using Moore's determinant. We show that the moduli space of $\mathcal{F}_1(n,m)$ is a real $2m^2-6m+5-\sum^{m-n-1}_{i=1}{m-2 \choose n-1+i}$ dimensional subset of a algebraic variety with the same real dimension when $m>n+1$.

Using several numerical invariants, we study a partition of the space of line arrangements in the complex projective plane, given by the intersection lattice types. We offer also a new characterization of the free plane curves using the Castelnuovo-Mumford regularity of the associated Milnor/Jacobian algebra.

We study homogeneous curves in generalized flag manifolds $G/K$ with ${\rm G}_2$-type $\mathfrak{t}$-roots, which are geodesics with respect to each $G$-invariant metric on $G/K$. These curves are calledequigeodesics. The tangent space of such flag manifolds splits into six isotropy summands, which are in one-to-one correspondence with $\mathfrak{t}$-roots. Also, these spaces are a generalization of the exceptional full flag manifold ${\rm G}_2/T$. We give a characterization for structural equigeodesics for flag manifolds with ${\rm G}_2$-type $\mathfrak{t}$-roots, and we give for each such flag manifold, a list of subspaces in which the vectors are structural equigeodesic vectors.

This paper is devoted to the study of hyperbolic systems of linear partial differential equations perturbed by a Brownian motion. The existence and uniqueness of solutions are proved by an energy method. The specific features of this class of stochastic partial differential equations are highlighted and the comparison with standard existence results for SPDEs is discussed. The small perturbations problem is studied and a large deviation principle is stated. A pathwise approximation result, similar to the stochastic differential equations case, is established, with an application to a support theorem.

Let $S=K[x_1, x_2, \dots, x_n]$ be the polynomial ring in $n$ variables over a field $K$. We fix integers $d$ and $t$. A monomial $x_{i_1}x_{i_2}\cdots x_{i_d}$ with $i_1\leq i_2\leq\dots \leq i_d$ is $t$-spread if $i_j-i_{j-1}\geq t$, for any $2\leq j\leq n$. Let $I_{n,d,t}$ be the ideal generated by all $t$-spread monomials of degree $d$ and let $K[I_{n,d,t}]$ be the toric algebra generated by the monomials $v$ with $v\in G(I_{n,d,t})$. This generalizes the classical (squarefree)Veronese algebras. The aim of this paper is to characterize the algebras $K[I_{n,d,t}]$ which are Gorenstein.

For a prime number $p \equiv 3 \;\mbox{mod $4$}$, we write $p=2n\ell^f+1$ for some power $\ell^f$ of an odd prime number $\ell$ and an odd integer $n$ with $\ell \nmid n$. For $0 \leq t \leq f$, let $K_t$ be the imaginary subfield of ${\Bbb Q}(\zeta_p)$ of degree $2\ell^t$ and let $h_t^-$ be the relative class number of $K_t$. We show that for $n=1$ (resp. $n \geq 3$), a prime number $r$ does not divide the ratio $h_t^-/h_{t-1}^-$ when $r$ is a primitive root modulo $\ell^2$ and $r \geq \ell^{f-t}-1$ (resp. $r \geq (n-2)\ell^{f-t}+1$). In particular, for $n=1$ or $3$, the ratio $h_f^-/h_{f-1}^-$ at the top is not divisible by $r$ whenever $r$ is a primitive root modulo $\ell^2$. Further, we show that the $\ell$-part of $h_t^-/h_{t-1}^-$ stabilizes for ``large" $t$ under some assumption.

We investigate the local differential geometric invariants of cuspidal edge and swallowtail from the view point of singularity theory. We introduce finite type invariants of such singularities (see Remark 1.5 and Theorem 2.11) based on certain normal forms for cuspidal edge and swallowtail. Then we discuss several geometric aspects based on our normal form. We also present several asymptotic formulas concerning our invariants with respect to Gauss curvature and mean curvature.