Latest Posts

An Explicit Example of the Proof of the Nullstellensatz
I’m in an algebraic geometry class right now, and a friend was struggling conceptually with the proof of the strong nullstellensatz. I thought it might be helpful to see a concrete example of the idea, since the proof is actually quite constructive! Which brings us to this post:

A Follow Up  Explicit Error Bounds
In my previous post I talked about how we can use probability theory to make certain scary looking limits obvious. I mentioned that I would be writing a follow up post actually doing one of the derivations formally, and for once in my life I’m going to get to it quickly! Since I still consider myself a bit of a computer scientist, it’s important to know the rate at which limits converge, and so we’ll keep track of our error bounds throughout the computation. Let’s get to it!

Using Probability to Make Difficult Limits Obvious
Hands down one of my worst subjects is probability, which is a shame because it’s the source of a lot of powerful arguments, both formal and heuristic. I remember struggling with the analysis of randomized algorithms, and the closely related probabilistic method in combinatorics was no easier. Of course, the more you work with these things the easier they become, and I’m starting to feel a bit better about the basics at least[^1]. One pattern that I’ve seen a few times recently was so slick that I wanted to write something up about it in my first “quick analysis tricks” post in a little while!

Rational maps $\mathbb{P}^1 \to \mathbb{P}^n$ are automatically regular
The other day a friend of mine was asking for help with an algebraic geometry problem. I was happy to walk her through it, and because she and I were doing this online, I actually have a record of what I sent her! She seemed to think it was helpful, so I figured I would post (a slightly edited version of) it ^_^.

Slaughtering Competition Problems with Quantifier Elimination
Anytime I see questions on mse that ask something “simple”, I feel a powerful urge to chime in with “a computer can do this for you!”. Obviously if you’re a researching mathematician you shouldn’t waste your time with something a computer can do for you, but when you’re still learning techniques (or, as is frequently the case on mse, solving homework problems), it’s not a particularly useful comment (so I usually abstain). The urge is particularly powerful when it comes to the contrived inequalities that show up in a lot of competition math, and today I saw a question that really made me want to say something about this! I still feel like it would be a bit inappropriate for mse, but thankfully I have a blog where I can talk about whatever I please :P So today, let’s see how to hit these problems with the proverbial nuke that is quantifier elimination!

Some Questions about the Mathematics of Hitomezashi
A little while ago I made a post with some sage code that makes pretty pictures based on a certain Japanese stitching based off of Hitomezashi Sashiko. It’s fun to play with, and I love making pretty pictures, but it also raises some interesting combinatorial questions! Questions which, unfortunately, I’m not sure how to answer. This post is about some partial progress that I’ve made, and also shows how I’ve been using sage to develop conjectures for these problems. I’ll end with the statement of a few of these conjectures, and if I’m lucky some readers will have ideas for attacking these problems!

Topological Categories  A Unifying Framework
I think there is an obvious analogy that most mathematicians notice, perhaps subconsciously, when learning about topological spaces and measure spaces (often within a year or two of each other). The definitions look similar, as do the associated maps (continuous maps pull back open sets to open sets, and measurable maps pull back measurable sets to measurable sets). Seeing this, it’s reasonable to ask if measure spaces and topological spaces share any similarities. More abstractly, one can ask whether their categories share any similarities. The answer turns out to be a resounding yes, and the study of these Topological Categories will be the subject of today’s post!

A Hitomezashi Maker
It’s been a hot second since I’ve done something artistic, and a recent numberphile video gave me something small and simple to do! Plus, a really close friend of mine is a talented fibre artist, and since this type of artwork (hitomezashi) is originally a kind of decorative mending, I thought it would be fun to code this up and send it to her! It makes a great demo even without the fibre background, and I hope you all have fun playing with it ^_^.

Talk  The Weil Conjectures and Topos Theory
This quarter I took a class on analytic number theory with Dr Lapidus, and as a final project I wrote up a survey about the Weil Conjectures, which were the impotus for Topos Theory, an extremely deep subject that I’m always trying to learn more about. For bonus points, we could give a talk on our project, and since I’m
an attention whorealways happy to give a talk, I went for this option. 
Using Ultrapowers to Solve Problems
An Ultraproduct is a construction from mathematical logic that lets us prove (first order) statements about some family of structures by studying a single object which somehow captures the behavior of the family. In this post we’ll talk about ultraproducts – what are they, what do they do, and most importantly how can we actually use them to solve problems?

Talk  The Univalence Axiom
Today I gave my first proper invited talk, which was really exciting! I spoke at CMU after I graduated, but I knew the organizers so it didn’t really feel like an invitation. Jake Park at the University of Florida, though, invited me to give a talk about HoTT type theory in their department. I was and am super grateful for the opportunity, and I think the talk itself turned out fairly well too!

