I'm an undergraduate mathematics and computer science major at Carnegie Mellon University. I'm currently in Russia participating in the Math in Moscow program at the Independent University of Mathematics.
My email is adsmith [at] andrew [dot] cmu [dot] edu.
My CV can be found here.
While in Moscow, I'm keeping sporadic lecture notes here for my classes. I make no promises about keeping these up-to-date, but the general goal is to post them shortly after each lecture. This way I can make sure there are no details I missed, and my notes might be a more useful second viewpoint to anyone who didn't catch something during lecture.
Since more people have started read these, you should be warned that while I'm trying to cover all of the material from lecture, I don't always cover it how it was presented in lecture. Instead, writing these gives me a chance to work out how I think this material would be most clearly presented -- sometimes this means I spend a lot of time restructuring sections or coming up with cleaner proofs.
- Representation Theory, IUM Fall 2014: 1 2 3 4 5 6 7 8 (9-11 are being combined into one discussion of symmetric groups).
- Algebraic Number Theory, IUM Fall 2014: 1 2 3 4 5 6 (7-9 are building up the Dirichlet unit theorem, and the course's approach needs to be significantly reworked; 10-11 will probably be done first).
- Commutative & Homological Algebra, IUM Fall 2014: 7 8 10/11 (9 will be up shortly, I need to clean up a few lemmas).
- Riemann Surfaces, IUM Fall 2014: 1 2
- Algebraic Geometry, IUM Fall 2014: I tried this for like a day, but there's basically no way I can write better notes than these
Random notes: I'm also going to be keeping some notes here of random things I felt like writing and pretending someone will read. These are likely to vary between expositions and me just trying to think things through.
- Accessible categories and Grothendieck topologies: I wanted to make a thing that looked like a sheaf into a sheaf. It turned out to be an accessible category, but before I learned that I gave it a topology that determines accessibility. Some material repeated from my REU writeup from 2014.
- Group actions in an abelian category: In which I was curious about the canonical splitting of invariant subspaces when you can average representations.
- The dual group functor: Looking into why the group of one-dimensional representations is isomorphic to the abelianization, but not naturally.
- Categorical Jordan-Holder: I kept seeing Jordan-Holder pop up so I tried to figure out what the proper setting for it is. Turns out it's conormal categories with lattices of subobjects an epi-mono factorization.
- Generalized Homotopy: I wanted to see what homotopy should mean for schemes. At the moment all this note does is extend some results from Galois theory and covering spaces to coequalizer-preserving "fundamental groupoid" functors. If I get more time I'll add to it to show how to construct a fundamental groupoid functor from a category with a nice "functor of points" (I believe this is a Galois category?), and see how it interacts with a Grothendieck topology (i.e. can we make a fundamental groupoid functor whose maps with unique path lifts correspond to the covering spaces in the Grothendieck topology?)
- I've been working on trying to figure out how to define a connection on a module and thence a sheaf; once I get through lecture notes that should be up, hopefully with nice constructions like de Rham cohomology.
I've also been working on typing up solutions to Ravi Vakil's book Foundations of Algebraic Geometry. It's a very exercise-punctuated exposition style, and I'm challenging myself to understand the whole text by working through all of the exercise. At press time I'm in chapter 17 of 30, but it's getting longer every day!
In the summer of 2014, I participated in the SUMaR math REU at Kansas State University. My advisor was Alex Gonzalez. My project focused on generalizing a notion of Euler characteristic for finite categories to certain infinite categories arising from the homotopy properties of compact Lie groups. We also provided a geometric reinterpretation of these objects as certain sheaves over a Grothendieck site.
Our writeup can be found here.
In the summer of 2013 I did an REU at Clemson University. There my advisor was Mohammed Tesemma. My project studied rings of invariants in the Laurent polynomials of subgroups of the general linear group. We placed a natural topological structure on their space of different initial algebras under all possible monomial orders, and clasified all resulting topologies.
A preprint of the resulting paper can be found here. Here are slides on our results.
In the spring of 2014 I was a TA for 15-453 Formal Languages, Automata, and Computability. The course website can be found here.
- A package for painlessly drawing graphs on a torus while having to learn as little TikZ as possible. This was written to make a few assignments easier in 21-484 Graph Theory. It's available for anybody in the class (or I guess anybody outside the class who likes drawing graphs on tori?)
- A package for drawing plane algebraic curves cut out by a single polynomial. Very useful if your algebraic geometry professor loves making "draw all the patches of this projective curve" an assignment. Requires gnuplot.