The Barbican is worth visiting for it's brutalist architecture alone. The concert programming is really diverse; it's home to the BBC Syphony Orchestra, but you'll also find bands like the Dirty Projectors and the National amongst the classical and contemporary composer line up. Plenty of art, dance and theatre too, however, the Conservatory is the real gem that's not to be missed.
As a keen runner, running through Greenwich Park and on to the river side I see the Thames Barrier, which is not only the worlds second-largest movable flood barrier (after the Oosterscheldekering in the Netherlands) but is also an iconic site on what is fast becoming a recognised stretch for developers including the 02 and Ravensbourne College of Design and Communication.
A slice of Scandi in the heart of London. Great for lunch, with all the meatballs, salads and classic Scandi foods you could want, along with a great little deli shop too.