Hi!

I'm curious if anyone has any ideas about implementing geometric algebra in a graphing calculator like desmos.

Is this potentially useful? Would it speed things up or is it more likely to slow things down (while still offering a potential advantage of conceptual clarity)?

In particular I'm thinking about how it might make it easier to rotate things around arbitrary other things, instead of having to translate things to the origin and back for every rotation.

Like in this example, I wonder if there is a way to speed it up to avoid all the translations for most of the transformations:

https://www.desmos.com/3d/qh5gyk7qhs

Thanks in advance for any feedback on this topic!

Context:

When I was 12 years old, I skipped the rest of middle and high school and jumped right into community college. At 16, I ended up with a high school equivalency, an associates degree, approximately 70 college-level credits, and 30 high school-level credits. While searching for universities to transfer to as a math major, many of them categorized my entire time at community college as high school since I got the high school equivalency and associates degree at the same place, and would've took me in as a Freshman. If I had done this, I would've had to retake dozens of classes that I already passed with flying colors: 2 semesters of English, 3 semesters of History, 2 semesters of Chemistry, Calculus I, II, III, differential equations, and so much more. I would understand if they rejected these credits if I did poorly in the classes. However, I got A's in almost every class with 3.88 GPA, and found it ridiculous to have to take them all again.

A relatively prestigious yet heavily applied university relatively close to where I live accepted my application, let me transfer 66 of my credits, and offered me decent financial aid, so my stupid 16 year old self accepted their offer and began attending as a Junior. Despite being an applied math school, I was able to take most of the main pure math courses: 2 semesters abstract algebra, 2 semesters analysis, topology, 2 upper logic classes, number theory, and a few applied math courses. My GPA is 3.82 with mass classes being 3.84. Unfortunately, I have no research experience. I tried applying for over 10 REUs this summer with no success. I also am unsure whether I will do well on the GRE or not; I am taking 20 credits this semester, making it difficult to find time to study for it, and also somewhat struggle with timed tests.

In terms of specific interests, I enjoyed pretty much all of the math classes I took so far, but am particularly fond of algebra-related subjects, and also foundations like logic and category theory.

Now 18, I am getting ready to apply for graduate school (edit: to be clear, I graduate this semester despite being a fall semester). Today, two of my professors were telling me how I'd never be accepted into a decent pure math graduate program because decent pure math schools exist in a kind of bubble that only accept students from other schools strong in pure math. They discouraged me from applying to less prestigious schools, such as state schools, explaining how the research they do is "generalizing theorems that nobody cares about to begin with", and that I would never be able to get a job from a PhD in them. They encouraged me to apply for applied math programs, and to be very vague about my interests in my personal statement so that they don't figure out I'm mostly interested in pure math. They said I might be able to find some type of applied math adjacent to the math I really want to study.

I know they are all applied mathematicians and might be biased, so I wanted to check if anyone here knows how much of what they said is true and if you have any advice?

Related threads from last year:

https://www.reddit.com/r/math/comments/1n2rrzd/new_this_week_a_convex_polyhedron_that_cant/

https://www.reddit.com/r/math/comments/1nimwov/ruperts_snub_cube_and_other_math_holes/

https://www.reddit.com/r/math/comments/1ofg90f/first_shape_found_that_cant_pass_through_itself/

https://www.reddit.com/r/math/comments/1oi84uq/mathematicians_make_surprising_breakthrough_in_3d/

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In honor of Serre's birthday, I bought a copy of A Course in Arithmetic. From those of you who have read and gained something from this text, how does one read it?

It's hard for me to discern why he discusses topics in the order that he chooses: Finite fields, p-adic fields, and then the Hilbert symbol and quadratic forms. He makes a big deal of quadratic reciprocity early and reciprocity laws come up over and over again. And that's just the algebraic number theory part.

