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If you've selected "revert to default view" and added those columns, they should stay visible unless one or more of your perspectives are hiding them again. The built-in Project and Context perspectives should just be letting whatever columns were visible in the previous view stay visible. (Which means if the previous view hid the columns, they stay hidden.)

If you can figure out which perspectives are hiding those columns, the solution would be to show the Perspectives window, check the 'also restore settings for other mode' box, and then make sure the start and due date columns are visible in both views of that perspective.

If you're hiding those columns on purpose in some of the perspectives, you'll probably want to select the built-in Project context, show the columns, and then select Save Window as New Perspective. Ditto for Context view.

Of course, the next step would be to send email to requesting the ability to edit the built-in perspectives. :-)