Historian Mary Beard moves seamlessly from tackling aggressive dimwits on Twitter to talking about the politics of ancient Rome.

It's the kind of abrupt gear change we now expect from public intellectuals.

And Professor Mary Beard - who became Dame Mary in the most recent honours - has been highly successful at mixing the realm of ideas with a popular appeal, embodied in her ratings-friendly TV history shows.

She's also a Cambridge professor of classics who has the battle scars of the badlands of social media, facing violent threats and relentlessly misogynistic abuse.

But how should public thinkers make big ideas open and accessible? How do they avoid being sidelined in an era of fake news and attention-seeking punditry?

This is the challenge for the British Academy, the national body for the humanities and social sciences, which is throwing open the doors of its London headquarters for a summer showcase, promising "ideas for curious minds".

Getting people across the threshold is the first challenge, she says. The British Academy is based in an elegant building near St James's Park in central London. It's a place dedicated to public culture, but wants to stop looking like a private club.

"You can see why people might be a bit intimidated," says Dame Mary, a fellow of the academy.

"But what the television shows you is that ideas are not the preserve of posh white men." ...