Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explai …

I have 3 general abstract reasons to care about complex analysis in a single variable: The laplacian is, up to a constant multiple, the only isometry invariant PDO in the plane, …

Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important t …

What interesting applications are there for theorems or other results studied in first-year calculus courses? A good example for such an application would be using a calculus theo …

Dear All! There is a trend, for some people, to study representations of quivers. The setting of the problem is undoubtedly natural, but representations of quivers are present in …

$$\{A\in GL_n(\mathbb{C}) : |det(A)|=1\}$$ This seems to me to be a perfectly natural group to study; it is easy to define and contains $U(n), SL_n$, and all the torsion. Is th …

Dear everyone, Motivation : From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of el …

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral con …

Tropical geometry can be described as "algebraic geometry" over the semifield $\mathbb{T}$ of tropical numbers. As a set, $\mathbb{T}=\mathbb{R}\cup { -\infty}$; this is endowed wi …

This is, basically, me trying to generalize "Why should I care for sheaves and schemes?" into a reasonable question. Whether successfully, time will tell, but let me hope that if n …

I am teaching an advanced undergraduate class on topology. We are doing introductory knot theory at the moment. One of my students asked how do we know to use this skein relation t …

I am starting to dive into a study of Several Complex Variables. I would like to have a few guiding examples of "big payoffs". These should be very natural sounding theorems whic …

I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and w …

Apologies if my question seems overly naive, but I haven't seen/heard/read any good answers. What is modern computability theory "really" about? The study of feasible(even remot …

This is related to Anweshi's question about theories of noncommutative geometry. Let's start out by saying that I live, mostly, in a commutative universe. The only noncommutative …