TypeScript and Lightning?

Aura does not support TypeScript, so for the time being, you'll have to live with what we're given, which is Locker Service. Note that the Issue logged suggests that TypeScript might be something in the future, but you'll probably want to log an idea.

Note that Locker Service, while it doesn't necessarily provide the same language benefits of TypeScript, still ensures a very high level of security, including protecting the native DOM elements, preventing components from talking to each other in unapproved manners, secure resources, and so on. It's kind of "functionally" the same, but definitely not the same.