# All Questions

100,185 questions
2k views

282 views

5k views

### Mathematical Physics? (Particularly computational)

I just saw a post like this one, but particularly for statistical mechanics, I thought I'd ask the question in general. Where does a mathematically trained person go to learn mathematical physics? By ...
9k views

### Consequences of Geometric Langlands

So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki ...
738 views

### How to distinguish between natural and unnatural equivalences of categories

Some equivalences of categories are constructed by explicitly giving a pair of functors that are inverses up to isomorphism. For example, the equivalence between CRing^op and affine schemes is given ...
16k views

### Where does a math person go to learn statistical mechanics?

The more math I read, the more I see concepts from statistical mechanics popping up -- all over the place in combinatorics and dynamical systems, but also in geometric situations. So naturally I've ...
4k views

### What are the components of a transpose operator from $\mathbb R^{n\times n}$ to $\mathbb R^{n\times n}$?

Say I'm working in the space of linear transformations from $\mathbb R^n$ to $\mathbb R^n$ and I've picked a basis so I can identify with any operator a component matrix in $\mathbb R^{n\times n}$. ...
4k views

### Universal definition of tangent spaces (for schemes and manifolds)

Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the ...
714 views

### Why do branches of math vary in proof styles and what category are different branches in?

Some branches of math seem to have reasoning which is more global. There is a lot of efficiency in the proofs because the reasoning transfers easily between proofs. For other branches of math, a lot ...
6k views

### Classification problem for non-compact manifolds

Background It is well-known that the compact two-dimensional manifolds are completely classified (by their orientability and their Euler characteristic). I'm also under the impression that there is ...
Let $X$ be a complex analytic space. It is a 'well known fact' that the categories of local systems on $X$ (i.e. locally constant sheaves with stalk $C^n$), and of (holomorphic) vector bundles on $X$ ...