If $X,Y$ are vector fields and $\def\Fl{\operatorname{Fl}}\Fl^X_t$ and $\Fl^Y_t$ their local flows, let $[\Fl^X_t,\Fl^Y_t]:= \Fl^Y_{-t}\Fl^X_{-t}\Fl^Y_t\Fl^X_t$ denote the group co …

Assume we have a smooth Riemannian metric $g$ on a small one-sided neighborhood $U$ of $0$ on the plane, say $U_\epsilon=\lbrace(x, y): x^2+y^2<\epsilon, y\geq 0\rbrace$. When …

Given a n-by-n matrix $\mathbf{\phi}$ and a vector $\mathbf{X}$, solve for the two vectors $\mathbf{\Phi}$ and $\mathbf{\Omega}$ that satisfy: $$ \Phi_i = \sum_{l} \frac{\phi_{il} …

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory …

I'm trying to find an algebraic proof of irreducibility of the polynomial $x^n-x-1$ over rational numbers (or integers, which the same). I've read the Selmer's paper "On the irredu …

What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explaination, both talking about curves and surfaces...

Fix an algebraic integer $\alpha$ of degree $n$ such that the extension $K=\mathbf{Q}(\alpha)/\mathbf{Q}$ has intermediate fields. (We can assume $K$ is Galois with non-simple Gal …

Theorem 8.29 in "Combinatorial commutative algebra" by Miller and Sturmfels states the upper-semicontinuity property for Groebner deformations (say, over an algebraically closed fi …

In Aubin's book (nonlinear problems in Riemannian Geometry), starting from p. 106, it is shown that a Green's function of a compact manifold without boundary satisfies $$G(P,Q) \l …

One of the main players in the categorical geometric langlands correspondence is the moduli stack of rank n integrable connections on a complex curve. The reason for considering s …

I need to compute a set of multivariate definite integrals with infinite integration domain $$\displaystyle \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f(x_1,x_2, \ldo …

When do the notions of totally disconnected space and zero-dimensional space coincide? From what I gather, there are at least three common notions of topological dimension: coverin …

Given a set of distances between every pair of points of an integral point set P of n points; say D = {${d_i}$} Q1. What is the least time complexity possible/known for …

Let me begin with an example. Consider the computable function $f(x) = 2x$. A Turing machine can implement this function in $O(|n|)$ steps: simply walk to the end of the input st …

Let $R$ be a (commutative) noetherian integral domain. Let $I$ be a prime ideal of $R$. Let $S$ be a finitely generated $R$-subalgebra of $\mathrm{Frac}(R)$. Is $S \cap R_I$ nece …