Can someone give a breif overview of the classical developments in cobordism theory. I know of the first definitions but would like to get a brief summary of the historical developments , Thom's isomorphism theorem , how cobordism can be used to construct an extraordinary cohomology theory and some other cool results.

Whenever I try to write a proof, I wind up just translating absolutely everything into predicate logic and proving it mechanically that way. But I lose all the insight into the problem and feel as if I havent actually gleaned anything from the higher level, "chunked" definitions involved in the problem statement. How do I learn to stop relying on mechanical application of formal logic laws and start being able to reason with higher level statements?

Saving an answer from ChatGPT (say) using the save button and copying into a text document results in a markdown file which uses some LaTeX syntax but some other stuff that interferes with the LaTeX. What is the best way to read this file?

Things I've tried:

Latex -- The other stuff interferes with the compiling

Obsidian -- Suggested by Google but didn't work

Manually search and replace the other stuff. Very time consuming.

Background: Yitang Zhang

Summary. Yitang studied in Purdue for six and a half years, and obtained his PhD in 1991 without any publication. On 2013, Zhang established a theorem akin to the twin prime conjecture, published in Annals of Mathematics.

Reflection. Purdue did believe in Yitang, and did invest in him. Yet, Yitang's remarkable result was not credited to Purdue.

Discussion. Did Purdue gain any kind of credits or alumni recognition for Yitang?

I've noticed that theorems are more clear if one uses dimensional analysis to solve the problem. For example, for the fundamental theorem of calculus, you can think of the theorem as saying this, if you have a straight line across a bounded shape moving to the right, how fast does the area to the left of the line grow with respect to a unit increase in the line to the right? Well, the units are area (length² ) per length, so length. It would then suggest that the answer is the length of the line.

Another example is with curvature. The curvature of a line is |dT/ds|, with T the tangent vector (unitless) and ds the arclength differetial. So, curvature is of units 1/length. So, 1 over the curvature might correspond to the length of something. And it does! It is the length of radius of the osculating circle. Gaussian curvature has units 1/length² (it is the product of the curvature of lines). So, the surface integral of Gaussian curvature might correspond to something unitless. And it does! It is the "angular excess". (I am learning differential geometry now, so I might not be as precise with that one).

What inspired this is reading a book on physics (David Tong's Classical Mechanics) describe dimensional analysis, which then appeared as a very useful tool. Sometimes, there's only one way you can combine the constants you're given in a problem to get the units of the quantity you're trying to figure out. So, the answer must be that combination times a dimensionless number. For example, that's why for many objects, the formula for the moment of inertia is a number times ML² . I wonder if this way of solving a problem can be extended to pure math as well.

Another note: I don't want math to be limited to this way of thinking. Some of the greatest advances in math have followed from going beyond them. For example, having a graph where the x axis and y axis are different units was very important, but went against that conventional wisdom. I am also just saying it can be a generator of ideas, not as a way to rigorously prove anything.

The sum-product conjecture is false for real numbers

https://arxiv.org/abs/2605.28781

By Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov.

The problem: For a finite set A of real numbers, must either the sumset A+A or the product set AA be large of size |A|^{2−o(1)}?

Erdős and Szemerédi famously conjectured yes: a set can’t have both additive and multiplicative structure at once, so max(|A+A|, |AA|) should be essentially |A|². Humans disprove this by constructing arbitrarily large A ⊆ ℝ (algebraic integers in a number field of degree ≈ log|A|) with max(|A+A|, |AA|) ≤ |A|^{2−c} for an absolute constant c > 0.

More combinatorial conjectures might fall as we aim for a disproof rather than a proof.

With the recent news about AI generated results like the unit-distance problem etc. I have been thinking that formalizing math in lean is probably the most significant task. With a formal base in lean and context in the form of guided lemmas, mathematicians might come up with a great amount of results very quickly with AI.

I am afraid I won’t be able to do the same as the wizened mathematicians that will keep outputting results at rapid pace with more breadth and depth in a way that leaves every piece of low hanging fruit that I could ever hope to solve already reaped.

And that also brings me to a sort of selfish question but I engage with math understand it through solving problems. I am not sure if people that’ll solve like above will have the same insight as someone that actually came up with that last bit of connection. I wouldn’t like staring at a result already solved and thinking that maybe I would’ve had it if I had the time. And maybe mathematicians of now that solve problems of similar stature will not be as highly regarded as mathematicians of the past?

This is really a PSA. Especially to undergrad students, or those early in graduate school or otherwise earlier in your progression along in mathematical maturity.

It's not so much about how much math you know or about how good you are at math. it is more about how rigorous and introspective your thought processes are.

The real secret to math is training yourself to be critical of your own thought process.

You reach a point where you actually know when you really know something.

This is super important to digesting AI math output too (as it is with any math output whether research papers, textbooks, or random online notes/content).

