# Research

### 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

**A**¹-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

Artin-Schreier root stacks (2019). Also available at arXiv:1910.03146.

### 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

**A**¹-local degree via stacks (in preparation), with Libby TaylorArtin-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.

Abstract Algebra (Wake Forest, Fall 13 - Spring 14)

Algebraic Geometry of Curves (Virginia, Fall 16)

Algebraic Number Theory (Virginia, Spring 16)

Algebraic Stacks (in progress)

Algebraic Topology (Virginia, Spring - Fall 16)

Analysis of Banach Spaces (Wake Forest, Spring 14)

Analytic Number Theory (Wake Forest, Fall 13)

Arithmetic Fundamental Group (Virginia, Spring 17)

Class Field Theory (Wake Forest, Fall 13 - Spring 15)

Commutative Algebra (Virginia, Spring 16/19)

Complex Analysis (Wake Forest, Fall 10)

Complex Surfaces (in progress)

Derived Categories (in progress)

Differential Geometry (Wake Forest, Spring 15)

Differential Topology (Virginia, Fall 15)

Elementary Number Theory (Wake Forest, Spring 13)

Étale Cohomology (in progress)

Étale Theory (Virginia, Fall 18 - Spring 19)

Fibre Bundles (Virginia, Spring 17/18)

Galois Cohomology (in progress)

General Topology (Wake Forest, Fall 13 - Spring 14)

Generalized Jacobians (Virginia, Fall 16 - Spring 17)

Homological Algebra (Wake Forest, Fall 14 / Virginia, Spring 17)

Homotopy Theory (Virginia, Fall 17)

Hyperbolic Geometry (in progress)

L-Functions and Modular Forms (Virginia, Fall 17 - Spring 18)

Lie Groups (Virginia, Fall 15 / Spring 18)

Linear Algebra (Wake Forest, Fall 14)

Measure Theory (Virginia, Spring 16)

Modular Forms (in progress)

Noncommutative Algebra (Virginia, Fall 15)

Numerical Methods (Wake Forest, Fall 14)

Probabilistic Measure Theory (Wake Forest, Spring 15)

Real Analysis (Wake Forest, Fall 12)

Representation Theory (Wake Forest, Fall 14)

Sheaf Cohomology (Virginia, Fall 18)

Short Course on Schemes (Virginia, Summer 17)

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!