Can biplane or triplane designs be revived with modern materials?
Biplanes (and triplanes) became (nearly) extinct not because high strength materials (like composites) were not available, but because they became available. 

One of the major reasons for use of biplanes in the early days of aviation was that the materials available were of insufficient strength for the (wing) designs used. 

The major disadvantage of biplanes (or triplanes) is aerodynamic- it produces a lot of drag compared to the monoplane and the wings interfere with each other.

So, the biplane was an (aerodynamically) inefficient solution to a structural problem. As higher strength materials like Aluminum became widespread, the biplanes fell out of favor.

While they can be revived, I don't see why someone would do that, except for nostalgia (or some special uses, like aerobatics).

There are some companies who manufacture or restore biplanes, but they usually don't use (modern materials like) composites, though modern avionics are used.