Physicists Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of the Fourier transform as a canonical isomorphism $L^2(K) \to L^2(\hat K)$ where $K$ is a locally...

I am interested in the existence of a vector valued solution $y = y(x, t) \in\mathbb{R}^n$ to a system of $2n$ equations: there are twice more equations than unknowns. More precisely: Let $A$ and $B$ be matrix valued functions $A, B \in C^1([0, L]\times [0, T]; \mathbb{R}^{n \times n})$. Does...

Let $X$ be a smooth projective variety of dimension at least $2$ over $\mathbb{C}$. Let $\mathcal{O}_X(1)$ be a very ample line bundle and $E$ be a $\mu$-semistable sheaf on $X$. Then a theorem of Maruyama states if $r(E)<dim(X)$, them for general hyperplane section $H\in|\mathcal{O}_X(1)|$, the ...

I'm interested in the dual question to: continuous images of open intervals, about surjections onto open intervals. Namely, if $X$ is a topological space, when can we guarantee that there exists a topological embedding of $X$ into some Euclidean space?