Let $G$ be an Abelian group. Let $A \subseteq G$. In additive combinatorics, one of the primary measures of the additive structure of $A$ is its additive energy, defined as $E(A) = …

We can think of the "tangent category" of a category $\mathcal{C}$ at an object $A$ as being the abelian group objects in the overcategory $\mathcal{C}/A$ (with whatever conditions …

Fix a positive integer $d \geq 2$, and let $n,k$ be natural numbers with $k \leq n$. Let b(n,k) denote the number of monomials of degree kd-(n+1) in n+1 variables x_0,..x_n with …

Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G. …

First of all, apologies for a somewhat vague question but let me give a try. We know what the projective objects in the category of Banach spaces are: these are precisely $\ell_1(\ …

Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville: “The big experts in the field had already tried to make this approach w …

Let $X$ be a variety. Then, is $X$ path connected? And by path connected, I mean any two closed points $P, Q$ on the variety can be connected by the image of a finite number of n …

I need help understanding the following condition: $u_h\in L^2(\mathbb{T}^d)$, $\|u_h\|_{L^2(\mathbb{T}^d)}=1$, where $h$ is the semiclassical parameter and $\mathbb{T}^d$ is the …

The following problem arises in the analysis of Bloom filters. Consider $m$ bins and $N=nk$ balls placed uniformly at random into the bins. A query chooses $k$ bins uniformly at …

I have some naive questions about polynomial-count affine varieties over $\mathbb{C}$: Are all reductive algebraic groups strongly polynomial-count? Are products of strongly poly …

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?

For me second-countability always felt like to be the more important and fundamental concept from general topology than separability. I wonder whether there are any points which ca …

Let $A$ and $B$ square real matrices. I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1. Can we say something a …

A Morse funnction on a smooth manifold is usually intuitively interpreted as follows: Imagine the manifold to be a mountainous landscape and the Morse function as the elevation of …

Are there tetrahedra which can be subdivided into three parts similar to the original? I believe this would require splitting one face into three parts. I know some types of tetrah …