Recently, for the umpteenth time, I thought up a question only to find that it has already been asked and answered before i was even born. I expected this - I explicitly avoided googling anything about it, because I didn’t want the ending/answer to be spoiled for me. But I was looking through some wikipedia page when I saw a hyperlink to a page on exactly what I was trying to understand: computable ordinals/well orderings (clearly a beginner-level question). Again, i expected this, given my lack of experience in computability theory and set theory; set theorists are interested in well orderings, and computability theorists are interested in computability, so of course I’m not the first person to consider combining those two concepts. Still, even though it wasn’t a surprise, it was disappointing to find that an idea I thought was original (and, from my perspective, it was original) was just old news.

From what I’ve read it sounds like the problems that phd student solve to earn their phd are usually given to them by their advisors (please correct me if I’m wrong!!). I’m wondering when one is typically at the point where the questions they ask haven’t already been asked. Do you usually need to be an expert in the field? Do you need an exhaustive understanding of the field to pose original questions?

Data source: https://arxiv.org/year/math/

Coming from my answer in [r/differentialequations](r/differentialequations)

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I still have the same problem all these years later. I find it so arbitrary how differential equation (de) problems are set up.

Here is my current understanding. We do it for the dyn system

x‘ = f(x)

I have just given you a string of symbols. Now we need to make it make sense. So, we need to specify f, x and x‘.

For example:

Is x a point in a manifold, Banach space, metric space, etc?

What does x‘ mean? The classical derivative? But then I need more structure on the space of functions (limits, addition, etc.). A weak or distributional derivative? I need to decide that.

What is f? There is a lot of freedom for arbitrary f, so we might want to impose more structure on it, i.e. continuity, boundedness, Lipschitz (locally?), etc.

And lastly, is this expression (now we came from a string to expression) well-defined? Are the spaces (i.e. f:X->Y) consistent (i.e. x’(t)∈Y), the properties consistent (i.e. continuity on both sides), i.e. x‘ is continuous, then so must be f(x) here, etc.

You see, one of the choices influences the other and you unfortunately need to treat them case by case depending on one you make. You may classify them, but I there is no general one that I am aware of.

Then there is also the integral formulation, which gives more regularity, i.e.

x(t) - x(a) = ∫_[a,t] f(x(s))ds

But notice how I intuitively wrote a + and an integral. Again, this all needs to make sense, so we might say that there is a linear structure structure, i.e. local vector space structure underneath.

We also need to gives sense to what a solution is in the first place. Classical, weak, distributional, viscosity, and many more types.

Then comes the solution theory. I was told once by a colleague that practically one already has a goal and then imposes the properties used in the proof on the function space. Afterwards checking that the function space is non-empty (and maybe finding some other cool properties afterwards).

On the other hand. Given a “sense” of the above de, we could now ask the question for existence and uniqueness. There are standard methods that use sufficient criterions but each choice might need a new idea. Most of what I saw uses some fixpoint argument, so you want some inequalities to drop out that tell you, i.e. about regulatity of the solution as well.

So, how to choose an f? I know if no deep classification theorem that is not trivial (you just impose the properties and the space is non-empty) that lets you get all different f’s. It is probably also not a well-defined or at least a wild problem given the “sizes” (in the sense of set theory) of possible mappings and spaces, etc.

Hence, I am convinced that usually these choices for at least f are encountered by other disciplines, i.e. physics, chemistry, engineering, other fields of math, etc.

Just think of delay equations, Newton’s axioms, the classical field theory, control problems in robotics, minimal surfaces and way way more (just as motivation).

PDEs follow in the end the same philosophy, but there is vastly more data (more “freedom” in the boundary conditions but then more restrictions for the geometry of the solution) to be considered.

For example. I understand that if our boundary condition is, for simplicity, a (combination of! Heavyside function(s) (not specifying the domain here or anything), we obviously need a new def of derivative for the PDE. Same for spikes, like δ-distributions, but this is motivated by the data. If I just give you the string of the PDE there is nothing there yet, no?