Dense Pythagorean Triples  A Cute Problem in Arithmetic Geometry
I follow a lot of other math blogs, and it’s always fun reading about what various people are thinking about. Particularly fun for me is when blogs pose problems to their readers, and the other day Shiva Kintali posted a really cute problem: For any rational numbers $\alpha \lt \beta \in (0,1)$, can we always find a pythagorean triple $a^2 + b^2 = c^2$ so that $\alpha \lt \frac{a}{b} \lt \beta$?

Talk  Bring Out the Crayons  A Survey of Descriptive Combinatorics
Last week Jacob asked me to give a talk in the topology seminar for UCR. I’ve been pretty busy lately, but I have a reputation to uphold, so I agreed, even though I didn’t have a great idea of what I wanted to talk about. My first idea was to talk about modal logic, which admits a nice topological semantics, but I wasn’t able to come up with any ways the logic was able to give us extra information topologically. It was always to topology helping to say things about the logic, and that’s not really the direction I wanted the ideas to flow for this talk. I considered talking about locales, but I’m still not super familiar with locales and I was worried I wouldn’t be able to answer some of the questions that might get asked. Most worryingly, I don’t have a good answer for “why should I care?”. I personally think locales are cool, and there’s good reasons to care if you’re already sold on constructive mathematics and toposes… But it’s hard to justify to a generic mathematician on the street[^1]. Eventually, I settled on Descriptive Combinatorics, an extremely interesting topic in its own right, and one which has lots of interesting ties to geometric group theory.

What Do Wirtinger Derivatives Do?
I was chatting with a friend about complex analysis earlier today, and I realized that I was never really told why we should care about Wirtinger Deritatives, at least in one complex variable[^1]. I figured I would write up a quick blog post about them, and explain how they can help explain (intuitively) which functions are holomorphic.

A Nonarithmetic Example of a Noneuclidean Principal Ideal Domain
This blog post changed dramatically over the course of writing it… I’m keeping the title, because I like the alliteration of “nonarithmetic noneuclidean”, but by the end I actually understood the arithmetic example better too. Perhaps a better title would be “In which algebraic geometry explains noneuclidean PIDs in a way that finally makes sense to me”.

Talk  Top $5$ Undecidable Problems  Number $4$ will shock you!
The quarter is starting to get into swing, and that means the grad student seminar has started back up! True to form I volunteered to give the first talk, but the scheduling led to a bit of a fiasco. I didn’t realize I was supposed to teach during gss, so on Monday I emailed the organizers to say that I couldn’t give the talk after all. Then,
yesterdaylast Thursday (it took me just over a week to finish writing this blog post), I heard that they had moved the gss especially for me, so I could make it! Unfortunately, since I thought I’d cancelled, I hadn’t worked on a talk throughout the week, and had to come up with something on the fly. It ended up being fine, but I have high standards for myself, and I do like to have a bit of time to prepare. The great news, though, is that we’re back in person, so I could do a blackboard talk (or whiteboard, as it were) and I didn’t need to make slides. 
Qual Recap
I passed ^_^! Now that I’ve gotten my two exams back, I want to talk a bit about what the experience was like, as well as explain my thought process for some of the problems. I’ve had another friend (Tim Rechen) over the past few days, so I still haven’t gotten around to writing up the last analysis qual prep post about Fourier Analysis. I promise I’ll do that soon, though!

Interlude  The Baire Category Theorem
We’ve had a few really detailed posts in a row – summarizing the main theorems of some major objects of study in analysis, and putting these theorems in contexts which (I hope) make them feel coherent and memorable. Today’s post is going to be a bit more relaxed. Two of my best friends (Remy and Alyss) are visiting me, so I want a slightly shorter form post to make sure I have time to spend with them. The Baire Category Theorem seems like the perfect topic!

Hilbert Spaces
Hilbert Spaces are banach spaces whose norms come from an inner product. This is fantastic, because inner product spaces are a very minimal amount of structure for the amount of geometry they buy us. Beacuse of the new geoemtric structure, many of the pathologies of banach spaces are absent from the theory of hilbert spaces, and the rigidity of the category of hilbert spaces (there is a complete cardinal invariant describing hilbert spaces up to isometric isomorphism) makes it extremely easy to make an abstract hilbert space concrete. Moreover, this “concretization” is exactly the genesis of the fourier transform!

