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Using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques, L^p boundedness theorems for p > 2 are obtained for maximal operators over a wide range of…

We study the universal family of odd hyperelliptic curves of genus $g \geq 1$ over $\mathbb{Q}$. We relate the heights of $\mathbb{Q}$-points of Jacobians of curves in this family to the reduction theory of the representation of…

In this note we construct a "restriction" map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that…

In this paper, we introduce and characterize a class of parabolically extended structures for relatively hyperbolic groups. A characterization of relative quasiconvexity with respect to parabolically extended structures is obtained using…

We prove an asymptotic formula for a special case of the Gauss hypergeometric function which arises in explicit formulas for moments of Maass form symmetric square L-functions. The resulting formula is uniform in several variables, which is…

Let $X,X_1,X_2,\ldots$ be i.i.d. ${\mathbb{R}}^d$-valued real random vectors. Assume that ${\mathbf{E}X=0}$, $\operatorname {cov}X=\mathbb{C}$, $\mathbf{E}\Vert X\Vert^2=\sigma ^2$ and that $X$ is not concentrated in a proper subspace of…

We prove two "large images" results for the Galois representations attached to a degree $d$ Q-curve $E$ over a quadratic field $K$: if $K$ is arbitrary, we prove maximality of the image for every prime $p >13$ not dividing $d$, provided…

The adjoint Fourier restriction inequality for the sphere $S^2$ states that if $f\in\lt(S^2,\sigma)$ then $\widehat{f\sigma}\in L^4(\reals^3)$. We prove that all critical points $f$ of the functional…

In this paper we consider the question of when a strongly regular graph with parameters $((s+1)(st+1),s(t+1),s-1,t+1)$ can exist. These parameters arise when the graph is derived from a generalized quadrangle, but there are other examples…

Let $\overline{\rho}: G_{\mathbf{Q}} \rightarrow {\rm GSp}_4(\mathbf{F}_3)$ be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equivalent, the Galois representation associated to the…

In this note, we continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid $z=xy,$ which are of the form $z=xy+h(y),$ where $h(y)$ is a smooth function of finite type.…

In the present paper we derive Liouville type results and existence of periodic solutions for $\chi^{(2)}$ type systems with non-homogeneous nonlinearities. Moreover, we prove both universal bounds as well as singularity and decay estimates…

We study a class of degenerate parabolic and elliptic equations in divergence form in the upper half space $\{x_d>0\}$. The leading coefficients are of the form $x_d^2a_{ij}$, where $a_{ij}$ are bounded, uniformly elliptic, and measurable…

A birationally liftable Galois section s of a hyperbolic curve X/k over a number field k yields an adelic point x(s) in the smooth completion of X. We show that x(s) is X-integral outside a set of places of Dirichlet density 0, or s is…

We study the properties of the principal symbol of the 3+1 equations of motion in Teleparallel Equivalent of General Relativity (TEGR) and assess the conditions for hyperbolicity. We use the Hamiltonian formulation based on the vectorial,…

We prove that relatively hyperbolic groups do not have Lafforgue strong Property $(T)$ with respect to Hilbert spaces. To do so we construct an unbounded affine representation of such groups, whose linear part is of polynomial growth of…

In this paper, we explore the geometric properties of unbounded extremal domains for the $p$-Laplacian operator in both Euclidean and hyperbolic spaces. Assuming that the nonlinearity grows at least as the nonlinearity of the eigenvalue…

We consider the Gauss curvature type flow for uniformly convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}\ (n\geqslant 2)$. We prove that if the initial closed hypersurface is smooth and uniformly convex, then the smooth…

In this paper, we establish Schr\"{o}dinger maximal estimates associated with the finite type phases \begin{equation*} \phi(\xi_1,\xi_2):=\xi^m_1+\xi^m_2,\;(\xi_1,\xi_2)\in [0,1]^2, \end{equation*} where $m \geq 4$ is an even number.…

In this paper, we theoretically and numerically verify that the discontinuous Galerkin (DG) methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the…