abstract: The roots of random polynomials and of random polynomial systems have been extensively studied in the past. While the picture is quite complete in the case of polynomials, the case of systems is much harder. Kostlan, Shub, and Smale random polynomial systems were introduced in the nineties and the expectation of the number of their real roots is known since then. Only recently the asymptotic

abstract: This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\Bbb{R}^2$ under the $\alpha$-curve shortening flow for exponents $\alpha>{1\over 2}$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $\alpha$-curve shortening flow to the unique translating soliton whose ends are asymptotic to

abstract: We show that if a compact, connected, and oriented $n$-manifold $M$ without boundary admits a non-constant non-injective uniformly quasiregular self-map, then the dimension of the real singular cohomology ring $H^*(M;\Bbb{R})$ of $M$ is bounded from above by $2^n$. This is a positive answer to a dynamical counterpart of the Bonk-Heinonen conjecture on the cohomology bound for quasiregularly

abstract: Answering a question posed by Conway and Norton in their seminal 1979 paper on moonshine, we prove the existence of a graded infinite-dimensional module for the sporadic simple group of O'Nan, for which the McKay-Thompson series are weight $3/2$ modular forms. The coefficients of these series may be expressed in terms of class numbers, traces of singular moduli, and central critical values

abstract: In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our construction converts certain graphs in Euclidean $(n+1)$-space to CMC $n$-hypersurfaces with asymptotically Delaunay ends in two steps: First appropriate

abstract: We study infinite time blow-up phenomenon for the half-harmonic map flow $$ \casesno{ u_t=-(-\Delta)^{{1\over 2}}u+\bigg({1\over 2\pi}\int_{\Bbb{R}}{|u(x)-u(s)|^2\over |x-s|^2}ds\bigg)u&\quad {\rm in}\ \Bbb{R}\times(0,\infty),\cr u(\cdot,0)=u_0&\quad {\rm in}\ \Bbb{R}, } $$ for a smooth function $u:\Bbb{R}\times [0,\infty)\to\Bbb{S}^1$. Let $q_1,\ldots,q_k$ be distinct points in $\Bbb{R}$

IN MEMORIAM: J. MICHAEL BOARDMAN 1938–2021 Photo courtesy of JHU Mathematics J. Michael (“Mike”) Boardman was a Professor Emeritus in the Mathematics Department of the Johns Hopkins University, and a leading expert in homotopy theory, a field in which he is most widely known for his construction of the first rigorously correct model of the homotopy category of spectra. Professor Boardman was born in

abstract: We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition.

abstract: We develop Hida theory for Shimura varieties of type~A without ordinary locus. In particular we show that the dimension of the space of ordinary forms is bounded independently of the weight and that there is a module of $\Lambda$-adic cuspidal ordinary forms which is of finite type over $\Lambda$, where $\Lambda$ is a twisted Iwasawa algebra.

abstract: We develop a new abstract derivation of the observability inequalities at two points in time for Schr\"{o}dinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator $H$ on $L^2(\Bbb{R}^n)$. In the second step we use results on asymptotic behavior of $e^{-itH}$, in particular

abstract: We establish results on unique continuation at the boundary for the solutions of real analytic elliptic partial differential equations of the form $Lu=\Delta u+a(x,y)\partial_{x}u+b(x,y)\partial_{y}u+c(x,y)u=0$ in two dimensions.

abstract: We prove that the Gromov-Witten theory (GWT) of a projective bundle can be determined by the Chern classes and the GWT of the base. It completely answers a question raised in a previous paper. Its consequences include that the GWT of the blow-up of X at a smooth subvariety Z is uniquely determined by GWT of X, Z plus some topological data.

abstract: In this paper we study the class of compact K\"ahler manifolds with ${\rm Ric}^\perp>0$. First we illustrate examples of K\"ahler manifolds with ${\rm Ric}^\perp>0$ on K\"ahler C-spaces, and construct ones on certain projectivized vector bundles. These examples show the abundance of K\"ahler manifolds which admit metrics of ${\rm Ric}^\perp>0$. Secondly we prove some (algebraic) geometric

abstract: We prove explicit rationality-results for Asai- $L$-functions, $L^S(s,\Pi',{\rm As}^\pm)$, and Rankin-Selberg $L$-functions, $L^S(s,\Pi\times\Pi')$, over arbitrary CM-fields $F$, relating critical values to explicit powers of $(2\pi i)$. Besides determining the contribution of archimedean zeta-integrals to our formulas as concrete powers of $(2\pi i)$, it is one of the advantages of our approach

