Latest Posts

• Finiteness in Sheaf Topoi

  The notion of “finiteness” is constructively subtle in ways that can be tricky for people new to the subject to understand. For a while now I’ve wanted to figure out what’s going on with the different versions of “finite” in a way that felt concrete and obvious (I mentioned this in a few older blog posts here and here), and for me that means I want to understand them inside a sheaf topos $\mathsf{Sh}(X)$. I’ve thought about this a few times, but I wasn’t able to really see what was happening until a few days ago when I realized I had a serious misconception about picturing bundles and etale spaces! In this post, we’ll talk about that misconception, and spend some time discussing constructive finiteness in its most important forms.

• Life in Johnstone's Topological Topos 3 -- Bonus Axioms

  In the first post of the series, we talked about what the topological topos is, and how we can think about its objects (and, importantly, how we can relate computations in the topos $\mathcal{T}$ to computations with topological spaces in “the real world”). In part two, we talked about algebraic structures, and how (for example) groups in $\mathcal{T}$ are related to topological groups.

• Life in Johnstone's Topological Topos 2 -- Topological Algebras

  In the first post, we introduced Johnstone’s topological topos $\mathcal{T}$ and talked about what its objects look like. We showed how the interpretation of type theory in $\mathcal{T}$ gives us an “intrinsic topology” on any type we construct. We also alluded to the fact that, by working in $\mathcal{T}$ as a universe of sets, we’re able to interact with topological gadgets by forgetting about the topology entirely and just manipulating them naively as we would sets!

• Life in Johnstone's Topological Topos 1 -- Fundamentals

  I’ve been thinking a lot about the internal logic of topoi again, and I want to have more examples of topoi that I understand well enough to externalize some statements. There’s more to life than just a localic $\mathsf{Sh}(B)$, and since I’m starting to feel like I understand that example pretty well, it’s time to push myself to understand other important examples too!

• Talk -- What is Factorization Homology?

  I was recently invited to speak at the AMS Sectional in Tallahassee, Florida. In particular, at the special session on Homotopy Theory and Category Theory in Interaction. The conference was this weekend, and I’m typing this up on my plane ride home. I had a great time, and met a lot of great people! The special session was pretty small, so we all got to talk to each other, and I got surprisingly close with the people I went out for lunch (and later, drinks) with!

• Proving Another "Real Theorem" with Topos Theory

  Another day, another post that starts with “So I was on mse…”, lol. Somebody asked whether maximizing over a compact set is a continuous thing to do. That is, given a continuous function $f : K \times X \to \mathbb{R}$ is the function $x \mapsto \max_{k \in K} f(k,x)$ continuous?

• Internal Group Actions as Enriched Functors

  Earlier today this month on the Category Theory Zulip, Bernd Losert asked an extremely natural question about how we might study topological group actions via the functorial approach beloved by category theorists. The usual story is to treat a group $G$ as a one-object category $\mathsf{B}G$. Then an action $G \curvearrowright X$ is the same data as a functor $\mathsf{B}G \to \mathsf{Set}$ sending the unique object of $\mathsf{B}G$ to $X$. Is there some version of this story that works for topological groups and continuous group actions?

• Talk -- 2-Categorical Descent and (Essentially) Algebraic Theories

  A few weeks ago I gave a talk at the CT Octoberfest 2023 about some work I did over the summer that I’m really proud of. Unfortunately, while writing up the result I found a 1999 paper by Pedicchio and Wood that proves the same theorem (with roughly the same proof), so I wasn’t able to publish. Thankfully, the work is still extremely interesting, and I was more than happy to talk about it at a little online conference for other category theorists ^_^.

• A truly incredible fact about the number 37

  So I was on math stackexchange the other day, and I saw a cute post looking for a book which lists, for many many integers, facts that Ramanujan could have told Hardy if he’d taken a cab other than 1729. A few days ago OP answered their own question, saying that the book in question was Those Fascinating Numbers by Jean-Marie De Koninck. I decided to take a glance through it to see what kinds of facts lie inside (and also to see just how many integers are covered!). Not only was I overwhelmed by the number of integers and the number of facts about them, the preface already includes one of the single wildest facts I’ve ever heard, and I have to talk about it here! Here’s a direct quote from the preface:

• Preprint -- The RAAG Functor as a Categorical Embedding

  After almost a year of sitting on my hard drive, I finally had time in August to finish revising my new preprint on Right Angled Artin Groups (Raags). And in September I had time to put it on the arxiv for people to see! Within 24 hours I had an email from somebody who had read it, and was interested in reading it closely! It’s super exciting to see that people are actually reading something I wrote, and it’s really validating to feel like the math I did last year was worth it ^_^. In this post, I’d love to give a quick informal description of what’s going on in that paper.

