## Motivation for studying group of homeomorphisms of topological spaces [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed 4 years ago. Improve this question Currently I am reading a paper titled “On the Group of Homeomorphisms of an Arc” by N.J Fine and G.E. Schweigert, that was published … Read more

## Relative cohomology in algebraic topology vs algebraic geometry

There is a notion of relative cohomology in both algebraic topology and algebraic geometry and the flexibility granted by this relative viewpoint is quite powerful in both cases. However, the two notions of relativeness are different: In algebraic topology, we consider cohomology (or homology or homotopy) of the form $H^k(X,A;R)$ where $A \subset X$ is … Read more

## Motivation for the covariant model structure on SSet/S

I was reading HTT 2.1.4, and I just totally lost what was going on. Could someone provide some motivation for this section? Why do we want another model structure? I’m sorry for not providing motivation, but as you can see, I’m unable to do so. Here’s a link to the ArXiv version: http://arxiv.org/pdf/math/0608040v4 Answer If … Read more

## Examples of high level math that can be motivated to laypeople

One of the difficulties of mathematics over other sciences is that our problems are harder to motivate to a general audience. A biologist studying a particular pathway in the body can say that he’s looking to understand or cure some disease, a physicist at CERN can say he’s trying to understand why particles have mass, … Read more

## Snazzy applications of Several Complex Variables techniques

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 which depend on a lot of the machinery of SCV to prove them in an efficient manner. Here are a few examples: 1. Let … Read more

## Motivations for the study of dual connections

I am intrigued by the notion of dual connections: two affine connections ∇ and ∇∗ are called dual if they satisfy X(g(Y,Z))=g(∇XY,Z)+g(Y,∇∗XZ) for a given (pseudo)-riemannian metric g. What is the motivation and the deep results behind this notion? What are the main fields of application: information geometry, riemannian foliations, webs …? I am interested … Read more

## Why the circle for Pontryagin duality? [duplicate]

This question already has answers here: When does Pontryagin duality generalize? (4 answers) Closed 4 years ago. For a locally compact group G, we define the Pontryagin dual as ˆG=Hom(G,T) where T is the circle group and the homomorphisms are continuous group maps. This duality has a lot of nice properties and shows up all … Read more

## Why doesn’t this group have a name?

$$\{A\in GL_n(\mathbb{C}) : |det(A)|=1\}$$ 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 there any good reason why this group isn’t among the usual classical groups that are so well-understood and thoroughly discussed? I understand that most of … Read more

## What is the significance of Friedlander-Iwaniec and related theorems?

On p.177 of Number Theory Revealed: A Masterclass by Andrew Granville, the author states that “One can ask for prime values of polynomials in two or more variables.” (though he later mentions Landau’s n2+1 conjecture so I’m not sure why single-variable polynomials are omitted). For example, there are some classical results, like Fermat’s Christmas theorem … Read more

## applications of Tate-Poitou duality

What are nice applications of Tate-Poitou duality? Answer One of the main themes of current number theory research, instigated by the work of Wiles and Taylor–Wiles on the Shimura–Taniyama modularity conjecture for elliptic curves, is the study of modularity (or more generally automorphy) of representations of Galois groups. The basic tool for proving modularity is … Read more