This is a very cynical and bad faith reading of the OP. I don’t see how knowing about HoTT or category theory gives one away as an absurdly pre-prepped kid who is shoehorned into academic success by tiger parents. I went to a Bay area high school full of specifically that kind of kid, and cannot think of a single one that knew what curry- was or had heard of prolog. Of course all anecdata. But let me tell you that the pre-prepped kids are not writing about wanting to do type theory research.
It’s a shame because the OP sounds like someone who would thrive at an mit/cmu, but likely got screwed due to an opaque system optimizing for a completely unknown objective (let me be cynical now—-the objective is optimizing us news rankings by only accepting top gpa/sat that have high yield, a subset of overprepared foreign students that will pay full ticket, and a scattering of “african americans” ie wealthy nigerians).
The system is bullshit. The only thing us universities have going for them is the professors and the fact that you can drop by their office and start a research relationship by extending your hand in good faith.