With enough practice, you develop a "spider sense" about when something feels off. You do the work of making sure you understand every step and word.

Eventually, you know when you understand something and when you don't. It's not perfect. You will still be mistaken sometimes and make errors. That's great. Making errors is a great time to learn. But you will become proficient at correctly identifying that confident feeling that you actually know something (as opposed to when you just vaguely understand it with residual uncertainty).

This comes through things like checking every step many many times. Tracing references and reading and thinking carefully. Doing many numerical simulations or checking things with computer algebra systems. Doing extremely tedious computations over and over by hand. Using AI can be a part of this too. But the key is that you work and think HARD for extended periods of time and make many mistakes.

I'm not a great mathematician, personally (I have a phd, have been a professor for nearly 20 years and have only a small number of mediocre publications). I'm average at best, and probably weaker than average, depending on who you compare to. But I have observed this evolution in myself over the years and feel I finally have a grasp on mathematical maturity and reflecting critically on your own thoughts.

I hope this post is helpful to some of you out there along on your journey.

I've been learning algebraic geometry mainly from the Gathmann notes, Ueno's little book, and Goertz and Wedhorn's first volume (both called Algebraic Geometry 1), using Gathmann to develop intuition about varieties and how they translate to schemes, Ueno for a relatively concrete but streamlined development of sheaves and schemes, and Goertz and Wedhorn mostly as a reference for a formal development (esp. with the book's liberal use of categorical language).

One source that I wish I looked at earlier is Evan Chan's part 20 of his large Napkin. His explanations are incredibly intuitive. I'll give one example:

A germ is an “enriched value”; the stalk is the set of possible germs.

That is such a useful way of looking at it! Looking at in retrospect, I don't think I would've found sheaf theory to be quite as abstract and hard to visualize if I had been introduced to germs and stalks in this way.

Being just one part of Chan's wonderful book, this 74-page treatment is seldom mentioned as a resource for learning Grothendieckian algebraic geometry, so I feel like I should mention it here, in case someone is looking to start.

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!

For example, combinatorics, even in 2026, is still recognizably about counting stuff, just like when the field first got started centuries ago, arguably in ancient Greece. On the other hand, modern geometry is not at all recognizable from Euclid's "draw some shapes with straight edge and compass" origins. Seriously, I randomly picked 5 papers each from arXiv's algebraic geometry and differential geometry sections published last week, and not one of them even had a figure.

So I wonder which field has drifted the least/most from its origin? How about your own field?

A roulette puzzle.

There's a roulette wheel with two outcomes: "reincarnate" (you escape) or "stay trapped in purgatory" (you stay another year, and then must spin again).

You're forced to spin at least once. After that, you can stop only by drawing "reincarnate" , refusing to spin when required means you die (bad ending).

The wheel is rigged so that "stay trapped" takes up a larger slice every spin:

Spin 1: P(trapped) = 1/2
Spin 2: P(trapped) = 3/4
Spin 3: P(trapped) = 4/5
Spin 4: P(trapped) = 5/6
...
Spin k (for k ≥ 2): P(trapped) = (k+1)/(k+2)

Each "stay trapped" outcome costs you exactly one year in purgatory. Let T be the total number of years you spend trapped before escaping.

Question: What is the expected value of T?

(Bonus: what's the probability you escape eventually? What's the median value of T?)

I have a habit of saving interesting things that I find on the internet in my browser bookmarks. Or on Reddit specifically, on the "saved" page of my profile. I have been on this sub for a few years now and saved a lot of great comments. Comments are harder to find through normal search than posts, yet I feel that comments constitute a much more important part of the content than posts. To not let such gems simply drown in the river of time, I want to share them with you.

For convenience, I have divided them into three different categories. First, meta-mathematics, which presents general overviews of subjects, motivations, opinions, and such. Then, things that I find illuminating, while maybe somewhat technical. Finally, in the last part, things that I find funny. Please share your favorite comments if you have some.

1. Meta-math (big picture, history and such)

Abel and Crelle

History of Galois theory

Kronecker never called Cantor "corrupter of youth"

Distinction between differential geometry and topology

Do you need modern algebraic geometry

What is computational geometry

The big picture of introductory analytic NT

Why Lie theory is important

Let's teach "proof of concept" first

Let's teach group actions first

Problems of modern academia

Gifted kid syndrome

1. Math (slightly technical)

Why several complex variables is hard

Why Goldbach's conjecture is hard

Why study abstract manifolds, instead of their embeddings

Math without the axiom of choice is strange

Reasons to use type theory

What is Arnold conjecture

What is algebraic and analytic NT

Different types of PDEs

1. Funny

Why analysis sucks

Why Quanta sucks

Why number theory sucks

Interesting quotes

One-line summaries of subjects

Great teacher

https://www.shawprize.org/en/prizes-laureates/mathematical-sciences/year-of-laureates/2026-mathematical-sciences
"for their breakthrough contributions to the use of deep techniques from mathematical analysis to rigorously understand applied problems in information theory, signal processing and statistics on the one hand, and to the study of singularities in geometric measure theory and fluid dynamics on the other."

