Related papers: A proof of Carleson's $\varepsilon^2$-conjecture

We consider polynomials that are orthogonal over an analytic Jordan curve L with respect to a positive analytic weight, and show that each such polynomial of sufficiently large degree can be expanded in a series of certain integral…

The subject of limit curve theorems in Lorentzian geometry is reviewed. A general limit curve theorem is formulated which includes the case of converging curves with endpoints and the case in which the limit points assigned since the…

We prove that the generalized Carleson operator with polynomial phase function of degree two is of weak type (2,2). For this, we introduce a new approach to the time-frequency analysis of the quadratic phase.

We show an Iwasawa functional equation for a two dimensional $p$-adic representation of the absolute Galois group of $\mathbf{Q}_p$. This allows us to complete Nakamura's proof of Kato's local $\epsilon$-conjecture in dimension $2$.

We prove $L^p$ bounds, $\frac{d^2 + 4d + 2}{(d+1)^2} < p < 2(d+1)$, for maximal linear modulations of singular integrals along paraboloids with frequencies in certain subspaces of $\mathbb{R}^{d+1}$, for $d \geq 2$. This generalizes…

We solve a randomized version of the following open question: is there a strictly convex, bounded curve \gamma in the plane such that the number of rational points on \gamma, with denominator $n$, approaches infinity with $n$? Although this…

We compute the exact values of the Jordan constants of abelian surfaces over finite fields.

We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger…

This work is devoted to study the asymptotic behavior of critical points $\{(u_\varepsilon,v_\varepsilon)\}_{\varepsilon>0}$ of the Ambrosio-Tortorelli functional. Under a uniform energy bound assumption, the usual $\Gamma$-convergence…

We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its…

We study normal analytic compactifications of C^2 and describe their singularities and configuration of curves at infinity, in particular improving and generalizing results of (Brenton, Math. Ann. 206:303--310, 1973). As a by product we…

We give a formal Laurent series proof of Andrews's $q$-Dyson Conjecture, first proved by Zeilberger and Bressoud.

In this paper a necessary and sufficient condition is deduced for the close-to-convexity of a cross product of Bessel and modified Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about…

We prove an explicit surjectivity result for products of non-isotrivial, non-isogenous elliptic curves over a function field of arbitrary characteristic. This is by way of an isogeny degree bound in this setting, generated from bounds for…

We consider certain determinants with respect to a sufficiently regular Jordan curve $\gamma$ in the complex plane that generalize Toeplitz determinants which are obtained when the curve is the circle. This also corresponds to studying a…

Let $K$ be a number field not containing a CM subfield. For any smooth projective curve $Y/K$ of genus $\geq2$, we prove that the image of the "Selmer" part of Grothendieck's section set inside the $K_v$-rational points $Y(K_v)$ is finite…

We give a characterization of $BMO^\alpha$-martingale spaces by using fractional Carleson measures. We get the boudedness of martingale transform and square function on $BMO^\alpha$-martingale spaces easily by using this characterization.…

We consider the so called Calder\'on problem which corresponds to the determination of a conductivity appearing in an elliptic equation from boundary measurements. Using several known results we propose a simplified and self contained proof…

We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.

Consider the discrete maximal function acting on $\ell^2(\mathbb Z)$ functions \[ \mathcal{C}_{\Lambda} f( n ) := \sup_{ \lambda \in \Lambda} \left| \sum_{m \neq 0} f(n-m) \frac{e^{2 \pi i\lambda m^2}} {m} \right| \] where $\Lambda \subset…