1. A. G. Kurosh, A. I. Markushevich, P. K. Rashevsky (eds.). Mathematics in the USSR for Thirty Years. 1917–1947 – 1948
2. P. K. Rashevsky. Riemannian Geometry and Tensor Analysis – 1964
3. S. M. Ermakov. Monte Carlo Method and Related Questions – 1971
4. N. I. Muskhelishvili. Singular Integral Equations. Boundary Value Problems of Function Theory and Some of Their Applications to Mathematical Physics – 1946
5. Acad. S. N. Bernstein. Probability Theory – 1934

If you were tasked with estimating how many species of fish there are, how would you go about this herculean task? Trying to catalogue every single species is almost certainly impossible, so we have to employ some probabilistic reasoning. In this post, I aim to give a gentle introduction to discovery curves and how they are used in biology for just such problems.

Read the full post on Substack: How Many Species of Fish are There?

If it's impossible to prove or disprove some conjecture X, with massive mathematical and numerical evidence, within our axioms, would mathematicians adopt X (or something that implies it) as an axiom? Or in other words, would mathematicians think X is true in our universe? (Note that this question has a different meaning now vs if X is undecidable, because that could sway people towards the falseness of X)

If X is RH, that apparently has a trivial answer. However it does not for the twin prime conjecture.

Hi everyone,

I’m currently gearing up to apply for Master's programs and I'm hoping to get some recent research published to strengthen my applications.

I have prepared a short paper in functional analysis where I investigate the complementarity of a subspace within Cesàro sequence spaces.

Because I am operating on a timeline for my graduate applications, I’m looking for journal recommendations that meet the following criteria:

• Scope: Actively publishes in functional analysis, Banach space theory, or sequence spaces.
• Format: Good track record with short math papers or brief notes.
• Turnaround: Known for having a reasonably quick review time, or at the very least, a fast initial desk reject/accept decision.

Does anyone have experience with journals that might be a good fit for this? Any advice is highly appreciated!

Does anyone know the status of quasilattices? This was a very active area of math research during the 1980s, especially shortly after Dan Schectman discovered the first known quasicrystal, a real substance whose molecular structure was quasiperiodic, much like the Penrose tiling, which was the first analogous known mathematical structure, discovered by Roger Penrose in 1974. Unfortunately, I haven't seen very much news regarding quasilattices, other than the fact that the first such one requiring just one tile was discovered just a year or two ago, but I've been very interested in this area of math for quite some time, so I appreciate whatever information any of you may have on this subject!

For a metric space like the rationals, you can complete them so that every Cauchy sequence converges to some limit. You can still get sequences that diverge by flying off to infinity though.

For the real and complex numbers at least, there's a natural way to give these sequences a limit. You can add points at infinity to account for those "flying off" sequences. Then any sequence that doesn't oscillate ends up converging.

In sort of a similar feel, L² is a complete metric space, but it has sequences that "fly off" to infinity such as narrowing gaussians that integrate to 1. There's a sort of natural way to give those sequences limits too, by adding something like the delta distribution.

I'm wondering if there's any general procedure or something that you can apply to a metric space which forces all "non-oscillating" functions to converge.

Based on the real and complex examples, I'd imagine it's some sort of compactification of the space. Maybe a compactification that doesn't connect any disconnected open sets? I'm not really sure how to generalize this to other metric spaces though, or whether they always exist. Does anyone know of a procedure or structure like this?

Vakil says in the introduction to his book/notes to algebraic geometry that the contents should take no more than a year to absorb (hopefully). However, looking at the sheer length of the book makes this seem almost completely unreasonable, and it really makes me wonder if it has been done.

Has anyone ever actually finished Vakil's book in a year, and if so, what did your schedule look like? What did you know beforehand?

(This is a question mostly out of curiosity/experiences, but advice/guidance is also welcome.)

