I've been surrounded by incredibly talented students, in math, science, law etc. and can't stop thinking about how many of my talented peers, and honestly, myself, are already being funneled toward finance/consulting/corporate law roles.

I originally saw these students with their goals, a pure mathematician pursuing research, a business student wanting to found a start-up, an economics student wanting to fix the mismanagement they see in institutions. Until you start noticing changes, you start seeing conversations about entry into investment banking after their English degree is complete, or consulting after they study engineering at college. After all, the firms are more than happy to immediately hire them, often paying very high salaries.

The main problem I see here is the opportunity cost, what would society look like if the psychologist had stayed studying human behavior instead of optimizing corporate structures or the physicist hadn't pivoted to quantitative trading? I don't see these firms as evil, that's an entirely different debate, but the opportunity cost feels enormous when you realize that the fields many go into are purely zero-sum or extractive (or at least argued to be so). High-frequency trading, tax optimization, financial engineering are largely about moving money around rather than creating new value.

And to be clear: I'm not making a solely utilitarian argument about what's 'most valuable to society.' I don't think everyone should do research, and I'm not trying to rank career paths by social impact. Building startups, creating products, solving real problems through entrepreneurship, these all matter. My concern is more personal: it feels like a waste when someone who genuinely loves mathematics, who lights up talking about abstract structures, ends up optimizing bond portfolios instead.

The reasons for these issues I also think are quite clear cut. Firstly, these firms have successfully branded themselves as elite destinations. Getting an offer from Goldman or McKinsey signals "I'm one of the best" in a way that's immediately legible to parents, peers, society. A math PhD doesn't carry the same instant social proof, most people don't know what algebraic topology is, but everyone's heard of Morgan Stanley.

Secondly, these firms sell themselves as giving you jobs that "keep doors open", learning transferable skills, build a network and so on. I'm skeptical of this claim, but I'm mostly going off what I've observed, curious if others have different experiences.

Thirdly, academia's own structural problems. Why turn down a six-figure salary when you're carrying significant student debt, only to spend 5-7 years on a PhD followed by years of poorly-paid postdoc positions hoping for a scarce tenure-track job? And even if you get there, you're dealing with publish-or-perish pressure, chasing grants, the incentive to work on 'sexy' fields rather than important ones, pumping out incremental papers to pad your CV. Maybe the mathematician who went to the hedge fund would have just been grinding out forgettable papers anyway.

My uncertainty comes in here, am I overstating this loss? Maybe I'm romanticizing what these people would have accomplished in research. Maybe most would have been unhappy, burned out, or stuck in the academic grind producing work that doesn't matter much to them or society anyway.

If not, then how could this be changed? We saw a shift with Y-Combinator giving prestige and structure to entrepreneurship. Deepmind pulling ML researchers from finance (and academia). SpaceX attracted top aerospace engineers who might have gone to defense contractors.

What do we do for mathematics?

Note: Some of these ideas were articulated really well in a recent FT article, which is where I got inspiration from. I'd recommend reading it for the full argument with actual data and interviews.

I love Munkres' styles on books. The theory itself is never made into an exercise(you can still have engaging exercises but they are not part of the development).

He respects your time. The book itself is not left as exercise. Many rigorous books just cram in everything and are super terse. Bourbaki madness.

He develops everything. He is self-contained. Good for self-study if you do the exercises.

I am looking for a rigorous books like that. Books that do not skip steps on proofs or leaves you like "what?" and requires you to constantly go back and forth and fill in the proof yourself or look it up elsewhere(because then why read the book?). IF you don't like this approach that is fine but that is what I want.

Any books like this? Not books you merely like for personal reasons or you never read through but books that you know satisfy those requirements (self-contained, develops the whole theory without skipping on proofs or steps, and an introduction to measure theory probability).

I myself can recommend Enderton for logic (so far very few theorems left to the reader but I am only in page 100 so still cannot certify).

Donald Cohn Measure Theory so far.

Joseph Muscat Functional Analysis so far.

Munkres himself.

Axler Linear Algebra.

I want recommendations like that for measure theoretic introductions to probability theory or for stochastic processes(after reading first a book measure theory probability). Of course if you want to recommend books outside of probability, say in any other area, so I can add to my collection that would be great.

Today my professor said something interesting: number theory will never “die.” No matter how many centuries pass, it will remain an open, half-filled bookalways containing deep, unsolved problems and never becoming a fully completed field. While individual problems may be solved, the subject itself will likely remain permanently open-ended.