Banach Spaces and Preserving Finite Dimensional Theorems
Banach Spaces are ubiquitous in analysis, and they let us rein in analytic objects using algebra (in particular, vector spaces) and completeness. Infinite dimensional vector spaces can be pathological, but by restricting attention to continuous operations for our topology, we can recover analogues for a lot of the finite dimensional theory!

Examples of Syntax/Semantics Theorems Throughout Math
I’ve been promising a blog post on syntax and semantics for a long time now. There’s a lot to say, as this duality underlies much of mathematical logic, but I want to focus on one particular instance of the syntax/semantics connection which shows up everywhere in mathematics. I talked about this briefly in a talk last year (my blog post debriefing from the talk is here) but it’s a kind of squishy and imprecise observation. Because of that, this post is going to be less expository than my usual ones, and is instead going to be a “definition by examples” of sorts. I’ll try to show examples from as many brances of math as possible, and hopefully by the end it becomes clear what the flavor of these theorems is, as well as how ubiquitous they are!

$L^p$ Spaces
On to day $2$ of qual prep boogaloo.
YesterdayTuesday[^13] was all about measures and integrable functions. Today, then, let’s talk about spaces of integrable functions. It’s time for $L^p$ spaces! 
Measure Theory and Differentiation (Part 2)
This post has been sitting in my drafts since Feb 22, and has been mostly done for a long time. But, with my upcoming analysis qual, I’ve finally been spurred into finishing it. My plan is to put up a new blog post every day this week, each going through some aspect of the analysis that’s going to be on the qual. Selfishly, this will be great for my own preparation (I definitely learn through teaching) but hopefully this will also help future students who want to see a motivated treatment of the standard analysis curriculum.

Showing There is No Element of Some Order in a Group
I took a practice algebra qual the other day, and was totally stumped by a pretty basic group theory question:

A geometric proof that $D_{2m} \leq \mathfrak{S}_n$ is possible for $m > n$
To nobody’s surprise, I was on MSE tonight, and saw a simple question about group theory. The original question doesn’t matter as much as a question it made me wonder to myself:

Solving Solvable Polynomials with Galois Theory (Part 1)
I’m super excited to be writing this post up! It’s been haunting me for almost exactly a month now, and it feels good to be close enough to done that I can finally share my hard work with the world ^_^.

Iteration Asymptotics
I really like recurrences, and the kind of asymptotic analysis that shows up in combinatorics and computer science. I think I’m drawn to it because it melds something I enjoy (combinatorics and computer science) with something I historically struggle with (analysis).

How Many Group Structures on a Set?
And so ends my first year of grad school. I’m pretty tired, and my mental health has taken a turn for the worse, though it’s hard to piece together if the last few weeks were tiring because my mental health was declining, or if my mental health is in decline because the last few weeks were tiring. Probably a little bit of both. Anyways, I have some free time again and a backlog of ideas for blog posts. Speaking of, now that my life update is out of the way, let’s see a kind of cute computation!

A Wild Arctan Formula
Yesterday a good friend of mine sent me the following bizarre formula:

Making Pretty Pictures for Galois Theory
So in my algebra class we’re doing galois theory, a subject which never seems to really click with me. I know a lot of the theorems, and I can even solve a lot of the problems, but I always feel uneasy about it. The computational problems often feel like guesswork, and the theoretical problems feel either trivial or impossible with little in between. I used to feel this way about analysis, but it stings more to be struggling so much with a subject so near to my heart.

Finite Calculus, Stirling Numbers, and Cleverly Changing Basis
I’m TAing a linear algebra class right now, and the other day a student came to my office hours asking about the homework. Somehow during this discussion I had a flash of inspiration that, if I ever teach a linear algebra class of my own, I would want to use as an example of changing basis “in the wild”. When I took linear algebra, all the example applications were to diagonalization and differential equations – but I”m mainly a discrete mathematician, and I would have appreciated something a bit closer to my own wheelhouse.

Reducing to $\mathbb{Z}$  Permanence and Concrete Proofs
There are lots of ways in which good notation can make results seem obvious. There are also lots of ways in which “illegally” manipulating expressions can give a meaningful answer at the end of the day. It turns out that in many cases our illegal manipulations are actually justified, and this is codified in the principle of Permanence of Identities! This is one place where category theory and model theory conspire in a particularly beautiful (and powerful) way.

Two Sage Visuals
I’m in a reading group with Elliott Vest and Jacob Garcia (supervised by Matt Durham) where we’re talking about CAT(0) Cube Complexes. We’re reading a set of lecture notes by sageev (pdf here, for the interested) and we came across a fairly simple problem that we wanted to draw. In a completely different vein, Russell Phelan asked a fun topological question in the UCR math discord, and to solve it I ended up needing to draw something else. I figured I would write up a quick post about both visualizations, since these things can be a bit tricky to get right.

