Mathematical Conversations

What is the difference between algebraic and analytic geometry? Is there some way to construct moduli "spaces" in analytic geometry (in the Archimedean or non-Archimedean contexts)? Is there a common language for expressing the foundations of algebraic and analytic geometry?

About Mathematical Conversations: We meet in Harry's Bar at 6pm, where free drinks are provided. After 20 minutes, we move to the Dilworth room, where the speaker gives a 20-minute talk, followed by 15 minutes of discussion with the audience. After that we return to the bar for further discussions.