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: Answer If … 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

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