That depends entirely on her interests. Some enjoy math, some enjoy politics, some enjoy art, some prefer geology, some go for fan fiction (true story, I know a few), some just like rockets. You just have to figure it out on your own.
In general though, I think the best conversation starters are current events/politics (hey, did you hear about Trump’s latest about-face? What’s your opinion on the death penalty?
Assisted suicide? Religion in politics?) or philosophy (is mathematics invented, or does it exist independently of us? Is it possible to know something without interacting with the outside world?).
Nearly all smart people have an opinion; the idea is to get them talking about their opinions. Politics and philosophy are simply the subjects most prone to armchair opinions. For girls in particular, feminism might be a topic of interest, but I’d approach that topic with caution. Questions like “if you could learn one other language instantaneously right now, what would it be?” are a bit safer.
PS: This applies to all smart people, regardless of gender. Imo, very smart people are more likely to break gender stereotypes anyway; talking with very smart guys should be pretty similar to talking with very smart girls.
The same topics any other girl would like to talk about.
Ask yourself this: if a girl asked you out and took the lead in the date — picking you up and paying for the meal, etc. — what kinds of things would you be interested in talking to her about?
You would probably want her to ask you questions so that she got to know you better, right? She would ask about your work, or school. She would ask about your family and your hobbies. Those topics would lead to more topics, and you would ask her questions also.
So go and do that and you should be fine. Good luck!
There are a lot… and it depends on the girl. Ask her about herself, and if you’re interested in something she mentions, ask her more about it. The conversation will then go to that topic.