• Estimating a Difference of Products

  Wow, it’s been a long time! Both since my last blog post, and since my last quick analysis trick. But I’ve been itching to write more blog posts lately, and I thought that something quick and easy like this would be a good way to get back into it without the kind of effort that goes into some of my longer-form posts (which I’m still working on, of course).

• Talk -- What is Algebraic Geometry and Why Should You Care?

  So an embarrassing amount of time ago (Feburary 17?) I gave a talk for the undergraduate math club titled “What is Algebraic Geometry, and Why Should You Care?”. I think it went quite well, and the audience seemed like they had a good time. I really wanted to have the talk recorded, since this is exactly the kind of talk I would have wanted to see as an undergrad and I think it should be available to more people. Unfortunately we weren’t able to make it happen, so we’ll have to wait until the next time I give this talk[^1].

• A Quick Application of Model Categories

  Almost exactly a year ago (time flies!) I was thinking really hard about model categories in preparation for the HoTTEST summer school. I learned a TON doing this, but I’ve just today seen a really nice (and somewhat concrete) reason to care about the whole endeavor! I’d love to share it ^_^

• Building Objects with Category Theory

  Typically category theory is useful for showing the uniqueness of certain objects by checking that they satisfy some universal property. This makes them unique up to unique isomorphism. However, category theory can also be used in order to show that objects exist at all, usually with the help of the Adjoint Functor Theorems.

• Externalizing Some Simple Topos Statements

  Hey all! It’s been a minute. I’ve been super busy with the UC strike and honestly I haven’t done math in any serious capacity for almost the past month. It’s been a lot of hard work trying to get fair contracts out of the UC, but I had a lot of travel plans this December to see my friends, so I’ve taken a step back from the picket line until January. Right now I’m in Chicago, so here’s an obligatory bean pic:

• Talk - Where Are The Open Sets?

  I was invited to give a talk at HoTTEST 2022, and was more than happy to accept! Ever since I was first learning HoTT I was curious how we could be sure that theorems in HoTT give us corresponding theorems in “classical” homotopy theory. Earlier this summer I spent a lot of time wrestling wtih $\infty$-categories (culminating in a series of blog posts) and I realized I was quite close to finally understanding the connection between HoTT and classical topology. I think this is a question that should have a more accessible answer (the machinery is fairly intense, but that shouldn’t prevent a high level overview) and I’m happy I was able to help fill this gap!

• Monoidal Monoidoidoids

  So I was on the nlab the other day, and I saw a fantastic joke: A 2-category is “just” a monoidal monoidoidoid! Here’s a screenshot in case the nlab page for 2-categories changes someday:

• Embedding Dihedral Groups in Vanishingly Small Symmetric Groups

  After the long and arduous process of writing my previous posts on homotopy theories and $\infty$-categories, it’s nice to go back to a relaxed post based on an mse question I answered the other day. Nature is healing ^_^.

• Why Care about the "Homotopy Theory of Homotopy Theories"? (Homotopy Theories pt 4/4)

  It’s time for the last post of the series! Ironically, this is the post that I meant to write from the start. But as I realized how much background knowledge I needed to provide (and also internalize myself), various sections got long enough to warrant their own posts. Well, three posts and around $8000$ words later, it’s finally time! The point of this post will be to explain what people mean when they talk about the “homotopy theory of homotopy theories”, as well as to explain why we might care about such an object. After all – it seems incredibly abstract!

• An Interlude -- Quasicategories (Homotopy Theories pt 3/4)

  In the previous post, we defined $\infty$-categories as being categories “enriched in simplicial sets”. These are nice, and fairly quick to introduce, but if you start reading the $\infty$-category theoretic literature, you’ll quickly run into another definition: Quasicategories. In this post, we’ll give a quick introduction to quasicategories, and talk about how they’re related to the $\mathcal{S}$-enriched categories we’ve come to know and love.

• Why Care About Infinity Categories? (Homotopy Theories pt 2/4)

  The title of this post is slightly misleading. It will be almost entirely about $\infty$-categories, but it will have a focus on how $\infty$-categories solve some of the formal problems with model categories that we outlined in part 1 of this trilogy.

• What are Model Categories? (Homotopy Theories pt 1/4)

  I’m a TA at the HoTTEST Summer 2022, a summer school about Homotopy Type Theory, and while I feel quite comfortable with the basics of HoTT, there’s a ton of things that I should really know better, so I’ve been doing a lot of reading to prepare. One thing that I didn’t know anything about was $\infty$-categories, and the closely related model categories. I knew they had something to do with homotopy theory, but I didn’t really know how. Well, after lots of reading, I’ve finally figured it out, and I would love to share with you ^_^.

• Chain Homotopies Geometrically

  The definition of a Chain Homotopy has always felt a bit weird to me. Like I know that it works, but nobody made it clear to me why it worked. Well, the other night I was rereading part of Aluffi’s excellent Algebra: Chapter 0, and I found a picture that totally changed my life! In this post, we’ll talk about two ways of looking at chain homotopies that make them feel more like their topological namesake.