abstract: We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain $X\subset\Bbb{R}^{2m-1}$, we find geometric significance in the function $\scr{M}_X(R)={\rm mag}(R\cdot X)$. The function $\scr{M}_X$ extends from the positive half-line to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit

abstract: The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter $\alpha$ in the regime connecting the first nontrivial Neumann and Steklov eigenvalues, and even somewhat beyond the Steklov regime. The result is close to optimal, since the ball is not maximal when $\alpha$ is sufficiently large negative

abstract: We prove that the spectral radius of an i.i.d. random walk on ${\rm GL}_d(\Bbb{C})$ satisfies a strong law of large numbers under finite second moment assumption and a weak law of large numbers under finite first moment. No irreducibility assumption is supposed.

abstract: This brief note describes an error in the following paper: Multiparameter Riesz commutators, {\it Amer. J. Math.} {\bf 131} (2009), no. 3, 731--769. A correction of this error seems to require new ideas, and has not been produced as of this note.

abstract: We consider the prehomogeneous vector space of pairs of ternary quadratic forms. For the lattice of pairs of integral ternary quadratic forms and its dual lattice, there are six zeta functions associated with the the prehomogeneous vector space. We present a conjecture which states that there are simple relations among the six zeta functions. We prove that the coefficients coincide on fundamental

abstract: We construct the three-variable $p$-adic triple product $L$-functions attached to Hida families of elliptic newforms and prove the explicit interpolation formulae at all critical specializations by establishing explicit Ichino's formulae for the trilinear period integrals of automorphic forms. Our formulae perfectly fit the conjectural shape of $p$-adic $L$-functions predicted by Coates and

abstract: We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the more general class of non-trapping compact Riemannian surfaces with no conjugate points, we show lens rigidity. We also prove the injectivity of the $X$-ray

abstract: We prove upper and lower bounds for the number of lines in general position that are rich in a Cartesian product point set. This disproves a conjecture of Solymosi and improves work of Elekes, Borenstein and Croot, and Amirkhanyan, Bush, Croot, and Pryby. The upper bounds are based on a version of the asymmetric Balog-Szemer\'{e}di-Gowers theorem for {\it group actions} combined with product

abstract: We prove that solutions of the cubic nonlinear Schr\"odinger equation on $\Bbb{R}^2$ can be approximated by a finite-dimensional Hamiltonian system, uniformly on bounded sets of initial data. This is despite the wealth of non-compact symmetries: scaling, translation, and Galilei boosts. Complementing this approximation result, we show that all solutions of the finite-dimensional Hamiltonian

abstract: We develop a theory of $p$-adic automorphic forms on unitary groups that allows $p$-adic interpolation in families and holds for all primes $p$ that do not ramify in the reflex field $E$ of the associated unitary Shimura variety. If the ordinary locus is nonempty (a condition only met if $p$ splits completely in $E$), we recover Hida's theory of $p$-adic automorphic forms, which is defined

abstract: Let $\Omega\subset\Bbb{R}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $\Omega=-\Omega$ with fixed Gaussian volume. Let $A=A_x$ be the second fundamental form of $\partial\Omega$ at $x$, i.e., $A$ is the matrix of first order partial derivatives of the unit normal vector at $x\in\partial\Omega$. For any $x=(x_1,\ldots,x_{n+1})\in\Bbb{R}^{n+1}$, let $\gamma_n(x)=(2\

abstract: We define a new condition on relatively hyperbolic Dehn filling which allows us to control the behavior of a relatively quasiconvex subgroups which need not be full. As an application, in combination with recent work of Cooper and Futer, we provide a new proof of the virtual fibering of non-compact finite-volume hyperbolic $3$-manifolds, a result first proved by Wise. Additionally, we explain

abstract: Let $p$ be a fixed odd prime number, $\mu$ be a Hida family over the Iwasawa algebra of one variable, $\rho_{\mu}$ its Galois representation, $\Bbb{Q}_\infty/\Bbb{Q}$ the $p$-cyclotomic tower and $S$ the variable of the cyclotomic Iwasawa algebra. We compare, for $n\leq 4$ and under certain assumptions, the characteristic power series $L(S)$ of the dual of Selmer groups $\textrm{Sel}(\Bbb{Q}_{\infty}

abstract: We prove some weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates. As application we obtain improved spherical average decay rates of the Fourier transform of fractal measures, and therefore improve the results for the Falconer distance set conjecture in three and higher dimensions.

