I am currently thinking about arithmetic statistics of curves over finite and global fields.

Even the notion of what it means to be a ``general," genus g curve over these fields

is an interesting problem and I would love to understand the arithmetic of 

non-superelliptic curves over (say) F_q. My other research interests on my mind

(but perhaps not at the forefront) include:

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• Explicit number theory/diophantine geometry​

  • Point counting on elliptic curves & abelian varieties​

  • Applications to post-quantum cryptography

• Diophantine equations and approximation​

  • S-units equations and algebraic tori ​

  • Continued fractions/badly approximable numbers

• Definability and decidability 

  • Hilbert's Tenth Problem over arithmetically significant rings​

  • Computational complexity of decision problems

• Algebraic Model Theory​

  • Transfer principles and o-minimality as applied to algebraic geometry

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