Now that I have finished my paper on higher reciprocity, I have begun thinking about my next project. Even though On Finite Fields and Higher Reciprocity is on the arXiv, I would like to also submit a portion to a journal as mentioned before (the choices are between the College Mathematics Journal and the Mathematics Magazine - I have to choose between them as their targetted audiences are slightly different). I have mentioned this before, but I will likely cover some results leading up to cubic reciprocity in a more expository style: I’m using playful prose in combination with rigorous language to place the law of cubic reciprocity in an entertaining light.

For a long time now I have been interested in Artin reciprocity. Artin reciprocity is the final generalization of reciprocity laws, and understanding even its statement requires substantial knowledge of class field theory, genus theory, algebraic number theory, and more. Cubic reciprocity is a good first step, but so much more is needed to approach Artin reciprocity. I believe a long term mathematics goal would be to read up on more algebra (preferably Dummit and Foote), a lot more algebraic number theory, and finally class field theory. This is usually something taught in an upper-level graduate course, so I don’t expect to study much now.

In the next academic year I will be working on a project concerning knot theory. I’m not exactly sure in which direction this project will go, but I have some particular interest in mosaic knots, and I have a few resources and ideas for open problems to approach. We’ll see where that goes.

Finally, university applications are fast approaching, and I have yet to start. Might as well start soon.

(19 July, 2024)