I started up the BOINC manager today, and the first thing I get is a message box offering to attach to a new project. Clicking on "Cancel", I look at the "Tasks" and "Projects" tabs, but find that I am apparently not connected to any projects. Why could this be happening, and is there a way to fix it?

I see from your other thread that you recently upgraded. If you do not install the new version of BOINC to the same folder, it will not pick up your old projects.