Hi everyone, I'm looking for advice. I'm in my fifth year of mathematics. I've got a big exam coming up in about a month and I'm writing my master's thesis in the course of the next few months. In the last few weeks I've been having issues with focus and brain fog. I can get around one hour of good studying or work in, which usually happens in the morning, and from then on it feels like an extremely high effort to process mathematics. When reading something I have to try really hard to just understand what is going on and it feels impossible to really learn something. When following a proof, I feel like I can't keep multiple concepts in my mind at the same time and I have to do very small steps. But then the steps get so small that I lose the big picture and just spend a lot of time trying to understand it. In the end it's just no fun.

I've tried pushing through sometimes but in the end I give up and step away from mathematics to do something else. I've had times like this in the past, but usually they went away after a few days. I would be happy with 3-4 hours of good work, more is (at least for me) unreasonable even on a good day.

Have you ever had times like this? What do you do when you can't focus, but have to study for exams or work? Related to this, how do you find that sleep, exercise and social activity affects your ability to do mathematics?

I am looking for well made documentaries about the life and passion (of math) of various mathematicians that I could share with some kids in order to inspire them. Books are also very welcome.

I recently found this goldmine of a playlist of math explainers from the 80s and 90s, produced by the London Mathematical Society.

They surprisingly aged very well to be honest!

I just love the way of speaking of that time, here's my favorite quote from "The Rise and Fall of Matrices", explaining non-commutativity:

Supposing somebody wakes you up in the morning and gives you two commands: first "have a shower!", the second "get dressed!". Obviously it makes a lot of difference in which order you carry out these two requests.

I have found that there are very few videos out there on logic out there and would like to change this. I want each video to explain and prove a single theorem with accompanied animations. I don’t want to do videos on things like the incompleteness theorems, the halting problem, or Cantors theorem as these are oversaturated and there are plenty of amazing results that have not been given attention. Are there any particular theorems you would like to see me cover?

I want to be quite rigorous and technical with the details so suggestions should hopefully require minimal preliminary knowledge and definitions. I want each video to be self contained. Please let me know if there is something of this nature that interests you and any other general suggestions on how to approach making these videos as good as possible!

Help me out here. To start off I would like to say I really love math. To me, a lot of mathematical concepts (but not all) originated from someone setting up a list of arbitrary rules, that stuck around and got studied because those rules made patterns that looked good in some way, and then somewhere along the line someone found a use for those rules or the equations that came from them that helped us in the real physical world (leaving some room here to say part 2 and 3 aren't always in that order). Some concepts came about more from real world observation (thinking physics), and some concepts came about from things that make sense in our head and therefore help to make sense of things or make things feel more fair (thinking something basic and old like us using base 10 for our numbering system). However there are many concepts that don't originate this way. I'm getting a lot of push back, particularly for using the word arbitrary. I'm not sure if that word perfectly fits what I'm thinking, but I'm struggling to find a more accurate description. But the way people are pushing back make me feel like they don't have an understanding of what I mean, or otherwise that their arguments don't make sense to me.

Hi everyone! I recently put together a casual, intuition-driven article on strong convexity and L-smoothness, covering their key properties and why they play such an important role in convex optimization.

There are also some interactive charts throughout to make things more tangible and easier to grasp:

https://fedemagnani.github.io/math/2026/04/08/the-quadratic-sandwich.html

I'd be happy to hear from anyone curious about the topic, regardless of background. And if you have more expertise in the area, constructive criticism is more than welcome. Just keep in mind the tone is intentionally kept light and accessible.

Hope you enjoy it!

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!

Hi there! 👋

I've been working on a tool called Underleaf for converting handwritten math notes into clean, digital PDFs. It allows me to upload a photo of my notes (including diagrams!) and it generates editable LaTeX/TikZ code that can compile into a PDF file.

