# Research

### Adviser: Andrew Obus

### Areas of Interest

Arithmetic geometry (covers of curves, wild ramification, étale fundamental groups, formal orbifolds)

Algebraic geometry (stacky curves, deformation theory)

Number theory (modular curves, class field theory, analogies between number fields and curves)

Topology (homotopy theory in algebraic geometry, analogies between algebraic topology and algebraic geometry)

### Publications

(with Hugh Howards) Crossing number bounds in knot mosaics.

*Journal of Knot Theory and its Ramifications*(July 2018). Available at arXiv:1405.7683v2 [math.GT] (https://arxiv.org/abs/1405.7683)

### Projects

(with Libby Taylor)

**A**¹-local degree via stacks (in preparation)Wild ramification in stacky curves (dissertation in progress)

*Class field theory and the study of symmetric n-Fermat primes*(Master’s thesis)*Saturation in knot mosaics*(senior thesis)

### 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)

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)

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!