abstract: We prove a generalized version of Schmidt's subspace theorem for closed subschemes in general position in terms of suitably defined Seshadri constants with respect to a fixed ample divisor. Our proof builds on previous work of Evertse and Ferretti, Corvaja and Zannier, and others, and uses standard techniques from algebraic geometry such as notions of positivity, blowing-ups and direct image

abstract: We study geometric properties of the Lagrangian self-shrinking tori in $\Bbb{R}^4$. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a {\L}ojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori and then combining with a compactness theorem

abstract: On a bounded strictly pseudoconvex domain in $\Bbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local curvature invariant of the boundary. For bounded strictly pseudoconvex domains in $\Bbb{C}^2$ which are diffeomorphic to the ball, we motivate and consider the problem of determining whether the global

abstract: In this paper we obtain some extensions of the classical Krylov-Safonov Harnack inequality. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. We require that at scale larger than some $r_0>0$ (small) the functions satisfy the comparison principle with a standard family of quadratic polynomials, while

Abstract: We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\overline{\sigma}$ with $\overline{\sigma}^{{\rm ss}}=\chi^{k-2}\oplus\rho\oplus\chi^{k-1}$, where $\chi$ is the mod $p$ cyclotomic character and $\rho$ is an absolutely irreducible reduction of the Galois representation $\rho_f$ attached to a cusp form $f$ of weight

Abstract: We establish optimal Lebesgue estimates for a class of generalized Radon transforms defined by averaging functions along polynomial-like curves. The presence of an essentially optimal weight allows us to prove uniform estimates, wherein the Lebesgue exponents are completely independent of the curves and the operator norms depend only on the polynomial degree. Moreover, our weighted estimates

Abstract: Let $p$ be a prime, $K$ a finite extension of $\\Bbb\{Q\}_p$, and let $G_K$ be the absolute Galois group of $K$. The category of \\'etale $(\\varphi,\\tau)$-modules is equivalent to the category of $p$-adic Galois representations of $G_K$. In this paper, we show that all \\'etale $(\\varphi,\\tau)$-modules are overconvergent; this answers a question of Caruso. Our result is an analogue of

Abstract: We study the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of \{\\it a priori\} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of the area and the height in terms of the boundary. In case that the boundary is a circle, we study under what conditions the surface is rotational. Finally

Abstract: Let $X$ be a smooth projective variety defined over $\\overline\{\\Bbb\{Q\}\}$, and $f\\colon X\\dashrightarrow X$ be a dominant rational map. Let $\\delta_f$ be the first dynamical degree of $f$ and $h_X\\colon X(\\overline\{\\Bbb\{Q\}\})\\rightarrow [1,\\infty)$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds

Abstract: We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^\{n-1\}$-estimates for the regularity scale of the level set flow with two-convex initial data. The results, which seem new even in the most classical case of mean convex surfaces evolving by mean curvature

Abstract: We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of self-shrinkers of the mean curvature flow asymptotic to a given cone

Abstract: In an earlier paper, one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincar\\'e duality groups for each $n\\geq 4$. We show that those groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\\geq 4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous

Abstract: We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $|Q(x)-\\xi|<\\epsilon$ for a generic ternary form $Q$ is $|x|\\ll\\epsilon^\{-1\}$. We also establish an optimal rate of density for the values of polynomials maps in a number of other natural problems, including the values of linear forms restricted to suitable

Abstract: We obtain compact orientable embedded surfaces with constant mean curvature $0

AMERICAN JOURNAL OF MATHEMATICS Founded in 1878 by Johns Hopkins University INDEX TO VOLUME 142 2020 PAGE BECEANU, M. and W. SCHLAG. Structure formulas for wave operators . 751 BERGER, TOBIAS, KRZYSZTOF KLOSIN, CRIS POOR, JERRY SHURMAN , and DAVID S. YUEN. Deformations of Saito-Kurokawa type and the Paramodular Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1821 BERNDTSSON

Abstract: We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=\xi_0+\xi_1 x+\cdots+{1\over\sqrt{n!}}\xi_n x^n$, where $\xi_i$ are iid Gaussian random variables. The main ingredients in the proof are new estimates for the correlation functions of the real roots of $P_n$ and a comparison argument exploiting local laws and repulsion properties of these real

Abstract: Let $F$ be a number field and let $\Bbb{A}_F$ be its ring of adeles. Let $B$ be a quaternion algebra over $F$ and let $\nu:B\to F$ be the reduced norm. Consider the reductive monoid $M$ over $F$ whose points in an $F$-algebra $R$ are given by $$ M(R):=\big\{\big(\gamma_1,\gamma_2\big)\in\big(B\otimes_F R\big)^2:\nu\big(\gamma_1\big)=\nu\big(\gamma_2\big)\big\}. $$ Motivated by an influential

