My Portfolio & Posts

This is a book I'm currently writing. The core idea is to cover a good chunk of ordinary & modular character theory/representation theory, but in a goal-focused way: introducing concepts as we go along, in service of computing the character tables of different groups. In my opinion, that gives something tangible and verifiable worth focusing on, but allows introducing a good amount of theory as it's needed. That means care in choosing the groups, and the ordering in which they're introduced, on which I spend a large amount of my time in writing this book: figuring out how to present the cool results I care about without being overwhelming and/or boring, always anchored to computing character tables by hand!

I currently have the first ten chapters mapped out, but still need to add details to most of them (basically expanding outlines to real prose). These 10 chapters are on the low-degree symmetric and alternating groups. A surprising amount of character theory can be talked about here, including: permutation characters; exterior products; generalized induction and restriction; bimodule adjunctions and reciprocity theorems; and Deligne-Lusztig theory! There's so much material I pushed some into appendices. The PDF linked here only contains the chapters/appendices I've actually finished!

Huppert proved that a finite group with all maximal subgroups of prime index must be supersolvable. I give 3 distinct but related proofs of this theorem, using local analysis, group homology, and representation theory.

This writeup contains a spectral-sequence-free proof of the Stallings exact sequence on low-dimensional group homology. It also contains a bunch of applications of this sequence to prove some non-trivial things about infinite groups.

I wrote up all the facts I always remember about topological groups, but can never remember how to prove. These include: separation axiom results; having abelian fundamental group; when they're metrizable; and Haar measure.

The trefoil knot complement is equivalent to the quotient space $SL(2,ℝ)/SL(2,\mathbb{Z})$. A wonderful paper of Rubinstein & Gardner shows how to prove this based on Seifert fiber space structure. I wrote up my understanding of their proof, including (as usual) a more group-theoretic approach. Plus some pictures!

Everyone knows the Sylow theorems. But comparatively few people know an amazing counting lemma that is incredibly powerful: the orbit-stabilizer theorem modulo a prime. Starting with that lemma, I wrote up proofs of all the Sylow theorems, and some generalizations. I think it works nicely to show how powerful such counting can be.

It's a classic result that every closed orientable 3-manifold bounds a 4-manifold; the proof I've always known used Dehn surgeries. A while ago on MathOverflow I saw the idea of using cohomology classes, so I wrote up the details.

I had a very hard time finding a proof that the diffeomorphism group of a compact manifold was an actual topological group. I was eventually able to cobble the proof together from various sources, and this note is the result.

A 1-page note looking at how the 3-sphere acts (as a Lie group) on multiple different subspaces of quaternion space, and the resulting topological implications.

A longer note looking at the same 3-sphere actions as the previous note, with more detail.

In a work-related math group I'm part of, there was a discussion about classifying plane bundles over the 2-sphere. Someone wrote a short argument that oriented plane bundles are classified by second cohomology, so there are infinitely many plane bundles, indexed by the integers, and these must be the usual complex line bundles. I wrote this note in response: yes there are infinitely many real plane bundles, but they are not in 1-1 correspondence with the integers indexing the complex line bundles. I like this note because it highlights the importance of not treating every countably infinite set equally!

The classic example of two non-homotopic manifolds with the same homotopy/homology/cohomology groups comes from Lens spaces in dimension 3, but the actual math behind proving all of this is somewhat complicated. I saw a much simpler example in one dimension higher on MathOverflow, so I wrote up the details in this very short note.

This is an older, but longer, writeup of the same result as above: finding two manifolds with the same homotopy/homology/cohomology groups. I include it here because it also contains a detailed description of the cohomology ring of a connected sum.

This note gives two different proofs that $G{r}^{+}(\mathrm{2,4})$ is a product of spheres.

A short note containing a proof that ${\pi }_1$ acts on higher homotopy groups in a well-defined way.

An expanded math blog post I did a lifetime ago, about using the transfer map in group theory to prove some things about groups and manifolds.

When solving an exercise in Hatcher's Algebraic Topology, I needed to prove a certain pair had the HEP. I ended up working it out myself, and the nuance of the topological argument was super fascinating, so I wrote it up. Turns out there's a cleaner proof in Bredon's Topology and Geometry, but I like mine too!

In graduate school we compiled solutions to lots of old comprehensive exam problems. I have hundreds of these solutions written out, but the problems were cataloged in a different file. I'm currently rewriting the problems and solutions to one file, while also cleaning up some of the typography and math. This is kind of boring, so progress is slow!

Solutions to all problems in Algebra: A Graduate Course by Isaacs.

Solutions to all problems in Finite Group Theory by Isaacs.

Various golang implementations of different tree types.