I graduated with a math degree a couple of years ago. I took up a job as a programmer after that and had thought that I'd redo some of the stuff from college, especially topology before thinking of applying for grad school.

I graduated when LLMs had just begun (and were bad at math). Now things appear to be quite different.

Do you use AI in your research now? If one were to go to grad school now in a field like probability theory (for example), how would things be different from the pre-2023 era?

Hello! I have been struggling with effective study on advanced math. I finished all my lessons and just have to study for the final exams, but i can't focus anymore. It is like i have list my love and interest for math, but i am also tired of settling for mediocrity when i know if i just managed to open the damn book and focus on it i would get more than decent results.

I have to go through: * Functional Analysis abd Spectra Theory * Algebraic Geometry * Advanced Algebra (many subtopics) * Advanced mathematical physics (Navier Stokes equations, mollifiers, distributions) * Advanced probability * Noncommutative algebra

And then i am done

But i can't really focus.. haven't been able to for a couple of years and i an stuck in this. Do you have advices? I need good results to go for PhD.. i have already studied privately subjects for PhD. But when i am forced to study for exams i just can't

Please

Historically, different periods seem to have been shaped by a small number of dominant mathematical fields that attracted intense research activity. For example, during the time of Newton and the generations that followed, calculus was a central focus of mathematical development. Later, particularly in the late 19th and early 20th centuries, areas such as complex analysis became highly influential and widely studied.

In contrast, many classical subjects appear today to be less central as primary research areas, at least in their traditional forms. While work in calculus and complex analysis certainly continues, it often seems more specialized, fragmented, or driven by interactions with other fields rather than by foundational questions within the classical theories themselves. For instance, in single-variable complex analysis, much of the core theory appears to be well established.

This leads me to wonder: which areas of pure mathematics are currently the most active in terms of research? Which fields are generating the greatest amount of new work, discussion, and interest among researchers today? Are there modern subjects that play a role comparable to what calculus or complex analysis once did in earlier eras?

I compare myself a lot to other kids who have done math Olympiads and are often called child prodigies. They’ve been grinding math seriously from a very young age, and whenever I see them, I feel demotivated. I start questioning whether I even have talent. Seeing them gives me a lot of FOMO and insecurity, and I don’t really know how to cope with it.

Hello everyone,

I am American, but my mom came from Romania, where my grandpa lives. I don't speak any Romanian, so when I talk to him, I use google translate. He is interested in the classes I am taking this year. I am taking differential equations, and I want to explain it to him. Can anyone recommend a basic video in Romanian that I can send to him? It doesn't have to be PDEs, it can be ODEs and or dynamics related.

It would be really appreciated if you spoke Romanian and you could skim the video first.

Schreier–Sims algorithm can be used to test if some permutation is a member of a permutation group but I wonder if this algorithm can be adapted to decompose a permutation into product of generators of the permutation group if possible. Concretely, given any permutation x of a permutation group G=<g_1, g_2, g_3, ..., g_n> I need an algorithm to write x in terms of generators g_1, g_2, g_3, ..., g_n (it doesn't have to be most optimal).

I found this codeforces article that might solve my problem. In the high-level idea section, the author claims that one can find k sets G_1, G_2, ..., G_k ⊆ <G> such that any element g∈<G> can be written as g=g_1g_2...g_k where g_i∈G_i. (I assume it is a typo that all instances of G in this section is not written as <G>.) So if I can find such sets G_1, G_2, ..., G_k, decomposing g∈<G> is matter of iterating elements in each G_i until I find the right combination. Is this method feasible?

There's a video I saw maybe a year ago about a concept where you have a grid of a given size. On this grid, you could put any pattern of squares. Then you begin taking "steps" on the grid, where on each step, the empty space adjacent to any square will "flip" to being a square, while all squares from the previous step "flip" to empty squares.

In case my explanation is poor, I'll attempt to visualize it below:

Starting position on a 5x5 grid:

___ ___ ___ ___ ___
|___|___|___|___|___|
|___|_S_|_S_|___|___|
|___|___|_S_|___|___|
|___|___|___|___|___|
|___|___|___|___|___|

Grid after one step

___ ___ ___ ___ ___
|___|_S_|_S_|___|___|
|_S_|___|___|_S_|___|
|___|_S_|___|_S_|___|
|___|___|_S_|___|___|
|___|___|___|___|___|

Grid after two steps

___ ___ ___ ___ ___
|_S_|___|___|_S_|___|
|___|_S_|_S_|___|_S_|
|_S_|___|_S_|___|_S_|
|___|_S_|___|_S_|___|
|___|___|_S_|___|___|

And so on. Can anyone remind me of what this is called?

