Below you can find typed mathematical notes, which are byproducts of classes I have taken and attempts I have made to learn topics. Please be warned that there may be typos or errors - they are entirely my own - if you see any, I'd love it if you let me know. In no particular order:
- A Quick Git How-To.
I kept finding myself forgetting all the terminal commands for using Git, so I wrote a quick note to refer to.
- Introduction to Algebraic Geometry.
In Fall 2020 Alexander Duncan and I started a graduate student algebraic geometry learning seminar called STAG: Students Teaching Algebraic Geometry
. This continued until Summer 2022 and for Fall 2021 we decided to write guided lecture notes for the seminar. My contributions were sections 4, 5, 6, 8, 9, A, B, and numerous exercises throughout the rest of the notes.
- Eric Tries to Write Down the Yoneda Lemma.
The Yoneda lemma is a powerful categorical result that I use often. Textbooks tend to omit the proof, claiming it to be obvious (in the sense that if you follow your nose and do the only thing that makes sense at each step, you'll prove the lemma). I wanted to write out the proof for myself. After I did, I gained an appreciation for its obviousness. In the future I may take a second pass at these notes which emphasizes how obvious it is; the proof as I wrote it is quite spartan. For now: the key idea is that given a representable functor, it is very natural (excuse the pun) to plug the representing object itself in and take the identity map in that homset. I am pleased with the corollaries and examples though.
- Singularities in Positive Characteristic.
In Fall 2019 and Spring 2020 I took two semesters of singularities in positive characteristic with Lance Edward Miller. These notes are based off of the first semester's lectures. You can also read the second semester's notes
, but my progress on these was cut short due to covid.
- Snake Lemma.
Weibel's An introduction to homological algebra defers the proof of the snake lemma, so I strove to write the proof explicitly. In doing so, I learned why Weibel omitted it.
- Hom-Tensor Adjunction.
These notes prove in detail the tensor-Hom adjunction. "Hom-tensor adjunction" flows off the tongue better but tensor is the left adjoint and Hom the right.
- Construction of Ext.
These notes build the degree 1 Yoneda Ext module.
- Four Explicit Projective Resolutions.
These notes describe how to build projective (in fact, free) resolutions of modules by hand. It also includes notes on the commands needed to request resolutions from Macaulay2.
- An introduction to homological algebra Exercises.
On 28 June, 2019, I took my oral exam to become a PhD candidate. (My committee included Lance Edward Miller, Paolo Mantero, and Mark R Johnson.) My syllabus concerned topics in homological algebra and to prepare I spent the previous year studying from Weibel's An introduction to homological algebra. I completed almost every exercise in the first four chapters, though if I were to take a second pass at this nowadays I'm sure I'd find many ways to improve these solutions.
In Fall 2017 and Spring 2018 I took two semesters of algebra with Matthew B Day. We used Dummit-Foote. These notes are based on the lectures but are currently unfinished because I found myself wanting to reorder the topics multiple times.
- Complex Analysis.
In Spring 2017 and Fall 2017 I took two semesters of complex analysis with Andrew Raich. We used Greene-Krantz. In preparation for my qualifying exams, I wrote these notes based on the lectures.
- ODE-PDE Questions.
In Fall 2016 I took a semester of ordinary differential equations with Mark R Johnson and in Spring 2017 I took a semester of partial differential equations with Phillip S Harrington. We used Hirsch-Smale-Devaney and Evans. In preparation for my qualifying exams, I wrote these notes based on the past quals. The numbering indicates, out of the nine exams sampled, how many times a given question appeared.
- Algebraic Topology.
In Fall 2016 and Spring 2017 I took two semesters of algebraic topology with Matt Clay. We used Munkres and Hatcher. In preparation for my qualifying exams, I wrote these notes based on the lectures.
I have also given several expository talks. Slides and/or notes and/or videos are posted under the Papers and Talks tab.