The classical four-vertex theorem describes a fundamental property of simple closed planar curves. It has been extended to space curves, namely a smooth, simple closed curve in $\mathbb{R}^3$ has at least four points with vanishing torsion if it lies on a convex surface. More recently, Ghomi [6] extended this property to curves lying on locally convex surfaces. In this paper we provide an alternative

In [CVX3] (T. H. Chen, K. Vilonen, and T. Xue, “Springer correspondence for the split symmetric pair in type A”, Compos. Math. 154 (2018), no. 11, 2403–2425), we have established a Springer theory for the symmetric pair $(\operatorname{SL}(N), \operatorname{SO}(N))$. In this setting we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These

In this short note, we prove that a complex Finsler vector bundle with positive Kobayashi curvature must be ample, which partially solves a problem of S. Kobayashi posed in 1975. As applications, a strongly pseudoconvex complex Finsler manifold with positive Kobayashi curvature must be biholomorphic to the complex projective space; we also show that all Schur polynomials are numerically positive for

The Eisenbud–Green–Harris (EGH) conjecture states that a homogeneous ideal in a polynomial ring $K[x_1, \dotsc, x_n]$ over a field $K$ that contains a regular sequence $f_1, \dotsc , f_n$ with degrees $a_i, i =1, \dotsc , n$ has the same Hilbert function as a lex-plus-powers ideal containing the powers $x^{a_i}_i , i = 1, \dotsc , n$. In this paper, we discuss a case of the EGH conjecture for homogeneous

Let $f : X \to Y$ be a proper, dominant morphism of smooth varieties over a number field $k$. When is it true that for almost all places $v$ of $k$, the fibre $X_P$ over any point $P \in Y (k_v)$ contains a zero-cycle of degree $1$? We develop a necessary and sufficient condition to answer this question. The proof extends logarithmic geometry tools that have recently been developed by Denef and Lo

We prove that termination of lower dimensional flips for generalized $\operatorname{klt}$ pairs implies termination of flips for $\operatorname{log}$ canonical generalized pairs with a weak Zariski decomposition. Under the same hypothesis we prove that the existence of weak Zariski decompositions for pseudo-effective log canonical pairs implies the existence of weak Zariski decompositions for pseudo-effective

We describe an operation which modifies a Lagrangian submanifold $L$ in a symplectic manifold $(M, \omega)$ such as to produce a new immersed Lagrangian submanifold $L^\prime$, which as a smooth manifold is obtained by surgery along a framed sphere in $L$. Intuitively, this can be described as collapsing an isotropic disc with boundary on $L$ to a point. The inverse operation generalizes classical

Given a smooth variety $X$ with an action of a finite group $G$, and a semiorthogonal decomposition of the derived category, $\mathcal{D}([X/G])$, of $G$-equivariant coherent sheaves on $X$ into subcategories equivalent to derived categories of smooth varieties, we construct a similar semiorthogonal decomposition for a smooth $G$-invariant divisor in $X$ (under certain technical assumptions). Combining

We show that any one-relator group $G = F / {\langle \negthinspace \langle} w {\rangle \negthinspace \rangle}$ with torsion is coherent — i.e., that every finitely generated subgroup of $G$ is finitely presented — answering a 1974 question of Baumslag in this case.

Let $M$ be a compact Riemannian manifold without boundary. We investigate the integrals of $L^2$-normalized Laplace eigenfunctions over closed submanifolds. General bounds for these quantities were obtained by Zelditch [23], and are sharp in the case where $M$ is the standard sphere. However, as with sup norms of eigenfunctions, there are many interesting settings where improvements can be made to

The goal of this paper is to study the subspace of stability condition $\mathcal{E} \subset \operatorname{Stab}(X)$ associated to an exceptional collection $\mathcal{E}$ on a projective variety $X$. Following Macrì’s approach, we show a certain correspondence between the homotopy class of continuous loops in $\Sigma_\mathcal{E}$ and words of the braid group. In particular, we prove that in the case

We prove the nonvanishing of the twisted central critical values of a class of automorphic $L$‑functions for twists by all but finitely many unitary characters in particular infinite families. While this paper focuses on $L$‑functions associated to certain automorphic representations of unitary groups, it illustrates how decades-old non-archimedean methods from Iwasawa theory can be combined with the

Given a three-dimensional projective log canonical pair over a perfect field of characteristic larger than five, there exists a minimal model program that terminates after finitely many steps.

We prove that the entropy map for countable Markov shifts of finite entropy is upper semi-continuous on the set of ergodic measures. Note that the phase space is non-compact. We also discuss the related problem of existence of measures of maximal entropy.

Let $S$ be a Burniat surface with $K^2_S = 6$. Then we show that $\operatorname{glct}(S, K_S) = \frac{1}{2}$ by showing that $\operatorname{glct}(S, 2K_S) = \operatorname{lct} (S, E) = \frac{1}{4}$ for some divisor $E \in \lvert 2K_S \rvert$. This implies that Tian’s conjecture (which fails in general) holds for the polarized pair $(S, 2K_S)$, since the corresponding graded algebra is generated by

We prove that the mapping class group of a closed connected orientable surface of genus at least eight is generated by three involutions.

We completely calculate the $RO(G)$-graded coefficients of ordinary equivariant cohomology where $G$ is the dihedral group of order $2p$ for a prime $p \gt 2$ both with constant and Burnside ring coefficients. The authors first proved it for $p = 3$ and then the second author generalized it to arbitrary $p$. These are the first such calculations for a non-abelian group.

We prove that the following endpoint Kato–Ponce inequality holds:\[{\lVert D^s (fg) \rVert}_{L^{\frac{q}{q+1}} (\mathbb{R}^n)}\lesssim{\lVert D^s f \rVert}_{L^1 (\mathbb{R}^n)}{\lVert g \rVert}_{L^q (\mathbb{R}^n)} \\+{\lVert f \rVert}_{L^1 (\mathbb{R}^n)}{\lVert D^s g \rVert}_{L^q (\mathbb{R}^n)}\; \textrm{,}\]for all $1 \leq q \leq \infty$, provided $s \gt n/q$ or $s \in 2 \mathbb{N}$. Endpoint estimates

In this paper we study submanifolds of contact manifolds. The main submanifolds we are interested in are contact coisotropic submanifolds. They can be viewed as analogues to symplectic contact coisotropic submanifolds, and can be defined by the symplectic complement with respect to the symplectic structure $d \alpha \vert_\xi$, the restriction of $d \alpha$ on the contact hyperplane field $\xi$. Based

In this paper, we prove some differentiable sphere theorems and topological sphere theorems for submanifolds in Kähler manifold, especially in complex space forms.

We prove that any non-isotrivial elliptic K3 surface over an algebraically closed field $k$ of arbitrary characteristic contains infinitely many rational curves. In the case when $\operatorname{char} (k) \neq 2, 3$, we prove this result for any elliptic K3 surface. When the characteristic of $k$ is zero, this result is due to the work of Bogomolov–Tschinkel and Hassett.

On a compact Kähler manifold with semi-ample canonical line bundle and Kodaira dimension one, we observe a relation between the infinite-time singularity type of the Kähler–Ricci flow and the characteristic indexes of singular fibers of the semi-ample fibration.

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