• Locale Basics

  This quarter I’ve been running a reading group for some undergraduates on pointless topology. I want to take some time to write down a summary of the basics of locales (which are the object of study of pointless topology), both so that my students can have a reference for what we’ve covered, and also because I think there’s still space for a really elementary presentation of some of these topics, with a focus on concrete examples.

• How many symbols can $f'(x)$ have if $f$ has $n$ symbols?

  The other day SMBC put up a lovely comic which did a great job nerdsniping me. I knew that I wanted to try to solve it as soon as I saw it, but I didn’t have the time for a little while (it’s midterm season and I had grading to do). It’s a cute problem, and I want to share my solution with all of you! First, here’s the comic that started it all:

• Using Geometry in Logic

  One thing that I talk a lot about is the (surprisingly tight) connection between geometry and logic. I feel like this is something that one usually gains an appreciation for by seeing lots of examples, and I found a particularly simple example today on mse.

• How Holomorphic Functions are Just Like Polynomials

  I took a complex exam last week a while ago, and while I was studying I realized that a lot of the theorems were saying that holomorphic functions behave like polynomials. This makes sense, since a holomorphic function, which locally has a power series, looks like a polynomial of infinite degree, but there’s actually quite a bit to say here! With that in mind, I decided to write up some quick thoughts about this, in line with my post from a while ago talking about banach space theorems generalizing finite dimensional linear algebra (see here). Now, on with the show!

• Why is the Completion of a Local Ring "More Local" than just Localizing?

  An oft-repeated piece of intuition I’ve seen while trying to learn algebraic geometry is that localizing a ring at a prime is like “zooming in” on that point. But if you want to zoom in “really close” then you have to take the completion of this ring… Why is that?

• Talk (?) -- Why Care about Lie Algebras?

  I gave my second (and last) presentation/talk in my lie algebras class today, and it was on a topic that was and is really important to me. When learning something new, I think it’s worth asking yourself what it does for you. What problems does it solve? How do the structures we’re learning about arise? There’s a lot of people who enjoy abstraction for its own sake, but I ultimately care about solving problems, and while I’m not going to shy away from high abstraction mathematics to do so (I’m still a category theorist after all) I think it’s important to be aware of concrete examples that can ground your theory in things that obviously matter.

• Talk -- Let's Solve a Simple Analysis Problem. Together.

  Last Friday I gave a talk in GSS where I tried to give a super concrete application of topos theory to “mainstream” mathematics. The idea was to solve a simple analysis problem, streamlining the argument by using the internal logic of a sheaf topos. The good news is that I think I was quite successful in making my point that “ordinary mathematicians” should care about topos theory and constructive logic. The bad news is that the last 10 minutes of my talk were false… It didn’t end up mattering, but I was still pretty torn up about it. Anyways, in this post I’ll give an overview of the talk, which should double as a nice description of how to actuallly use topos theory to solve problems. I’m planning to start a new series soon where I go over this proof in more detail, explaining the relevant aspects of topos theory along the way!

• Talk (?) -- Universal Enveloping Algebras

  This barely counts as a talk, but I want it catalogued with the rest of the talks I’ve given, because this is going to have a retrospective aspect to it like all of my post-talk posts do. A while ago now, I gave a 30 minute presentation in my Lie Algebras class where I answered two questions that I’d brought up over the course of the class, with the unifying thread being this: both questions are naturally answered by the Universal Enveloping Algebra.

• A New (to me) Perspective on Jordan Canonical Form

  Lie Algebras have been on my to-learn list for a fairly long time now, and I’m finally taking a class focusing on them (and their representation theory). Our professor is very concrete in how she presents things, and so this class is doubling as a higher level review of some matrix algebra, which I’ve been enjoying. In particular, last week we talked about the Jordan Canonical Form in a way that I quite liked, and which I’d never seen before. I’m sure this will be familiar to plenty of people, but I want to write up a thing about it anyways just in case!

• Automatic Asymptotics with Sage

  Recurrence relations show up all the time in combinatorics and computer science, and even simple objects give recurrences that are difficult or impossible to solve. Thankfully, we can often find a generating function for our objects, and through the power of complex analysis and algebraic geometry, we can use the singularities of the generating function in order to get good asymptotic estimates. Software like Maple can automatically compute asymptotic expansions for a lot of generating functions using the asympt function… Can sage?

• 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 whore always 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, yesterday last 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. Yesterday Tuesday[^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 commands aa and nn 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 yesterday last week a 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 pre-made slides. I typically take a more improvisatory approach to my talks, and I plan out 4-5 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 fixed-point theorems in analysis. In particular, we used a leveled-up 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 ^_^.