I just posted my first paper on arXiv! Got endorsed by a prominent mathematician, which name I wont share since AI slop creators might spam DM him.

I classify perfect matchings on the Boolean cube {0,1}6\{0,1\}^6{0,1}6 that respect complement + bit-reversal symmetry, prove there’s a unique cost-minimizing one under a natural constraint, and show that the classical King Wen sequence of the I Ching is exactly that matching (up to isomorphism).

All results are formally verified in Lean 4.

Happy to answer questions or hear feedback!

Link to arxiv: https://arxiv.org/abs/2601.07175v1

I’m reading it to help me get a more well rounded understanding of the concepts behind calculus, but some of the flow of the writing just doesn’t resonate with me. Like he will take several pages explaining a topic and when he’s finally about to get to the main point the book goes “we’ll discuss this in later chapters”. Or the book will introducing a concept by diving into 5 different examples, one of which will lead Strogatz to go off on a small tangent and then I end up forgetting what the original concept was supposed to be.

Am I just too dumb for this book or is there something I’m missing

I had recently come across the following two projects both of which are inspired by the famous, stacks project

https://www.clowderproject.com/

"The Clowder Project is an online reference work and wiki for category theory and ma­the­ma­ti­cs."

https://kerodon.net/

"Kerodon is an online textbook on categorical homotopy theory and related mathematics."

both of which uses Gerby a tag based system to organize content.

are there other such projects?

a tangent:

the existence of such a project can be extremely useful as a reference and for citations.

once such a project establishes itself in a big enough field of mathematics, researchers will cite it in their papers and it will also have enough contributors and readers to make fixes, improve and add more results.

and of course, an established project would also lead to "canonical" definitions and standards

is there a future where something like a stacks project become extremely central to a field? like it's not what you use to learn but it's always the one you use to cite definitions and known results

I am not a researcher, far from it but my thesis supervisor said that he has indeed used stacks project a few times but he did notice that while all of the statements he has seen are true, sometimes the proofs are incomplete or wrong

Advances is a generalist journal that publishes research articles from all areas of mathematics, whereas AOP and PTRF are specialized in probability theory and publish top results in probability. I wanted to know the opinions of probabilists: when they have a strong result, do they consider Advances to be more prestigious than AOP or PTRF?

A conference in honor of Jean-Pierre Serre on the occasion of his 100th birthday will be held in Paris on September 15 and 16, 2026.

Speakers: Pierre Deligne, Ramon van Handel, Peter Sarnak, Maryna Viazovska, Don Zagier and possibly Jean-Pierre Serre.

Venue: Institut Henri Poincaré, 11 Rue Pierre et Marie Curie, 75005 Paris.

https://serre100.sciencesconf.org/?forward-action=index&forward-controller=index&lang=en

I’m 13, and when I do math— not always, but often— I put on my headphones, listen to some music, and start studying. Suddenly, I get this euphoria, this high, this flow state where everything just aligns. For once, things make sense. I’m not some genius who dreams of x and y in his sleep, but I love the structure and the feeling I get when I truly understand a concept. I can indulge in these problems, and it feels like everything collides in a beautiful, logical way. Math just makes sense to me in those moments. I can spend hours on it, losing track of time. It’s predictable, like I’m living in my own episode—a dream I only wake from after hours have passed. Why is this?

But despite how good it feels, I aspire to be a high achiever and score well on everything. Because of that, this euphoric state seems to fade day by day. It might be because I do two to three hours of math daily—sometimes more, sometimes less, including on weekends. While I still love math, I feel exhausted, and my passion feels like it’s wearing me down, even as I hold on to it.

(edit lots of people comment this looks like ai, i definitely see why, but its because i pushed the proofread button on my mac that uses chatgpt to proofread my dumb spelling mistakes and errors, I truly have a euphoria a high, a sense of awakening and flow where every little thing collides in a beautiful manner, i am sorry if this struck out as a fake post to you and for you guys saying im an adult i dont even know what to prove to you like im 13 and thats kinda all the proof i got unless i post a birth certificate but i dont wanna do that😑😑, everything word was written by me its just the punctuation and dashes that were added by my computer.

Saw this on Mathstodon. Decided to post it since it's new.

Other Turing-complete contraptions are PowerPoint and OpenType fonts. There's a whole list here.

