Abstract:

Over the complex numbers, a variety can be considered as a complex analytic space, which reflects the geometry of the variety through "GAGA"-type theorems. Attempting to copy this process naïvely over the p-adics fails horribly: for example, the p-adics are totally disconnected as topological spaces, while the algebraic affine line is connected. This necessitates a more subtle definition of a non-Archimedean analytic space. In this talk, I will introduce Berkovich spaces, which are one way to overcome this problem. These spaces find many uses across both number theory and geometry, have nice topological properties and admit similar GAGA theorems. After constructing the analytic affine line, we will consider the key notion of a "skeleton", draw an explicit picture of the analytification of an elliptic curve and briefly discuss the links of the theory with mirror symmetry.

Video: