Adjaye was born in Tanzania to Ghanaian parents, his father a diplomat, so these formative places included parts of Africa, Saudi Arabia, and eventually London. It was in London that he studied architecture, launched his practice, and designed his first projects.

The designs for the Smithsonian, which Adjaye calls "a new kind of new museum which is really about a narrative about people and a country", suggest both the range of influences and the sensory properties of which Simpson speaks. They promise dazzle and glitter, which Adjaye likes to alternate in his work with severity and darkness. There will be a ceiling, subject to budget, in the form of "a shower" of thousands of pieces of split pine, and a black room with a circular oculus above, which admits a glittering cylinder of falling water. There will be light and shadows filtered through filigree bronzework.

Adjaye sees it partly as a monument, like the other museums and the memorials to presidents that occupy the Mall, and like them it has a formal symmetry, but he also wants to be different from them. So it has a distinctive tiered shape, with walls inclining outwards as they rise, which he says is derived from the forms of Yoruba craftsmen in the region of Africa from which slaves were mostly extracted. There is bronze cladding, which "takes on the cast metal architecture of the American south". One of the first trades adopted by freed slaves was, he tells me, metalwork. Adjaye likes to tell stories about his projects, and this one is no exception.

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The Museum of African-American History and Culture is plainly a career-defining project. Pending its completion, however, Adjaye's most significant finished building is his Sugar Hill housing development in Harlem, New York, whose first residents will shortly move in. This is built for low-income and in some cases formerly homeless families, but also includes an early education centre and a Children's Museum of Art and Storytelling. Here, working with the housing specialists SLCE Architects, the aim was to bring pride to this essential use, within the brutal budgets of such projects. At no point, according to Adjaye, did anyone say "that seems like a nice thing, Mr Architect, just do it". The cost of everything had to be justified.

The building occupies a commanding site in upper Manhattan, in an area rich in the history of the Harlem Renaissance, with some handsome streets of century-old houses. It is one of Adjaye's tougher buildings, a blocky concrete citadel in charcoal grey, given drama by a cantilevered offset two thirds of the way up. The windows are quite small, because that is what the budgets allow, but the concrete is relieved by a pattern of roses etched into it, which Adjaye says is inspired by the decorations on older buildings in the neighbourhood. There is also a serration in the plan, which derives from the stepped profile of local terraces.

The project hasn't been universally popular in the neighbourhood. Despite the inspirations drawn from historic buildings, some saw it as inappropriate for its context, and a recent review has called it "a dead-eyed guard tower", its patterns "the result of an evening spent fiddling with Photoshop". Baxter admits that she was herself "challenged" by some of its aspects, but that "its inherent beauty stands out". She says that "David doesn't change his mind. He stays stable. He convinces in a steady way. Clients come around to see his position." And she did.