We prove that the degrees of the iterates $\deg (f^n)$ of a birational map satisfy $\liminf(\deg (f^n)) \lt + \infty$ if and only if the sequence $\deg (f^n)$ is bounded, and that the growth of $\deg (f^n)$ cannot be arbitrarily slow, unless $\deg (f^n)$ is bounded.

We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrödinger equation\[i \partial_t u + \partial^2_x u + i {\lvert u \rvert}^{2 \sigma} \partial_x u = 0.\]The equation has a two-parameter family of solitary wave solutions of the form\[\phi_{\omega , c} (x) = \varphi_{\omega ,c} (x) \exp\left \lbrace i \frac{c}{2} x - \frac{i}{2 \sigma + 2} \int^x_{-\infty}\varphi^{2

We show Fujita’s spectrum conjecture for $\epsilon \textrm{-}\log$ canonical pairs and Fujita’s log spectrum conjecture for log canonical pairs. Then, we generalize the pseudo-effective threshold of a single divisor to multiple divisors and establish the analogous finiteness and the DCC properties.

In this paper we study the bilinear multiplier operator of the form \begin{align*}&H^t(f,g)(x)=\int_{\mathbb{R}^d}\!\int_{\mathbb{R}^d} m(t\xi,t\eta)\,\mathrm{e}^{2 \pi \mathrm{i} t|(\xi,\eta)|}\, \widehat{f}(\xi)\, \widehat{g}(\eta)\, \mathrm{e}^{2 \pi \mathrm{i}x(\xi +\eta)}\,d \xi d \eta, \\&1 \le t \le 2\end{align*}where $m$ satisfies the Marcinkiewicz–Mikhlin–Hörmander’s derivative conditions

The purpose of this article is to give a simple and explicit construction of mock modular forms whose shadows are Eisenstein series of arbitrary integral weight, level, and character. As application, we construct forms whose shadows are Hecke’s Eisenstein series of weight one associated to imaginary quadratic fields, recovering some results by Kudla, Rapoport and Yang (1999), and Schofer (2009), and

Let $M$ be a geometrically finite rank one locally symmetric manifolds. We prove that the spectrum of the Laplace operator on $M$ is finite in a small interval which is optimal.

Let $X$ be a smooth projective curve of genus $g \geq 2$ over an algebraically closed field $k$ of characteristic $p \gt 0 , \mathfrak{M}^s_X (r, d)$ the moduli space of stable vector bundles of rank $r$ and degree $d$ on $X$. We study the Frobenius stratification of $\mathfrak{M}^s_X (3, d)$ in terms of Harder–Narasimhan polygons of Frobenius pull-backs of stable vector bundles and obtain the irreducibility

Let $X$ be a projective variety admitting a polarized (or more generally, int‑amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is $\mathbb{Q}$-factorial normal, every minimal model program is equivariant relative to the monoid $\mathrm{SEnd}(X)$ of all surjective endomorphisms, up to finite index. Further, when $X$ is rationally connected and smooth

In this paper we obtain a finite set $S$ of separating invariants for the variety of Hopf algebras of a fixed dimension. In dimension $p^2$ where $p$ is a prime or when dimension is $\lt 18$, except $8, 12, 16$, these invariants determine isomorphism classes of Hopf algebras, i.e., two Hopf algebras of a given dimension are isomorphic if and only if each of the invariants in $S$ take the same values

In this short article, given a smooth diagonalizable group scheme $G$ of finite type acting on a smooth quasi-compact separated scheme $X$, we prove that (after inverting some elements of representation ring of $G$) all the information concerning the additive invariants of the quotient stack $[X/G]$ is “concentrated” in the subscheme of $G$-fixed points $X^G$. Moreover, in the particular case where

We establish $\ell^p (\mathbb{Z})$ boundedness of $r$-variational seminorm for operators of Radon type along subsets of prime numbers of the form $\lbrace p \in \mathbb{P} : \lbrace \varphi_1 (p) \rbrace \lt \psi(p) \rbrace$ As an application we obtain the corresponding pointwise ergodic theorems.

Starting from a suitable set of self-diffeomorphisms of a closed Riemann surface, we present a general branched covering method to construct surface bundles over surfaces with positive signature. Armed with this method, we study the classification problem for both surface bundles with nonzero signature and closed simply connected smooth $\operatorname{spin} 4$-manifolds.

This article establishes, in an analytic framework and in two dimensions, the first WKB constructions describing the eigenfunctions of the pure magnetic Laplacian with low energy when the magnetic field has a unique minimum that is positive and non-degenerate.

In this paper, we develop the theory of Perelman’s $W$-functional on manifolds with isolated conical singularities. In particular, we show that the infimum of $W$-functional over a certain weighted Sobolev space on manifolds with isolated conical singularities is finite, and the minimizer exists, if the scalar curvature satisfies certain condition near the singularities. We also obtain an asymptotic

We give new lower bounds for $L^p$ estimates of the Schrödinger maximal function by generalizing an example of Bourgain.

We prove that for a polynomial diffeomorphism of $\mathbb{C}^2$, uniform hyperbolicity on the set of saddle periodic points implies that saddle points are dense in the Julia set. In particular $f$ satisfies Smale’s Axiom A on $\mathbb{C}^2$.

We define an annular version of odd Khovanov homology and prove that it carries an action of the Lie superalgebra $\mathfrak{gl}(1 \vert 1)$ which is preserved under annular Reidemeister moves.

We prove upper bounds for the Bergman kernels associated to tensor powers of a smooth positive line bundle in terms of the rate of growth of the Taylor coefficients of the Kähler potential. As applications, we obtain improved off-diagonal rate of decay for the classes of analytic, quasi-analytic, and more generally Gevrey potentials.

We prove that the exponential decay rate in expectation is well defined and is equal to the Lyapunov exponent, for supercritical almost Mathieu operators with Diophantine frequencies.

We consider an involution on the affine Weyl group of type $A$ induced from the nontrivial automorphism on the (finite) Dynkin diagram. We prove that the number of left cells fixed by this involution in each two-sided cell is given by a certain Green polynomial of type $A$ evaluated at $-1$.

In this paper, we exhibit an example of a smooth proper variety in positive characteristic possessing two liftings with different Hodge numbers.

We completely solve the inverse Galois problem for del Pezzo surfaces of degree $2$ and $3$ over all finite fields.

Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the Bartnik mass of such surfaces, without assumptions on the integral of the squared mean curvature. If the ambient manifold has negative scalar curvature, our method also

Let $R = k[x_1, \dotsc , x_n]$ be a polynomial ring over a field $k$ of characteristic zero and $\widehat{R}$ be the formal power series ring $k[[x_1, \dotsc , x_n]]$. If $M$ is a $\mathcal{D}$-module over $R$, then $\widehat{R} \otimes_R M$ is naturally a $\mathcal{D}$-module over $\widehat{R}$. Hartshorne and Polini asked whether the natural maps $H^i_{\mathrm{dR}} (M) \to H^i_{\mathrm{dR}} (\widehat{R}

This paper studies a class of Abelian varieties that are of $\mathrm{GL}_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (1) a totally real field $K$ or (2) a totally indefinite quaternion algebra over a totally real field $K$. Among the isogenous class of such an Abelian variety, we identify one whose Galois conjugates