Most of my current work is about arithmetic statistics of curves over finite and global fields. 

By coincidence, I think about a lot of double eponyms: Brill-Noether, Cohen-Lenstra, and

Katz-Sarnak. I am working to write down Brill-Noether general curves over arbitrary fields, understand q-Frobenius distributions on large-genus curves, and predicting

cohomology in the moduli space of curves. 

 

My other research interests on my mind (but perhaps not at the

forefront) include:​​

• 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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