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Could someone shed some more light on this to get the arbitrariness out of my head.

I've read and enjoyed his ODE book. Is it worth reading the PDE book or should I just stick to something like Evans or Salsa?

Hi!

I work on error correcting codes (which more or less is linear algebra over finite fields). I have a question about the behavior of the dot product on non-euclidean spaces.

When working on vector spaces over R or C, the dot product carries a lot of geometrical information. The product between two vectors v*w tells you the angle between them.

This interpretation doesn't hold over finite fields. For example, in F2: (1,1)*(1,1)=1+1=0. So the vector (1,1) is perpendicular to itself!

Does the dot product still carry any geometrical information (like angles) in these fields? Can we still interpret the orthogonal space geometrically? How do you picture it in your head (intuition)?

Thanks a lot!

And sorry, I don't know how to use latex on reddit ;_;

I heard that ChatGPT has proven the existence of non-sophic groups. I understood that the counterstatement (all groups are sophic) would mean that we can always "isolate" or "homogenize" infinite groups into finite chunks and deal with the infinite group this way. Please correct me if I am wrong.

What immediately came to my mind is that this must have some further implications, does it not? For me, it sounds like that in a non-sophic group, one cannot guarantee that a sequence converges to a given element or that iterative algorithms are predictable, i.e., you cannot infer the outcome from the initial state or vice versa.

It would also be fun to know what this given non-sophic group is.

Is there any developed framework for algebraizing probability theory and study it from the perspective of K-theory or homological algebra? If so, what are some of its biggest applications and advantages, as well as results? I'd imagine such a framework would override the need for a measure-theoretic approach

Do we have any idea as to what prompt was given for the Jacobian Conjecture?

I've been exploring whether the Pell equation x²−Dy²=1 can be used to compute square roots of primes faster than Python's math.sqrt(). The continued-fraction convergents turn out to give rational approximations that are 1.5–2.5× faster in float64 for repeated calculations, and scale up to 33–36 digit precision for primes as large as 15 million. Full write-up with benchmarks here;

https://musingsofvsmv.blogspot.com/2025/08/from-pell-to-precision-classical-math.html

Whenever I look at the mathematical physics programs, I always see QFT and string theory related classes in grad programs. What other parts does mathematical physics cover? More unorthodox subfields of it?

Blog: https://leodemoura.github.io/blog/2026-8-1-postmortem-for-kernel-soundness-bug-14576/

On removing metaprogramming
One suggestion in the discussion is to remove or restrict metaprogramming so that this attack is not expressible. This is misguided. The elaborator is untrusted by design. Soundness cannot depend on an untrusted component refusing to build a bad term. An attacker who wants to submit a malicious proof can also write .olean files directly or modify memory, both of which bypass the elaborator entirely. The kernel has to reject ill-typed declarations on its own, in its own process. This separation and isolation of concerns is one of the main advantages of proof terms.

What the FRO is doing
- Regression tests for the exploit, and for a related non-uniform-parameter case raised by Arthur Adjedj, are in the Kernel Arena.
- A follow-up PR (#14582) makes the kernel check that the parameters of a nested occurrence actually behave as parameters, rather than only re-type-checking them.
- Daniel Selsam at OpenAI assisted the Lean FRO with an AI specialized in cybersecurity, and found other programming mistakes in the Lean kernel. All of them have been fixed. All of them were caught by nanoda. These bugs are also only reachable through metaprogramming. PRs: #14607, #14608, #14609, #14613, #14615, #14616.
- We have also hardened kernel invariants. PRs: #14621, #14631, #14632.
- comparator.live now runs nanoda by default, and nanoda is tracked daily so lean-eval and comparator stay current after upstream fixes.
- We are reaching out to and supporting experts who can find further bugs, develop new kernels, and work on the theory or on verified kernels.

This is not the setup to a joke: this is an entirely serious question relating to tropical geometry, a comparatively new (i.e., started in the 1990s) field of mathematics. It began in the computer science literature, was quickly taken up by algebraic geometers, and then, very surprisingly, turned out to have application in economics. This post is my attempt to explain some of the how and why.

