WITH its Renaissance treasures and centuries-old stately palazzi, Florence is sometimes treated like a living museum rather than a vibrant city with contemporary culture. That myth is harder to sustain these days, thanks in part to Matteo Renzi, the city’s 35-year-old mayor, who was elected last year with promises to stir up the Tuscan capital. Evidence of a more youthful and revitalized Florence is everywhere. Dilapidated piazzas have been refreshed, contemporary art galleries have sprung up, and old-school palazzi have been turned into trendy restaurants. Traffic has also improved, making the city even more pleasant to navigate.