Modular Spaces Topology

Some result on modular hyperconvex spaces

Modular Forms and Topology

We want to discuss various applications of modular forms in topology. The starting point is elliptic genus and its generalizations. The main techniques are the Atiyah-Singer index theorem, the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, Kac-Moody Lie algebras, modular forms and theta-functions. Just as the representations theory of classical Lie groups has close connections with the...

some result on modular hyperconvex spaces

Topology and Sobolev Spaces

with 1 ≤ p <∞. W (M,N) is equipped with the standard metric d(u, v) = ‖u− v‖W1,p . Our main concern is to determine whether or not W (M,N) is path-connected and if not what can be said about its path-connected components, i.e. its W -homotopy classes. We say that u and v are W -homotopic if there is a path u ∈ C([0, 1],W (M,N)) such that u = u and u = v. We denote by ∼p the corresponding equiva...

Topology Notes: Ordered Spaces

Consider the following properties of a binary relation ≤ on a set X: (Reflexivity) For all x ∈ X, x ≤ x. (Anti-Symmetry) For all x, y ∈ X, if x ≤ y and y ≤ x, then x = y. (Transitivity) For all x, y, z ∈ X, if x ≤ y and y ≤ z, then x ≤ z. (Totality) For all x, y ∈ X, either x ≤ y or y ≤ x. A relation which satisfies reflexivity and transitivity is called a quasi-ordering. A relation which satis...