Abstract: The primal cohomology $\Bbb{K}_{\Bbb{Q}}$ of the theta divisor $\Theta$ of a principally polarized abelian fivefold (ppav) is the direct sum of its invariant and anti-invariant parts $\Bbb{K}_{\Bbb{Q}}^{+1}$, resp. $\Bbb{K}_{\Bbb{Q}}^{-1}$ under the action of $-1$. For smooth $\Theta$, these have dimension $6$ and $72$ respectively. We show that $\Bbb{K}_{\Bbb{Q}}^{+1}$ consists of Hodge

Abstract: We consider the nonlinear heat equation $u_t-\Delta u=|u|^\alpha u$ on $\Bbb{R}^N$, where $\alpha>0$ and $N\ge 1$. We prove that in the range $0<\alpha<{4\over {N-2}}$, for every $\mu>0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0(x)=\mu|x|^{-{2\over\alpha}}$. The construction is based on the analysis of the related inverted

Abstract: In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension $3$. We prove that the set of initial data for which solutions scatter to an unstable excited state $(\phi,0)$ forms a finite co-dimensional path connected $C^1$ manifold in the energy space. This manifold is a global and unique center-stable manifold associated with $(\phi,0)$. It

Abstract: We consider a rigidity problem for the spectral gap of the Laplacian on an ${\rm RCD}(K,\infty)$-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive $K$. For a weighted Riemannian manifold, Cheng-Zhou showed that the sharp spectral gap is achieved only when a $1$-dimensional Gaussian space is split off. This can be regarded as an infinite-dimensional

Abstract: Let $p$ be a prime number. Every $n$-variable polynomial $f(\underline{x})$ over a finite field of characteristic $p$ defines an Artin-Schreier-Witt tower of varieties whose Galois group is isomorphic to $\Bbb{Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a nontrivial finite character of $\Bbb{Z}_p$ and a generic polynomial whose convex hull

Abstract: We give a simple sufficient condition for a Verdier quotient $\scr{T}/\scr{S}$ of a triangulated category $\scr{T}$ by a thick subcategory $\scr{S}$ to be realized inside of $\scr{T}$ as an ideal quotient. As applications, we deduce three significant results by Buchweitz, Orlov and Amiot-Guo-Keller.

Let $\pi : E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P)\not=0$. In this article, we show that the function $t \mapsto \hat{h}_{E_t}(P_t)$ on $B(\overline{K})$ is the height induced from an adelically metrized line bundle with non-negative curvature on $B$

We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\overline{\sigma}$ with $\overline{\sigma}^{\rm ss}=\chi^{k-2} \oplus \rho \oplus \chi^{k-1}$, where $\chi$ is the mod $p$ cyclotomic character and $\rho$ is an absolutely irreducible reduction of the Galois representation $\rho_f$ attached to a cusp form $f$ of weight $2k-2$.

We prove that any complete immersed two-sided mean convex translating soliton Σ ⊂ R for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in R is the axisymmetric “bowl soliton”. We also show that if the mean curvature of Σ tends to zero at infinity, then Σ can be represented as an entire graph and so is the “bowl soliton”. Finally

Given a differential equation on a smooth $p$-adic analytic curve, one may construct a new one by pushing forward by an \'etale morphism. The main result of the paper provides an explicit formula that relates the radii of convergence of the solutions of the two differential equations using invariants coming from the topological behavior of the morphism. We recover as particular cases the known formulas

Abstract:We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of self-shrinkers of the mean curvature flow asymptotic to a given cone

Abstract:Let $X$ be a smooth projective variety defined over $\\overline\{\\Bbb\{Q\}\}$, and $f\\colon X\\dashrightarrow X$ be a dominant rational map. Let $\\delta_f$ be the first dynamical degree of $f$ and $h_X\\colon X(\\overline\{\\Bbb\{Q\}\})\\rightarrow [1,\\infty)$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds

Abstract:We establish optimal Lebesgue estimates for a class of generalized Radon transforms defined by averaging functions along polynomial-like curves. The presence of an essentially optimal weight allows us to prove uniform estimates, wherein the Lebesgue exponents are completely independent of the curves and the operator norms depend only on the polynomial degree. Moreover, our weighted estimates

Abstract:In an earlier paper, one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincar\\'e duality groups for each $n\\geq 4$. We show that those groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\\geq 4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous

Abstract:We study the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of \{\\it a priori\} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of the area and the height in terms of the boundary. In case that the boundary is a circle, we study under what conditions the surface is rotational. Finally