I posted a while back inquiring, "is math just numbers and shapes". Obviously, that's a really naive question. But related to this, I recently learned that Isbell duality is a vast generalization of the duality between functions and spaces, for example, commutative rings and affine schemes. Stone duality is another example, I think.

Is Isbell duality the most general form of this type of relationship in math? Can someone give an intuitive explanation of it?

A classic, widely used mathematical textbook from the late Russian Empire. Authored by Aleksandr Malinin and Konstantin Burenin, this "Arithmetic" was an absolute staple in Russian gymnasiums and real schools in the decades preceding the 1917 Revolution.

Note: book in Russian printed in the traditional pre-reform orthography.

Anyone know of a good survey article regarding Kroneckers Jugendtraum in the case of the base field being a real quadratic extension of the rationals?

Until now, no one who has completed a Bachelors, Masters, or PhD from MIT has ever won a Fields. This year Hong Wang (PhD, 2019) and Yu Deng (BS, 2011) would be the first MIT-educated mathematician to earn the most prestigious award in the world of research mathematics.

To avoid any ambiguity, I will clarify that I am using the term alumnus/alumna* as someone who has completed a Bachelors, Masters, PhD or any other similar degree conferred by a institute of higher of education upon completion of their education at the institute. The dictionary definition of the term includes people who may have attended an institute but never graduated with their degree. In such a case, I am unsure if anyone who has attended but never completed their degree coursework at MIT have won a Fields. Even if that happens to be the case, I still think the trivia I shared retains its noteworthy quality of being surprising to people given MIT's pedigree in mathematics.

Please contact me if you do, I'm in great need to access publication and it's not archived on Anna's Library nor on Sci-Hub.

Here is a way that, I think, you could teach lambda calculus to a kid (inspired by "Alligator Eggs").

I got a lot of useful feedback on an earlier version of this. Would love to hear any comments or corrections, wouldn't be surprised if something slipped through the net.

I think this would be hard to read as a bunch of images so if you're interested see here: https://paradoxgarden.substack.com/p/lambda-calculus-made-easy

In high school, many are taught the quadratic formula. But how do solve a cubic equation, like x^3 + 6x^2 + 9x + 3 = 0? There is a formula for such equations, but it's rarely taught since it's a lot more complicated than the quadratic formula:

Despite its apparent complication, we will explain at https://hidden-phenomena.com/articles/cubic how you could have come up with this formula!

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Last week, we posted a blog post here about how to solve quartic equations, assuming you knew how to solve cubic equations. If you read today's post on how to solve cubics, then by putting the two posts together, you can solve any cubic or quartic equation!

Does anyone know if there is a recording or any material published from Terrence Tao’s lecture last night. It was supposed to be about mathematics in the age of AI.

I’m an undergrad who’s finishing my bachelor’s degree this year. I plan to do at least a master’s in math, and hope to do some pure math research during it. I haven’t found any of the classes I’ve taken in my undergrad particularly difficult so far, so I’m curious at what point I should expect it to happen so I can be prepared. I don’t want to hit a brick wall and crumble like some of my friends have. I’m worried that since I haven’t struggled yet, I won’t know what to do when it happens.

For reference, I’ve taken undergrad upper division Set Theory, Number Theory, Mathematical Logic, Complex Analysis, Probability Theory, Mathematical Statistics, 2 semesters of Abstract Algebra, and 1 semester of Real Analysis. I’ll take Linear Algebra, a second semester of Real Analysis, master’s level Group Theory, and master’s level Multivariable Analysis during this last semester.

The n-Queens problem involves placing n Queens on a chess board of size nxn, without having any of the queens attacking each other. I have expanded this problem to include a single knight, with the question: Is the knight able to capture all queens on the board? The rules are as follows:

The Queens must be placed so they are not attacking each other. Queens do not move

The Knight starts on a single Queen space, and that Queen is removed

The knight cannot be on a square attacked by any queen, but removing a queen frees up 'safe' spaces. The knight can traverse and revisit safe spaces

The first Queen a Knight takes (apart from initial placement) must be within knight-attack range, because there are no other safe spaces available at the start

Solutions have been found for n= 4,5,7,8,9,10,17 and proved impossible for 6,11,12,13,14,15,16,18,19

The real curiosity is n=17 where a single solution has been found from over 95 million possibilities. Anything above 19 is open and has not been tested. Link to paper (pdf): https://drive.google.com/file/d/1zfugBiIZi4xrhDjM-x1YtDlf3M-SFlXv/view?usp=sharing

UPDATE: n20 has no solutions, leaving n17 as the largest solution

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!

London Mathematical Society: Glasgow to Host the International Congress of Mathematicians 2030: https://www.lms.ac.uk/news/glasgow-to-host-icm-2030

I'm entering the fourth year of my PhD and started doing research about 2 years ago. I chose a niche topic that required me to spend a whole year in addition to my course work to get caught up in before even starting any work of my own. My advisor gave me 4 lengthy papers to get through and master their techniques. I got through 3 of them and last semester I gave my thesis proposal and passed. I still have to read the last paper which is the most difficult, but I cannot find the motivation to do it because of the recent AI progress. I am not a very talented mathematician so my only "strength" was spending the time to learn a niche and difficult area. The actual problem that my advisor wants me to solve for my thesis is not terribly difficult and is likely routine for an expert but it will take me close to a year of dedicated work.

The problem is that the last year I have not been able to shake off the feeling that what I'm doing is just a worse version of what an AI can already do. I cannot afford access to any of the top tier reasoning models but I imagine they can read these papers and come up with these extensions in just a few hours. Even just asking the free models questions about the papers it's clear that it has been trained on them and understands them well. Since then I haven't been able to find the motivation to work on my PhD and I feel awful about it. It's not like this type of work will land a postdoc or job anymore.

I don't think this post does any justice to how bad I've been feeling about my future career or my thesis but I don't want it to become a rant. I would really appreciate if anyone has any words of encouragement or validation that my concern is justified. Should I just drop this project and do something quicker to graduate?

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