Research

 
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Research Interests

My research program touches on a wide variety of problems in algebraic geometry, number theory and topology. A unifying theme though is the theory of algebraic stacks. For me, these are essential tools for understanding some of the following topics:

  • Arithmetic geometry: covers of curves, wild ramification, étale fundamental groups, formal orbifolds

  • Algebraic geometry: classification of Deligne-Mumford stacks, moduli problems

  • Number theory: modular curves, modular forms, class field theory

  • A¹-homotopy theory: enriched enumerative geometry, homotopy theory of stacks

One of my main projects is to bring techniques from finite characteristic arithmetic geometry (e.g. Artin-Schreier-Witt theory) into the world of stacks. This has opened a number of exciting doors for future research.

Preprints

Publications

  • Crossing number bounds in knot mosaics (with H. Howards). Journal of Knot Theory and its Ramifications, vol. 27, no. 10 (2018). Also available at arXiv:1405.7683v2.

Projects

  • A¹-local degree via stacks (in preparation), with Libby Taylor

  • Artin-Schreier-Witt theory for stacky curves (in preparation)

  • Wild ramification in stacky curves (dissertation in progress, University of Virginia, with Andrew Obus)

  • Class field theory and the study of symmetric n-Fermat primes (Master’s thesis, Wake Forest University, with Frank Moore)

  • Saturation in knot mosaics (senior thesis, Wake Forest University, with Hugh Howards)

Notes

I keep notes on many courses and seminars I have been a part of. There are sure to be some errors, both cosmetic and mathematical, so if you find any, please contact me at ak5ah (at) virginia (dot) edu. Also, LaTeX files are available upon request. 

Although my philosophy is that 'all math is connected', I have grouped many of the above notes by area for ease of cross-reference. These larger files are available below. 

Thanks to Matt Feller, George Seelinger, Richard Vradenburgh and many others for noticing errors!