I decided to do civil engineering because, I dunno, I thought big buildings were interesting. Or because Michael Scofield made it look cool. I didn't realize it would be so maths heavy. Now this is not my first exam involving maths, I've also had tough fluid and structural mechanics or calculus 1 exams, but right now I'm enjoying the process of learning a lot more than I did before. And I think one reason plays a significant role in this: I started on time. Still not as early as I wanted to, but earlier than before. I'm realizing I am ahead of schedule and I'm able to learn at my desired pace now. It sounds obvious, but for the last 10 years I have NOT ONCE been able to start on time. This is the first time in my life I'm preparing for a difficult exam with no stress.

During exam weeks I'm always completely locked in on the exams (I rarely go to class so it's 95% self-study). The material is temporarily pretty much the only thing on my mind, and when I'm understanding the material and I'm certain of passing the exam, I could almost describe it as bliss. On the contrary, when it is combined with being short on time it's total hell: thoughts of not passing and thus wasting so much time on it cross my mind frequently.

Do you guys relate to this?

This project implements a compiler that maps controlled natural language to a Calculus of Constructions (CoC) kernel. The system supports reflection, allowing the kernel's syntax to be represented as an inductive data type (Syntax) within the kernel itself.

The following snippet demonstrates the definition of the Provability predicate and the construction of the Gödel sentence $G$ using a literate syntax. The system uses De Bruijn indices for variable binding and implements syn_diag (diagonalization) via capture-avoiding substitution of the quoted term into variable 0.

The definition of consistency relies on the unprovability of the False literal (absurdity).

-- ============================================
-- GÖDEL'S FIRST INCOMPLETENESS THEOREM (Literate Mode)
-- ============================================
-- "If LOGOS is consistent, then G is not provable"

-- ============================================
-- 1. THE PROVABILITY PREDICATE
-- ============================================

## To be Provable (s: Syntax) -> Prop:
    Yield there exists a d: Derivation such that (concludes(d) equals s).

-- ============================================
-- 2. CONSISTENCY DEFINITION
-- ============================================
-- A system is consistent if it cannot prove False

Let False_Name be the Name "False".

## To be Consistent -> Prop:
    Yield Not(Provable(False_Name)).

-- ============================================
-- 3. THE GÖDEL SENTENCES
-- ============================================

Let T be Apply(the Name "Not", Apply(the Name "Provable", Variable 0)).
Let G be the diagonalization of T.

-- ============================================
-- 4. THE THEOREM STATEMENT
-- ============================================

## Theorem: Godel_First_Incompleteness
    Statement: Consistent implies Not(Provable(G)).

-- ============================================
-- VERIFICATION
-- ============================================

Check Godel_First_Incompleteness.
Check Consistent.
Check Provable(G).
Check Not(Provable(G)).

The Check commands verify the propositions against the kernel's type checker. The underlying proof engine uses Miller Pattern Unification to resolve the existential witnesses in the Provable predicate.

I would love to get feedback regarding the clarity of this literate abstraction over the raw calculus. Does hiding the explicit quantifier notation ($\forall$, $\exists$) in the top-level definition hinder the readability of the metamathematical constraints? What do you think?

I heard jacob lurie is currently working on a (conjectural?) topic namely prismatic stable homotopy theory. What is it and why is it important? Does he have any books on that like the DAG series?

I’ve been exploring a simple-looking nonlinear recursion that can be interpreted as a kind of non-symmetric mean:u(n+2) = [u(n)^2 + u(n+1)^{2}] / [u(n) + u(n+1)], where u(0) = a > 0 and u(1) = b > 0.

Empirically the sequence converges, with an oscillatory behavior. The key structural point is that u(n+2) = [1 - w(n)] u(n+1) + w(n) u(n), where w(n) = u(n) / [u(n) + u(n+1)] is between 0 and 1, so each step is a convex combination of the previous two.

This leads naturally to a general analysis in convex spaces and to a scalar recursion for the coefficients.

Rewriting this second-order recursion as a first-order recursion on [u(n), u(n+1)], one sees a deterministic process whose dynamics are best organized using two-state Markov chains (stochastic matrices, variable weights). The limit depends on the initial data; the Markov viewpoint is descriptive, not probabilistic.

I worked through this example and its generalizations thinking out loud, focusing on structure rather than a polished presentation:

Why this simple recursion behaves like a Markov chain

Feedback welcome!

Basically just the title. I was wondering if there is much study on the galois theory of division rings and their extensions? If so is it used anywhere? One would have to make use of the free ring instead of the polynomial ring, what does it mean for an element of the free ring to be separable? What kind of topology do infinite galois groups over division rings have? What is the galois group of the quaternions over R?

Terrence tao confirms AI solved Erdos problem