MathOverflow

Let $M$ denote the set of Borel measures on the complex plane with compact support and where $\mu(\mathbb{C})<\infty$. We endow $M$ with the weak$^*$-topology which is the smallest topology such that whenever $f:\mathbb{C}\rightarrow\mathbb{R}$ is...

I have a puzzle which needs some helps from the experts here.Let the Kloosterman sum of degree 4 be as follows:$$S_4(1,n;c)=\hskip 0.5em \sideset{_{}^{}}{^{\ast}_{}}\sum\limits _{x,y,z \bmod c} e \left (\frac{nx+y+z+\overline{xyz}}{c}\right).$$
My question is： could we expect a bound like$$\sum_{c\sim C} K_4(1,n;c)\ll C^{2-\varepsilon}\,\,?$$Note that the individual bound for $K_4(1,n;c)$ is $\ll c^{3/2+\varepsilon}$. As far as I know, this type of sum was considered for degree two case, i...

A definition of the connected sum of two $n$-manifolds $M$ and $M'$ begins by considering two $n$-balls $B$ in $M$, $B'$ in $M'$, and glueing the varieties $M\setminus \mathring B$ and $M'\setminus \mathring B'$ along their boundary (an $(n-1)$-sphere...

The Hamilton-Jacobi equation of symplectic geometry states:$$\frac{\partial S}{\partial t} + H\!\left(q,\frac{\partial S}{\partial q},t\right)=0\tag1,$$
it is a nonlinear PDE for the generating function $S(q,t)$ (the phase of the wavefunction if you will) and the graph of the gradient of $S$ is the Lagrangian submanifold in phase space:...