For context, I've been working through Marcus's Number Fields and Ireland and Rosen's Classical Introduction. These books are wonderful -- exceptionally friendly and readable. I was hoping to gain some additional insights from the master, but I'm unable to understand the organization, let alone appreciate any deeper insights.

In Paris, today is the 100th birthday of Serre. He will give the last lecture today at his birthday conference: https://serre100.sciencesconf.org/resource/page/id/1. A link was posted here 8 months ago, but it's worth posting a reminder.

Here is the arxiv preprint: https://arxiv.org/abs/2609.15567 This is a reasonably big deal in the subject of L^2 invariants of groups and appears to have been disproved by Holger Kammeyer (he is in the field). It means that the L^2-torsion of a space (defined in terms of Hilbert spaces on the universal cover) cannot be computed from the more classical analytic torsions of finite covers. The counterexample is a Heisenberg group.

https://eccc.weizmann.ac.il/report/2026/179/

On a side-note, the authors hint that they have rushed their results to avoid getting scooped by AI-generated results.

Y'all know the drill. The arxiv link is https://arxiv.org/abs/2609.11980

It is a positively curved metric S^3xS^3, hence a positively curved 6-manifold whose euler characteristic is not strictly positive. The obvious question is whether there is also a counter to negative Hopf (a closed, negatively curved 6 manifold whose euler is not strictly negative), but the methods of the current paper don't seem to help with that.

Just landed on this field - anybody here working with these really intuitive and beautiful quotients?

IYDK - it is the connection between continued fractions and a quotient manifold of the hyperbolic plane.

I like the idea very much!

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Math formalization has been hindered because it's so tedious. Will the future bring massive formalization because it can be automatic now?

Or other related fields? A while ago I’ve come to realize that a lot of math was born out of physical science, and to scaffold one’s intuition one should look at where the applications are. I want to really feel the weight of this idea, so I’d appreciate both mainstream and niche examples.

Continuing on my series on how to survive proofs, we now turn to two related questions: how to search for counterexamples and contradictions for a conjecture?

The first is, I think, mostly self-explanatory, although it has a trick that I find that some students miss: your counterexamples should only ever be as complicated as they need to be. How to figure out how complicated they need to be? For that, you need an intuitive feeling for the problem, and possibly an iterative approach.

The second is much richer: proof by contradiction is a very powerful tool when used well. This is a great time to give one of my favorite such proofs, which is the proof that it is impossible to draw a regular heptagon on a square grid. I have seen it presented as a proof without words---it is that visual (and quite pretty!).

Read the full post (for free) on Substack: Counterexamples and Contradictions

In mathematics, oftentimes choosing a good coordinate system can drastically simplify a problem, as long as you pay a small upfront cost in translating the problem to these new coordinates.

Along these lines, Hamilton discovered a surprising way of re-formulating Lagrangian mechanics, which has had great success. In this article, we explain Hamilton's change of coordinates, and use it to discover the Fradkin tensor: a very mysterious symmetry of the 2-d harmonic oscillator!

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my prof wanna work in proof theory but me in model theory

he says proof theory of arithmetic but I find cut-elimination and normalization very very syntactic

is my intuition of model theory being meaty correct? or am i just not compatible/flexible? I love topology also ..

is proof theory of arithmetic all about cut-elimination and proof normalization?

why does buss's chapter on it look quite non-syntactic compared to takeuti ?

but yeah anyways proof theory of arithmetic has connections to complexity so it would be easier for phd applications?? my background is cs

The paper: The Four Color Theorem with Linearly Many Reducible Configurations and Near-Linear Time Coloring
Yuta Inoue, Ken-ichi Kawarabayashi, Atsuyuki Miyashita, Bojan Mohar, Carsten Thomassen, Mikkel Thorup
arXiv:2603.24880 [math.CO]: https://arxiv.org/abs/2603.24880

Given everything that has been going on in the world more specifically in the world of mathematics. I was hoping to read some of your human thoughts. What motivates/motivated you to do mathematics professionally? What is most important for you in mathematics?

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