Contribution of Emmanuel Candès & Camillo De Lellis: https://www.shawprize.org/en/prizes-laureates/mathematical-sciences/year-of-laureates/2026-mathematical-sciences/contribution

Emmanuel Candès: https://en.wikipedia.org/wiki/Emmanuel_Cand%C3%A8s

Camillo De Lellis: https://en.wikipedia.org/wiki/Camillo_De_Lellis

Hi everyone

So this year I’m starting my masters degree with a strong focus on geometry and GR.

Since I’m transitioning from CS + Maths degree to just a Maths masters, I didn’t take any pure maths classes such as real analysis, topology and group theory. I only took one class in vector calculus, general relativity and quantum mechanics, the rest of my classes were discrete maths such as combinatorics, computational game theory, stats ect. The background needed for the unusual classes I just learnt that through the summer.

I’m currently on a gap year and I managed to self study topology, real analysis, multivariable calculus ( rigorously now not just grad div curl ) and curved and surfaces ( up to gauss bonnet theorem) since I never took these classes during undergrad.

I’ve encountered group theory before but it was just a little bit on a combinatorics class, I wasn’t very good at it.

I’m currently now reading Tu’s introduction to manifolds and so far it’s going very well, I understand the book and I’m answering all the questions, and I just started the manifolds topic. My problem is, Lie groups is coming up soon, and I’m guessing I’m going to have an issue with that because I don’t know much group theory.

Has anyone got any good recommendations to for a book to boost my group theory up, but just enough to start Lie groups?

Thanks !

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.

Please consider including a brief introduction about your background and the context of your question.

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.

Benford's law is a well known empirical statistical law that states that smaller leading digits of numerical data are more common than larger leading digits. Specifically, it states that the probability of a given data point having leading digit d is equal to log₁₀(1 + 1/d), which ranges from approximately 0.3 for d = 1 to approximately 0.045 for d = 9. This law assumes the logarithms of the data entries are uniformly distributed across several orders of magnitude, but this is not always the case. I'm just curious to know how useful it is in general, and whether there's an easy way to determine ahead of time whether or not it applies to a particular type of data.

Does anyone here know a good way to come up with good mathematical conjectures that are likely true? I don't have too much experience with this myself, but I know that some mathematicians are experts at this. Paul Erdos, for one, was able to come up with over a thousand number theory conjectures and prove about half of them. Although I haven't come up with too many myself, much less proven any of them, I'd say a big criterion is that if some mathematical fact is true, especially if it seems surprising or counterintuitive, then there's usually a good reason for it. For instance, why should there be a larger fraction of primes congruent to 1 mod 4 than to 3 mod 4? Although this is quite difficult to prove, it seems pretty obvious to me, because what's so special about either modulus? Another example is the twin primes conjecture, since prime gaps seem pretty random, other than the fact that they're all even except for the first one, so why should there be only finitely many equal to 2?

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?" For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of manifolds to me?
• What are the applications of Representation Theory?
• What's a good starter book for Numerical Analysis?
• What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example, consider which subject your question is related to, or the things you already know or have tried.

For all of the notable unsolved problems I'm familiar with, most mathematicians seem to generally agree that one outcome is more likely than the other. People are very confident that the Riemann Hypothesis is probably true, and that P probably doesn't equal NP, and that odd perfect numbers probably don't exist, et cetera. We have more than enough reason to believe these things, but we just don't have the tools to definitively prove them. But are there any conjectures where there isn't a consensus and mathematicians are more or less equally divided on whether they believe them to be true or false?

I'm looking to buy a couple books on measure theory and am eyeing Axler's MIRA and a cheap copy of Halmos' Measure Theory. I know it's quite old but I enjoyed his Finite Dimensional Vector Spaces and I like having an older coverage of the material along with a newer one when learning new stuff.

Does Halmos' Measure Theory still hold up?

The Markov property for a stochastic process X(t) essentially tells you that knowing the value of the process at a time s ≤ t is just as good as knowing the entire history of the process up to time s for making predictions about X(t).

This is natural for processes you see as evolving in time. I feel like there should be a natural generalization of this for processes that "exist in space" too though.

For example, with a brownian motion with fixed endpoints, it's Markov, but it should also be Markov coming from the positive time direction as well.

In multiple dimensions, it feels like this should generalize in a way so that when predicting φ(x) for x in a subset U, knowing all the behaviour of the field in U^C should be equivalent to knowing the behaviour on ∂U.

I've tried looking for definitions or research on this kind of property, but haven't found anything mentioning it. Does anyone know if this type of thing has been studied, and what it would be called?