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Let $f$ be a primitive form of weight $2k+j-2$ for $SL_2(Z)$, and let $\mathfrak p$ be a prime ideal of the Hecke field of $f$. We denote by $SP_m(Z)$ the Siegel modular group of degree $m$. Suppose that $k \equiv 0 \mod 2, \ j \equiv 0…

Supplying the missing necessary conditions, we complete the characterisation of the $L^p\to L^q$ boundedness of commutators $[b,T]$ of pointwise multiplication and Calder\'on-Zygmund operators, for arbitrary pairs of $1<p,q<\infty$ and…

We consider the following fully parabolic Keller--Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+ au-\mu u^2,\quad x\in \Omega, t>0, \disp{v_t=\Delta v- v +u},\quad x\in \Omega, t>0,…

This note contains a solution to the following problem: reconstruct the definition field and the equation of a projective cubic surface, using only combinatorial information about the set of its rational points. This information is encoded…

Let $F/\mathbb{Q}_p$ be a finite extension. We explore the universal supersingular mod $p$ representations of $\mathrm{GL}_2(F)$ through computing a basis of their invariant space under the pro-$p$ Iwahori subgroup. This generalizes works…

We study the arithmetic of certain del Pezzo surfaces of degree 2. We produce examples of Brauer-Manin obstruction to the Hasse principle, coming from 2- and 4-torsion elements in the Brauer group.

We prove a local smoothing result for the Schr\"odinger equation on a class of surfaces of revolution which have infinitely many trapped geodesics. Our main result is a local smoothing estimate with loss (compared to \cite{ChMe-lsm})…

We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of $$|\nabla u|^{\alpha}F(D^{2}u)=f $$ and, more generally, for viscosity supersolutions of $|\nabla u|^{\alpha}\,{M}^-_{\lambda,\Lambda}(D^{2}u)\le f$. The…

We establish weighted inequalities for $BMO$ commutators of sublinear operators for all $0<p<\infty$. For weights $w$ satisfying the doubling condition of order $q$ with $0<q<p$ and the reverse H\"{o}lder condition, we prove that $\bullet$…

In this paper we prove the 2-local part of the Beilinson-Lichtenbaum conjectures on tosion in motivic cohomology. In particular we prove the Milnor conjecture relating Milnor's K-theory and the Galois cohomology with Z/2-coefficients. This…

We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by…

In this paper, we consider the Cauchy problem of the cubic nonlinear Schr\"{o}dinger equation with derivative in $H^s(\R)$. This equation was known to be the local well-posedness for $s\geq \frac12$ (Takaoka,1999), ill-posedness for…

This is a survey focusing on the Hasse principle for divisibility of points in commutative algebraic groups and its relation with the Hasse principle for divisibility of elements of the Tate-Shavarevich group in the Weil-Ch\^{a}telet group.…

We initiate a study of the local invariant cycle theorem with integral coefficients for 1-parameter semistable families of varieties. We show that it always holds for $H^1$, and it holds for $H^2$ if the general fiber has trivial Albanese…

We establish two global subellipticity properties of positive symmetric second-order partial differential operators on $L_2(\Ri^d)$. First, if $m \in \Ni$ then we consider operators $H_0$ with coefficients in $W^{m+1,\infty}(\Ri^d)$ and…

We show that Lieb's concavity theorem holds more generally for any unitarily invariant matrix function $\phi:\mathbf{H}^n_+\rightarrow \mathbb{R}$ that is monotone and concave. Concretely, we prove the joint concavity of the function $(A,B)…

A coprime commutator in a profinite group $G$ is an element of the form $[x,y]$, where $x$ and $y$ have coprime order and an anti-coprime commutator is a commutator $[x,y]$ such that the orders of $x$ and $y$ are divisible by the same…

We construct non-Lipshitz flow in $H^s$ for the cubic nonlinear Schr\"odinger equation on the 2-torus of revolution with a Lipshitz or smooth metric . The non-Lipshitz property holds for all $s<2/3$ for Lipshitz metric and $s<1/2$ for…

The theorem of Lieb, Schultz and Mattis (LSM), which states that the S=1/2 XXZ spin chain has gapless or degenerate ground states, can be applied to broader models. Independently, Kolb considered the relation between the wave number $q$ and…

We are concerned with a class of nonconvex and nonsmooth composite optimization problems, comprising a twice differentiable function and a prox-regular function. We establish a sufficient condition for the proximal mapping of a prox-regular…