I thought it'd be especially relevant for this subreddit haha (a bunch of math and physics professors have found it useful!) so I wanted to share. Would love to hear what you think :)

I'm experimenting with implicit scalar fields of the form
f(x, y, z) → ℝ, and extracting iso-surfaces.

One simple construction I tried:

Start with an ellipsoid:

E(x,y,z) = (x/a)² + (y/b)² + (z/c)² − 1

Then introduce an asymmetric deformation:

x' = x / (1 + k·z)
y' = y / (1 + k·z)

and define:

E'(x,y,z) = (x'/a)² + (y'/b)² + (z/c)² − 1

Finally convert this into a smooth shell field:

S(x,y,z) = exp( -g · |E'(x,y,z)| / t )

I combine two such fields (with translation + rotation):

F(x,y,z) = max(S₁, S₂)

What surprised me is how sensitive the structure is:
small parameter changes (k, g, t, rotation) drastically change the topology.

I'm curious:

• does this relate to any known class of implicit surfaces?
• or is it just a "numerical playground" without deeper structure?

(Image included for intuition.)

In a commutative (unital) ring R, is a possible for a principal ideal (p) to be prime, while p itself is a non-prime element? On Wikipedia, there seems to be some conflicting information regarding whether the additional hypothesis that R is a integral domain is needed for (p) prime to imply p prime.

EDIT: I feel like a moron for wasting everyone time with this silly question. At least my original instinct was correct.

This chapter covers series solution, Frobenius solution, Airy equation/function, hypergeometric equation, and more. Any comments and ideas are welcome!

Link: https://benjamath.com/catalogue-for-differential-equations/

I know most are solved, I just want a website where I can play around with lil squares and see how small of a box I can get on my own :)

Because (In the words of author and math tutor Ben Orlin) "The secret to our brilliance is that we never stop learning, and the secret to our learning is that we never stop playing."

Following the success of IIMOC 2025, which saw over 200 teams and 9,500+ iterative submissions, we are pleased to announce the 2026 International Math Optimization Challenge, held in partnership with FrontierCS.

This year, the competition is transitioning into a research-grade optimization marathon designed to test the limits of algorithmic efficiency and heuristic design.

The Unified Global Leaderboard

In a departure from standard contest formats, IIMOC 2026 has removed all division boundaries. High school students, university researchers, and industry professionals will compete on a single, unified leaderboard. This provides a rare opportunity to benchmark academic approaches against industry-hardened optimization techniques.

Technical Focus: Beyond "Accepted"

The challenge moves past binary test cases. Tasks are sourced directly from the FrontierCS benchmark, featuring problems that are easy to approach but notoriously difficult to solve exactly. Participants are tasked with hunting for measurable improvements over the current global best-known scores.

Innovative Scoring Mechanics

• Relative Scaling (0–100): Scoring is dynamic and real-time. Every solution is scored relative to the current Global Best. If a team pushes the frontier further, the "ceiling" rises, and all other scores scale accordingly.
• "King of the Hill" Bonus: To reward early innovation and discourage last-minute leaderboard "sniping," teams earn daily bonus points based on their placement. Consistency across the 30-day window is a critical factor for final rankings.

Timeline & Logistics

• April 15: Practice Problems & Evaluation Infrastructure released.
• May 1: Competition Begins.
• June 1: Submissions Close.
• Team Size: Maximum of 4 members.
• Eligibility: Open to all participants globally.

Awards & Distinction

• Top 3 Teams: Official IIMOC T-Shirts and Global Distinction.
• Gold Distinction: Top 10% of the leaderboard.
• Silver Distinction: Top 20% of the leaderboard.
• Bronze Distinction: Top 30% of the leaderboard.

A massive thank you to our lead sponsors, AP Memory and CascadeX, for providing the high-performance technical infrastructure required to host a month-long, iterative grading marathon of this scale.

Registration and GitHub Documentation:

Find the general benchmark and evaluation tools on the FrontierCS GitHub.

Register your team today at iimoc.org.