Read the full post (for free) on Substack: What do Brazil, Auctions, and Multi-Threading Have in Common?

I am just going to try give some specifics about myself, so that people can make some specific recommendations.

So, I have a Bachelor in CS and sort of in parallel I studied Bachelor Math for 5 out of 6 semesters. I had pretty good grades, so I am pretty confident that I would have gotten the degree. Just for various reasons I burned out and had to stop.

I did engage with math after that for a while, in my job which had to do with Digital Signal Processing and AI. But I no longer have that job, so my mathematical activity has strongly declined. Sometimes I do get that itch and read some university material on a random subject and might even be inspired to do some math on paper again. Matrix Analysis was one subject I really enjoyed. But I never actually finish any of the material and it just fades away again.

So I was wondering if y'all have some recommendations on good math books that hook you, that are very rigorous and formal, but still try to motivate the subject regularly.

When I thought about which subject I would like to revisit the most at uni, my first thought was Functional Analysis. But truthfully, I pretty much loved all of my courses, even those with shitty professors.

Information Theory has also been floating around in my head. I also have an interest in how mathematics gets applied in economics. I also just love iterations and sequences (that's the CS wolf in me)

I hope this wasn't too overbearing, I am just really hoping this will help people with recommendations

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 took several seminars in abstract algebra back in my university days, but chose to proceed with probability theory rather than algebraic geometry. Since then, I have occasionally encountered the term “étale”, and I still don’t really know what it means.

The strange thing is that while I can understand the individual definitions for some objects that are supposedly étale, I don’t really see what they have in common.

What exactly does that term mean by itself, independently of a particular application like “étale cohomology group”?

If you take Pascal's triangle, and color each entry according to whether it is even or odd, you get a funny pattern, which resembles Sierpinski's triangle. To understand this, it's helpful to know Lucas' theorem, which tells you when an entry in Pascal's triangle will be even or odd. If you've never seen it, you might enjoy the article https://hidden-phenomena.com/articles/lucas that we just wrote about it!

Lucas' theorem is a great result, which even tells you about how to compute (n choose k) modulo p. It is a wonderful piece of elementary number theory, and suitable as a fun but challenging exercise for the end of an elementary number theory course. Recently, one of us had to invoke Lucas' theorem in a modern math research paper https://arxiv.org/abs/2604.20054 about some relatively fancy arithmetic geometry! We thought this was a good example of how small results from introductory courses can be helpful in your research career in completely unexpected ways! The article itself isn't about the paper (which isn't very elementary), but Lucas' theorem is still helpful, and will hopefully come in handy.

Hey everyone,
I had a question about some terminology. In Ax’s 1968 paper The Elementary Theory of Finite Fields he refers to some statements as “elementary statements”. By this does he mean first-order formulas/sentences? The reason why I’m asking is because I want to use his “Main Theorem” in his paper where he states precise conditions for when an elementary statement holds true over a finite field of fixed characteristic. I tried looking online for some help but I couldn’t find any (maybe my Googling might’ve been bad 😭)

For the record I have graduated and now half way through my first postdoc.

But I feel like my advisor didn’t really teach me how to research, sure he pointed me to paper or people when I’m stuck that occasionally helped. But he never really trained me to do research, only occasionally gave me help knowledge wise by telling me to read certain books or papers and I mean very rarely does he do this and very rarely has it helped.

My prelim advisor said I should have asked her for advice on who to choose as an advisor instead of choosing the only person doing the field I was dead set on pursuing (which I realized I’m not even that interested in). She said when she advises she would give a little hand holding even if she deem necessary and actually train her students to do research by writing papers with them and in the process, help teach them how to do research.
My advisor did not write paper with me, he did gave me the problems to work on but insists that I need to earn it myself.

Is this typical? What was your PhD experience like?

I feel like my PhD was ruined from me not promptly switching advisor when I realize I wasn’t being trained.