Remembering the Reverse Triangle Inequality
The quarter is over, and now that I’m vaccinated (twice!) I feel comfortable seeing people again. So I flew east coast to see my family and a bunch of friends. Before I left, I had a few ideas for blog posts, and figured I would get around to writing one now.

Checking Concavity with Sage
I haven’t been on MSE lately, because I’ve been fairly burned out from a few weeks of more work than I’d have liked. I’m actually still catching up, with a commutative algebra assignment that should have been done last week. I was (very kindly) given an extension, and I’ll be finishing it soon, though.

Talk  Problem Solving Without Ansibles  An Introduction to Communication Complexity
Wow, two talk posts in one day! Thankfully the actual talks were a week apart!

Talk  Categories, Modalities, and Type Theories: Oh My
Last week I gave a talk at CMU’s Graduate Student Workshop on Homotopy Type Theory (HoTT). You can see the schedule of talks here.

Cohomology Intuitively
So I was on mse the other day…

Linearly Ordered Groups and CH
Earlier today Jonathan Alcaraz gave a GSS talk about Linearly Ordered (LO) Groups, which are a fun topic with connections to dynamics, topology, geometric group theory, etc. This reminded me of a problem I told myself to think about a while ago, and so I decided to finally do that. After a bit of thought, a friend from CMU (Pedro Marun) and I were able to figure it out. This post is going to be somewhat more meandering than usual (if you can imagine such a thing), because I want to showcase what the flow of thoughts was in solving the problem. At the end I’ll clean things up and write them linearly.

Sage Sums
Today I learned that sage can automatically simplify lots of sums for us by interfacing with maxima. I also learned recently that the
init.sage
file exists, which let me fix some minor gripes with my sage. Notably, I was able to add commandsaa
andnn
which automatically get ascii art or a numeric answer for the most recent expression! This is going to be a short post just to highlight how these things work, since I had to figure them out for myself. 
Measure Theory and Differentiation (Part 1)
So I had an analysis exam
yesterdaylast weeka while ago (this post took a bit of time to finish writing). It roughly covered the material in chapter 3 of Folland’s “Real Analysis: Modern Techniques and Their Applications”. I’m decently comfortable with the material, but a lot of it has always felt kind of unmotivated. For example, why is the Lebesgue Differentiation Theorem called that? It doesn’t look like a derivative… At least not at first glance. 
Talk  Why Think  Letting Computers do Math for Us
Yesterday I gave my second talk at the Graduate Student Seminar at UCR. I decided to talk about something near and dear to me, a topic which first got me interested in logic: decidability. The idea of decidability is to look for theories which are simple enough to admit a computer program (called a decider) which can tell you whether or not a given sentence is true.

Talk  Syntax and Semantics (Trans Math Day 2020)
I’ve been busy with some assignments and grading, so it took me a while to post this. We got there eventually, though! I gave a talk at an online conference for Trans Math Day on December 5th. There were a lot of interested speakers, so the organizers gave us 5 minute, 10 minute, and 20 minute spots. I was given a 5 minute talk, which is a borderline impossible assignment – Obviously I was still exited to give it, there were just a slew of challenges to work out.

Automorphisms Don't Extend
I was on mse last night (later than I should have been…) when I saw a really interesting question. In the interest of keeping the blog post self contained, I’ll transcribe the question here (with some notational edits):

Talk  Programming and Category Theory
Yesterday I gave a talk at the UCR Category Theory Seminar. I ended up putting off making the slides for longer than I should have, because I wasn’t entirely sure what I wanted the talk to be. The connections between Cartesian Closed Categories/Proof Theory and Constructive Logic/Programming Languages run extremely deep, and ths kind of talk can kind of be arbitrarily abstract. I wanted to make sure this talk was easily approachable, though, and it was tricky to find that balance.

Quantitative Cesaro Sums
To the surprise of no one, I was on math stackexchange earlier and saw an interesting analysis question. I have a weird fascination with tricky limit questions because I feel like I’ve always been bad at them. I like working on them for the same reason I like practicing the difficult parts of pieces of music – it makes me feel like I’m improving (in a “no pain no gain” kind of way).

Limsups and Liminfs of Sets
I’m in a measure theory class right now, and I think it’s important to be properly comfortable with measure theory in a way that I’m currently not. It has deep connections with things that I find very interesting (Descriptive Set Theory, Ergodic Theory, their intersection in Amenable Groups, etc.) and it’s one of my go to examples of an “obviously useful” branch of math. If you can apply your interests to measure theory somehow, I think that’s a compelling argument to fend off questions of “why is this worthwhile” (at least from other mathematicians).

