Patching methods are usually used to construct global objects
from more local ones. On the other hand, algebraic objects can
sometimes be understood from their local behavior, i.e., they
satisfy a local-global principle. In this talk, we explain...
Quillen's higher K-groups, defined in 1971, paved the way for
motivic cohomology of algebraic varieties. Their definition as
homotopy groups of combinatorially constructed topological spaces
initially seems abstract and inaccessible. In...
Recently Nicolas and Serre have determined the structure of the
Hecke algebra acting on modular forms of level 1 modulo 2, and
Serre has conjectured the existence of a universal Galois
representation over this algebra. I'll explain the proof of...