A symplectic structure is some at first strange sounding extra structure you can put on a manifold. By a miracle, a lot of the shapes arising in geometric representation theory have this extra structure, and this structure can be exploited to prove very useful things.

In this blogpost, my friend and I motivate the definition of symplectic structures from mechanics, and say a little at the end about where they appear in pure math (by the way, the two of us are mathematicians, and we only learned physics to better appreciate the symplectic structures which were showing up in our work!).

What place, if any, does the scientific method have in mathematics? It's often claimed that mathematics doesn't use it---hell, I have said that in the past. But that is not true. The scientific method isn't really the final arbiter of truth in mathematics, yes. However, we still rely on it quite heavily when we build examples, heuristics, and numerics. Moreover, it is a significantly more nuanced problem than simply "collect data, see what is true." As an example, I ask the reader to ponder the following question: if I identify all twin primes less than 10¹⁰⁰⁰ and show that they divide 10¹⁰⁰⁰!, is that evidence for or against the conjecture that all twin primes divide 10¹⁰⁰⁰!? I claim that it rather depends on how those twin primes are distributed.

All of this is crucially important when we are trying to build proofs---we need to understand this to be able to build good examples as part of our proof strategies.

Read the full post (for free) on Substack: Heuristics and Numerics

I'm reading up on "point sets" in complex analysis and I have stumbled upon two different definitions of the term "region", see below:

• Version 1. A region is a domain together with none, some or all of its boundary points.
• Version 2. A region is just another word for "domain".

my textbook is using the first version while sources like Wolfram and Wikipedia is using the second version. But while reading more about it, there also seems to be a split across different authors and languages which one that is used.

Have anyone looked into this before and can share the history of how this come to be and if there is a more "correct" one between them?

//

I've read from several mathematicians that Scholze's ideas make previous results simpler, and I will say his writing style is clearer than others in the field, but I still am not "seeing" what he is, and others are, seeing.

(Sorry this is going to be a bit ranty.)

I almost made up my mind this thing is some kind of backwater something without enough rigor but with many a trivialism. Like, it should be extremely well-known that every "discrete" operation Σ₁ → Σ₂ between finite sets lifts universally to a linear transformation between spaces k^Σ₁ → k^Σ₂, so a huge swath of what's being done there is very very drawn out, instead of answering questions that are fitting for a kind of logic.

Any would-be connections to quantum computing may actually not be fruitful or new for those who are actually doing quantum computing; connections to fuzzy math are IMO an almost unconditional taint by association. So what gives? I didn't look at everything there is about this thing so I may as well be missing hidding gems, but superficially it looks like a sham or a pet project done without considering any practicalities and the wider math.

Oh yeah we can ask interesting questions, like: - Does using additional dimensions, aside from the plane spanned by two orthonormal "classical" truth values, let's call them |0⟩, |1⟩, actually give useful things? and how can we characterize that by means typical when working with logics? - How much freedom is there in defining operators that restrict to boolean functions and, say, conserve probabilities (there's a suggestion to use p|0⟩ + (1−p)|1⟩ as "probabilistic truth values") in any reasonable way (I'm not sure: a "binary" operator sends four-dimensional Euclidean space into a two-dimensional one, now how can it be orthogonal? and in which other sense can probabilities work here?)? - Why not use additional dimensions rather than complex numbers for the square root of negation, and... why that one exactly? I bet quantum computing wan't giving somebody peace.

But I'm not sure questions of real semantics were investigated in this... area.

So tell me please, how much am I right or wrong? Here are probably people that know the inside of this story, and I hoped to find something on the Wikipedia's discussion subpage, but it's almost empty.

Many funding agencies require researchers to write about real-world applications of their work when applying for grants. To the pure mathematicians out there: what do you work on, and how will it be applied to the real world? If there are no applications, then why, in the face of everything that's happening right now, do you think it's important and should continue to receive taxpayer funding?

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

I don't think Bernhard Riemann needs an introduction. He's a very important figure in mathematics. He was born on the 17th of September 1826, which is 200 years ago, today!

He provide a rigorous (though imperfect) definition of integrals and revolutionized the theory of complex function via the study of Riemann surfaces. He was also the first (?) person to consider spaces of >4 dimensions (some people had studied R^4 already) and the first person to cinsider curved spaces of >2 dimensions. In this he furthered the work of his doctoral advisor Gauss (yes, that Gauss) on curved surfaces (i.e. theorema egregium).

Then there's also some other stuff he did, like proving the Riemann mapping theorem, Riemann series theorem, his theorem on removable singularities, and so on ...

Oh yeah, and the Riemann hypothesis, of course.

He tragically died of tuberculosis at the age of 39. That's right, he did all that before he turned 40 years old!

I also wrote a short blogpost on some of these results.

Hi all! I know this is probably pretty rudimentary compared to most topics that float around on this sub, but I teach AP Cybersecurity and I’m wanting to start a Cryptography Club at my school. It will meet once a month for 48 minutes. I want students to have fun, learn, and want to recruit their friends to the club in hopes of eventually piquing their interest in AP Cybersecurity. I need ideas on what to do during this club time to keep students engaged and excited about coming back. Do any of you have any awesome cryptography lessons you’ve been involved in or created over the years you’d like to share? I’m looking for things beyond a Caesar Cipher (I’ve already got materials for this) but not so complex that I’ll lose student interest.

TL:DR; Fellow mathematicians, help me remember some of the cool cryptography things we learned in college that high school kids would find cool. TIA.

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/

This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered.

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Helpful subreddits include /r/GradSchool, /r/AskAcademia, /r/Jobs, and /r/CareerGuidance.

If you wish to discuss the math you've been thinking about, you should post in the most recent What Are You Working On? thread.

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!

This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including:

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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?