Nilpotentizing Groups
I really like group theory, and I’ve spent a lot of time reading about groups and their properties. Most of these properties seem like very natural things to consider ($p$groups, abelian groups, simple groups, etc.) and the ones that don’t typically seem motivated by some external factor (solvable groups come to mind). However, I have always been somewhat confused by nilpotent groups. I know that they are “almost abelian”, and I can rattle off some facts about them and sketches of proofs… But it was never made clear to me how to work with them in practice. If I come across a nilpotent group in the wild, how does that help me? Surely I should be able to leverage the “almost abelian”ness in a way that’s more general than “elements of coprime order commute”.

$H_1 \cong H_2$ doesn't mean $G / H_1 \cong G / H_2$!
I spend a lot of time on math stackexchange (mse), and I periodically see “simple” questions that totally shatter a misconception that I didn’t know I even held.

Talk  Model Theory and You
Today I gave my first talk at my new department. I was pretty nervous going into it for a few reasons. Obviously giving your first talk at a new institution is going to be stressful. This was going to be my introduction to a lot of the older grad students, and I really wanted to make a good first impression. This was also my first zoom talk, and my first proper talk using premade slides. I typically take a more improvisatory approach to my talks, and I plan out 45 different (similar) talks, and change what I’m presenting based on audience interest and time constraints. Since I almost always give blackboard talks, it’s easy to introduce or omit an example on the fly without my audience catching on. This is not unlike many magic tricks I perform where the ending depends on the spectator’s choices. Since the audience hasn’t seen the trick before (or the talk, in this case) you can totally change the ending without anyone knowing. Of course, this goes out the window when you’re writing slides in advance. If you skip an example, your audience sees you skip past a few slides. I’m glad I’m getting experience with this more structured setup, but it’s still out of my comfort zone. It forces you to, basically, set the talk in stone, and I’m worried that what I set will not live up to my expectations. Thankfully, in this instance, the talk was extremely well received. I’m trying to convince my department to care more about logic, and it seems this was a good first impression for both me and the subject I’m evangelizing. If anyone is interested in seeing the slides, I’ve attached them to this post.

Submodels vs Elementary Submodels
I’m giving a GSS talk next week in the hopes of introducing myself to my new department. More importantly, I am giving this talk to try and showcase the utility of model theory in combinatorics and algebra. While writing this talk, I’ve been thinking about the best way to discuss elementary submodels as opposed to regular old submodels. When it was taught to me, a focus was placed on “quantifying over extra stuff”, which is true, but I was never shown a simple example. While I was thinking about how to teach it, I realized we can already find elementary and nonelementary submodels in graph theory, and that these might serve as good examples for people new to the area.

Existence through Fixed Points
While watching this lecture (in case the link breaks one day: Steffano Luzzatto teaching Dynamical Systems at ICTP, lecture 11), I saw my first clever use of fixedpoint theorems in analysis. In particular, we used a leveledup version of the contraction mapping theorem to show that a certain map was actually a homeomorphism. The main takeaway is that by rearranging what you want to be true into a statement that some object is a fixed point, we can show that our dreams can be realized. This mode of argument is extremely common in algebra and logic, so it’s like a friendly face in the often stressful world of analysis.

Lower Bounds from Closed Sets
This is the start of a series of short posts which will come sporadically. To say that I struggle with analysis would be akin to saying that Stravinsky’s “Rite of Spring” wasn’t well received during its premiere. While technically true, the reality is much more dramatic than you might expect. I’m obviously being a bit hyperbolic here, but it really is something that I find quite difficult (this is probably due in part to lack of exposure).

Right Angled Artin Groups
I’m starting my PhD at UC Riverside this fall, and I’ve enrolled in a program that introduces me to some faculty and gets me familiar with the campus… Or, it would were it not for covid. Regardless, I’ve been meeting with a summer advisor (Matt Durham) over zoom every week for a while now to talk about hyperbolic surfaces and mapping class groups. I’m not sure I’m ready to make a blog post about that material, but last week we talked about Right Angled Artin Groups (RAAGs), which arise as subgroups of mapping class groups in a very natural way (cf. this article by Thomas Koberda). Raags, as we’ll soon see, are controlled by their combinatorial structure, which gives us a good way of creating counterexamples by converting algebraic conditions into combinatorial ones.

Visualizing Hyperbolic Isometries
Welcome to the inaugural post of this blog! As a quick preface, I plan to post about math, particularly math that I’m struggling with, and hopefully we can work through some stuff together ^_^.